Finite Group Databasebeta

← #33149112

Subgroups — #33149112

Order 18688.

932 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988874C211yesmaximal, characteristic, complemented
2C2×C4×#11924013C211yesmaximal, characteristic
3#22612842C211yesmaximal, characteristic, complemented
4#22611452C211yesmaximal, characteristic, complemented
5#22611412C211yesmaximal, characteristic, complemented
6#22611305C211yesmaximal, characteristic, complemented
7#22611256C211yesmaximal, characteristic, complemented
8#22610656C211yesmaximal, characteristic, complemented
9#22610490C211yesmaximal, characteristic, complemented
10#22610282C211yesmaximal, characteristic
11#22610282C211yesmaximal, characteristic
12#22610192C211yesmaximal, characteristic, complemented
13#22610191C211yesmaximal, characteristic, complemented
14#22610177C211yesmaximal, characteristic, complemented
15C73×#2452C211yesmaximal, characteristic, complemented
16C22×#11924013C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20#17988874—24no—
21C4×#11924013—24no—
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×#14556511C2211yescharacteristic
32C2×#14556511C2211yescharacteristic
33C2×#14556511C2211yescharacteristic
34C2×C4×#10596831C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988458—21no—
38#17988458—21no—
39#17988453C2211yescharacteristic
40#17988446—21no—
41#95×D73—21no—
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×C4⋊Q8×C73C2211yescharacteristic
51C2×C22⋊Q8×C73—21no—
52C2×C4×D4×C73C2211yescharacteristic
53#396×C73—21no—
54#393×C73—21no—
55#390×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#389×C73C2211yescharacteristic
59#385×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×D292C2311yescharacteristic
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C22×C4×D73—21no—
65C2×#11924013—22no—
66C2×#11924013—22no—
67C2×#11924013Q814yescomplemented
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C4×D292—22no—
73C4×D292—22no—
74C4×D292—22no—
75C4×D292—22no—
76C4×D292—41no—
77C4×D292—41no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86#14556524—22no—
87#14556524—41no—
88#14556522—22no—
89#14556522—22no—
90#14556522—22no—
91C4⋊#10596831—22no—
92#14556512—22no—
93#14556512—22no—
94#14556512—41no—
95#14556511—22no—
96#14556511—22no—
97#14556511D414yes—
98C4×#10596831—22no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103C22×Q8×C73—21no—
104C22×D4×C73C2311yescharacteristic
105C23×C4×C73—21no—
106C4⋊Q8×C73—22no—
107C4⋊Q8×C73—41no—
108C22⋊Q8×C73—44no—
109C4×D4×C73—22no—
110C4×D4×C73—41no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73D411yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73—21no—
123C2×C22⋊C4×C73—21no—
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C2×C42×C73C2311yescharacteristic
127#95×C73—21no—
128#95×C73—21no—
129#95×C73D411yescharacteristic
130#95×C73D411yescharacteristic
131C23×D73—21no—
132C2×D292—21no—
133C2×D292—21no—
134C2×D292—21no—
135C2×D292C2×D412yes—
136C2×D292C2×Q812yes—
137C2×D292#4112yes—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C2×C4×D73—24no—
142C2×C4×D73—42no—
143C2×C4×D73—42no—
144C2×C4×D73—42no—
145C2×C4×D73—42no—
146C2×C4×D73—81no—
147C2×C4×D73—81no—
148C2×C4×D73—81no—
149C2×C4×D73—81no—
150#11924013—28no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831C2×D412yes—
162C2×#10596831C2×Q812yes—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73#4112yes—
169#11924007—81no—
170#11924007—81no—
171#11924005—24no—
172#11924005—42no—
173#11924005—42no—
174C24×C73—21no—
175C2×Q8×C73—42no—
176C2×Q8×C73—42no—
177C2×Q8×C73—81no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73C2×Q812yes—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—42no—
188C22×C4×C73—42no—
189C22×C4×C73—81no—
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2411yescharacteristic
197C4⋊C4×C73—22no—
198C4⋊C4×C73—22no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—42no—
205C4⋊C4×C73—42no—
206C4⋊C4×C73—42no—
207C4⋊C4×C73C2×D412yes—
208C22⋊C4×C73—42no—
209C22⋊C4×C73—42no—
210C22⋊C4×C73—42no—
211C22⋊C4×C73—42no—
212C22⋊C4×C73—81no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—21no—
216C42×C73—21no—
217C42×C73—21no—
218C42×C73—21no—
219C23×C73—22no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73C22×D411yescharacteristic
227C22×D73—22no—
228C22×D73—22no—
229C22×D73—22no—
230C22×D73—41no—
231C22×D73—41no—
232C22×D73—41no—
233C22×D73—41no—
234Q8×C73—84no—
235D4×C73—22no—
236D4×C73—22no—
237D4×C73—41no—
238D4×C73—41no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—22no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—84no—
265C2×C4×C73#13512yes—
266C2×C4×C73AGL(1,ℤ/8)12yes—
267C2×C4×C73C22×D411yescharacteristic
268C2×C4×C73C22×Q811yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C2×C4×C73C2×#4111yescharacteristic
271C2×C4×C73H(5,𝔽2)11yescharacteristic
272C2×C4×C732−1+411yescharacteristic
273C2×Dic73—22no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282D292—22no—
283D292—22no—
284D292—22no—
285D292C22⋊Q814yescomplemented
286C4×D73—84no—
287C4×D73—84no—
288C4×D73—84no—
289#10596831—22no—
290#10596831—22no—
291#10596831—41no—
292#10596831—41no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73—42no—
296C22×C73—81no—
297C22×C73—81no—
298C22×C73—81no—
299C22×C73—81no—
300C22×C73#52511yescharacteristic
301C22×C73D4211yescharacteristic
302C22×C73D4×Q811yescharacteristic
303C22×C73C2×#13511yescharacteristic
304C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C73#57611yescharacteristic
306C22×C73#57711yescharacteristic
307C2×D73—24no—
308C2×D73—42no—
309C2×D73—42no—
310C2×D73—42no—
311C2×D73—42no—
312C2×D73—42no—
313C2×D73—42no—
314C4×C73—22no—
315C4×C73—22no—
316C4×C73—41no—
317C4×C73—41no—
318C4×C73—41no—
319C4×C73—42no—
320C4×C73—41no—
321C4×C73—81no—
322C4×C73—81no—
323C4×C73C2×C22⋊Q811yescharacteristic
324C4×C73#53911yescharacteristic
325C4×C73#54711yescharacteristic
326C4×C73#55411yescharacteristic
327Dic73—42no—
328Dic73—81no—
329Dic73—81no—
330#26380—731nomaximal, Sylow
331C2×C73—82no—
332C2×C73#244911yescharacteristic
333C2×C73#310611yescharacteristic
334C2×C73#311811yescharacteristic
335C2×C73#326411yescharacteristic
336C2×C73#328211yescharacteristic
337C2×C73#338011yescharacteristic
338C2×C73#338411yescharacteristic
339D73—44no—
340C2×#476—731no—
341C2×C4×D8—731no—
342#2452D7311yescharacteristic, complemented
343#2349—731no—
344#2092—731no—
345#2085—731no—
346#2065—731no—
347#2057—731no—
348#1941—731no—
349#1916—731no—
350#1879—731no—
351#1879—731no—
352#1864—731no—
353#1864—731no—
354#1862—731no—
355C73#2638011yescharacteristic, complemented, Sylow
356C22×D8—731no—
357C2×C4⋊Q8C2×D7311yescharacteristic
358C2×C22⋊Q8—21no—
359C2×C4×D4—731no—
360C2×C4×D4—731no—
361C2×C4×D4—731no—
362C2×C4×D4C2×D7311yescharacteristic
363#476—1464no—
364C4×D8—1464no—
365C2×#107—731no—
366C2×#107—731no—
367C2×#106—731no—
368C2×#106—731no—
369C2×#106—731no—
370C2×C4⋊C8—731no—
371C2×#102—731no—
372C2×#102—731no—
373C2×#102—731no—
374C2×#102—731no—
375C2×#102—731no—
376C2×#102—731no—
377C2×C4×C8—731no—
378#396—21no—
379#393—21no—
380#390C2×D7311yescharacteristic
381#390C2×D7311yescharacteristic
382#389—731no—
383#389—731no—
384#389C2×D7311yescharacteristic
385#389C2×D7311yescharacteristic
386#385—21no—
387#385—1461no—
388#385—1461no—
389#382—731no—
390#380—1461no—
391C4×C4⋊C4C2×D7311yescharacteristic
392C2×#95—1461no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#336—731no—
398#336—731no—
399#336—731no—
400#336—731no—
401C22×Q8—21no—
402C22×D4—731no—
403C22×D4C22×D7311yescharacteristic
404C23×C4—21no—
405C23×C4—1461no—
406C23×C4—1461no—
407C23×C4—1461no—
408C2×D8—734no—
409C2×D8—1462no—
410C2×D8—1462no—
411C22×C8—731no—
412C22×C8—731no—
413C22×C8—731no—
414C22×C8—731no—
415C4⋊Q8—22no—
416C4⋊Q8—41no—
417C22⋊Q8—44no—
418C4×D4—22no—
419C4×D4—41no—
420C4×D4—2921no—
421C4×D4—2921no—
422C4×D4—1462no—
423C4×D4—1462no—
424C4×D4—1462no—
425C4×D4—1462no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1461no—
434C2×C4⋊C4C22⋊D7311yescharacteristic
435C2×C4⋊C4C22⋊D7311yescharacteristic
436C2×C4⋊C4C22×D7311yescharacteristic
437C2×C4⋊C4C22×D7311yescharacteristic
438C2×C4⋊C4C22×D7311yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—1461no—
445C2×C22⋊C4—1461no—
446C2×C22⋊C4—1461no—
447C2×C42C22×D7311yescharacteristic
448C2×C42C22×D7311yescharacteristic
449C2×C42C22×D7311yescharacteristic
450#107—2921no—
451#107—1462no—
452#106—1462no—
453#106—1462no—
454#106—1462no—
455C4⋊C8—1462no—
456#102—2921no—
457#102—734no—
458#102—1462no—
459#102—1462no—
460#102—1462no—
461#102—1462no—
462C4×C8—1462no—
463#95—21no—
464#95—21no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95C22⋊D7311yescharacteristic
470#95C22⋊D7311yescharacteristic
471C24—21no—
472C24—1461no—
473C2×Q8—42no—
474C2×Q8—42no—
475C2×Q8—81no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—732no—
482C2×D4—732no—
483C2×D4—732no—
484C2×D4—1461no—
485C2×D4—1461no—
486C2×D4—1461no—
487C2×D4Q8×D7312yes—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—2922no—
495C22×C4—2922no—
496C22×C4—2922no—
497C22×C4—2922no—
498C22×C4—5841no—
499C22×C4—5841no—
500C22×C4—5841no—
501C22×C4—5841no—
502C22×C4—1464no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4C2×C22⋊D7311yescharacteristic
507C22×C4C2×C22⋊D7311yescharacteristic
508C22×C4C2×C22⋊D7311yescharacteristic
509C22×C4C2×C22⋊D7311yescharacteristic
510C22×C4C2×C22⋊D7311yescharacteristic
511C22×C4C2×C22⋊D7311yescharacteristic
512C22×C4C23×D7311yescharacteristic
513D8—1468no—
514C2×C8—732no—
515C2×C8—732no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8—1461no—
525C2×C8—1461no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—5841no—
537C4⋊C4—5841no—
538C4⋊C4—732no—
539C4⋊C4—1461no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4—1461no—
543C4⋊C4—1461no—
544C4⋊C4D4×D7312yes—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—2922no—
551C22⋊C4—2922no—
552C22⋊C4—1464no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—2921no—
567C23—2921no—
568C23—2921no—
569C23—2921no—
570C23—1462no—
571C23—1462no—
572C23—1462no—
573C23C22×C22⋊D7311yescharacteristic
574Q8—84no—
575D4—22no—
576D4—22no—
577D4—41no—
578D4—41no—
579D4—734no—
580D4—1462no—
581D4—1462no—
582D4—1462no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—84no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—5844no—
618C2×C4—5844no—
619C2×C4—5844no—
620C2×C4—1462no—
621C2×C4#1455661112yes—
622C2×C4#1455661712yes—
623C2×C4C2×D4×D7311yescharacteristic
624C2×C4C2×Q8×D7311yescharacteristic
625C2×C4#41×D7311yescharacteristic
626C2×C4#41×D7311yescharacteristic
627C2×C4#1455663611yescharacteristic
628C2×C4#1455663911yescharacteristic
629C8—2921no—
630C8—2921no—
631C8—1462no—
632C8—1462no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—2922no—
641C22—2922no—
642C22—2922no—
643C22—2922no—
644C22—2922no—
645C22—2922no—
646C22—1464no—
647C22#1798909411yescharacteristic
648C22D4×C22⋊D7311yescharacteristic
649C22Q8×C22⋊D7311yescharacteristic
650C22C2×#1455661111yescharacteristic
651C22C2×#1455661711yescharacteristic
652C22#1798939011yescharacteristic
653C22#1798939511yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—2922no—
664C4—5841no—
665C4—5841no—
666C4C22⋊Q8×D7311yescharacteristic
667C4#1798918311yescharacteristic
668C4#1798923911yescharacteristic
669C4#1798929111yescharacteristic
670C2—82no—
671C2—2924no—
672C2#2261373711yescharacteristic
673C2#2261998511yescharacteristic
674C2#2262010911yescharacteristic
675C2#2262140711yescharacteristic
676C2#2262163911yescharacteristic
677C2#2262279411yescharacteristic
678C2#2262282611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2452C2
socleC23×C73C22×D4
prime radicalC22×#11924013C22
hypercenter#2452D73
hypocenterC73#26380
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2452

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#17988874C2semidirect
3. #22612842C2semidirect
4. #22611452C2semidirect
5. #22611412C2semidirect
6. #22611305C2semidirect
7. #22611256C2semidirect
8. #22610656C2semidirect
9. #22610490C2semidirect
12. #22610192C2semidirect
13. #22610191C2semidirect
14. #22610177C2semidirect
15. C73×#2452C2semidirect
67. C2×#11924013Q8semidirect
285. D292C22⋊Q8semidirect
342. #2452D73semidirect
355. C73#26380semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179888742
C2×C4×#119240132
#226128422
#226114522
#226114122
#226113052
#226112562
#226106562
#226104902
#226102822
#226101922
#226101912
#226101772
C73×#24522
#2638073