Finite Group Databasebeta

← #33148899

Subgroups — #33148899

Order 18688.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988887C211yesmaximal, characteristic, complemented
2C2×#17988627C211yesmaximal, characteristic, complemented
3#22613644C211yesmaximal, characteristic, complemented
4#22611464C211yesmaximal, characteristic, complemented
5#22611433C211yesmaximal, characteristic, complemented
6#22611319C211yesmaximal, characteristic, complemented
7#22611269C211yesmaximal, characteristic, complemented
8#22610694C211yesmaximal, characteristic, complemented
9#22610502C211yesmaximal, characteristic, complemented
10#22610370C211yesmaximal, characteristic
11#22610254C211yesmaximal, characteristic
12#22610192C211yesmaximal, characteristic, complemented
13#22610124C211yesmaximal, characteristic, complemented
14#22610116C211yesmaximal, characteristic, complemented
15C73×#2361C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×#14556583C2211yescharacteristic
18C2×#14556550C2211yescharacteristic
19#17988887—24no—
20#17988627—24no—
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×C4⋊#10596831C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×C22⋊#10596831C2211yescharacteristic
34#17988523—21no—
35#17988510—21no—
36#17988485C2211yescharacteristic
37#17988485C2211yescharacteristic
38#17988484—21no—
39#17988470—21no—
40#17988457C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×#121×C73C2211yescharacteristic
51C2×C4×D4×C73—21no—
52C2×C4×D4×C73C2211yescharacteristic
53C22×C4⋊C4×C73C2211yescharacteristic
54#390×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#388×C73—21no—
57#386×C73—21no—
58#385×C73—21no—
59C4×C4⋊C4×C73C2211yescharacteristic
60C4×C22⋊C4×C73C2211yescharacteristic
61C2×#11924012—22no—
62C2×#11924012—22no—
63C2×#11924012—22no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556583—22no—
69#14556583—41no—
70#14556550—22no—
71#14556550—41no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C4×Dic73—21no—
85#14556524—22no—
86#14556524D414yescomplemented
87#14556522—22no—
88#14556522—22no—
89#14556522—22no—
90#14556522—41no—
91C4⋊#10596831—22no—
92C4⋊#10596831—22no—
93#14556512—22no—
94#14556512—22no—
95#14556512—22no—
96#14556512—41no—
97C22⋊#10596831—22no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101C22×D4×C73—21no—
102C22×D4×C73C2311yescharacteristic
103C23×C4×C73—21no—
104C23×C4×C73—21no—
105C23×C4×C73C2311yescharacteristic
106#121×C73—22no—
107#121×C73Q814yes—
108C4×D4×C73—22no—
109C4×D4×C73—41no—
110C4×D4×C73—44no—
111C2×C4⋊C4×C73—22no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—41no—
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73D411yescharacteristic
123C2×C22⋊C4×C73D411yescharacteristic
124C2×C42×C73—21no—
125C2×C42×C73C2311yescharacteristic
126C2×C42×C73C2311yescharacteristic
127#95×C73—21no—
128#95×C73—21no—
129#95×C73D411yescharacteristic
130#95×C73D411yescharacteristic
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135#11924012—28no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831C2×D412yes—
147C2×#10596831#4112yes—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—42no—
156C4⋊Dic73—81no—
157C4⋊Dic73C2×D412yes—
158#11924007—42no—
159#11924007—42no—
160#11924007—42no—
161#11924007—81no—
162C22⋊Dic73—42no—
163C22⋊Dic73—81no—
164C4×Dic73—42no—
165C24×C73—21no—
166C24×C73C2×D411yescharacteristic
167C2×D4×C73—24no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—42no—
174C2×D4×C73—42no—
175C2×D4×C73C2×Q812yes—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—42no—
184C22×C4×C73—42no—
185C22×C4×C73—42no—
186C22×C4×C73—42no—
187C22×C4×C73—81no—
188C22×C4×C73—81no—
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×Q812yes—
195C22×C4×C73#4112yes—
196C22×C4×C73C2411yescharacteristic
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—42no—
202C4⋊C4×C73—42no—
203C4⋊C4×C73C2×D412yes—
204C4⋊C4×C73C2×Q812yes—
205C22⋊C4×C73—24no—
206C22⋊C4×C73—22no—
207C22⋊C4×C73—41no—
208C22⋊C4×C73—42no—
209C22⋊C4×C73—42no—
210C22⋊C4×C73—81no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—42no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—22no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73—41no—
229C23×C73C22⋊Q812yes—
230C23×C73C22×D411yescharacteristic
231D4×C73—22no—
232D4×C73—22no—
233D4×C73—48no—
234D4×C73—41no—
235D4×C73—41no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
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—41no—
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—84no—
263C2×C4×C73—84no—
264C2×C4×C73C22⋊Q812yes—
265C2×C4×C73AGL(1,ℤ/8)12yes—
266C2×C4×C73C22×D411yescharacteristic
267C2×C4×C73C22×Q811yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C2×C4×C73H(5,𝔽2)11yescharacteristic
271C2×C4×C732−1+411yescharacteristic
272C2×Dic73—22no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284#10596831—22no—
285#10596831—22no—
286#10596831—41no—
287#10596831—41no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—42no—
296C22×C73—42no—
297C22×C73—42no—
298C22×C73—81no—
299C22×C73—81no—
300C22×C73—81no—
301C22×C73—81no—
302C22×C73C2×C22⋊Q811yescharacteristic
303C22×C73D4211yescharacteristic
304C22×C73#54711yescharacteristic
305C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C73#57511yescharacteristic
307C22×C73#57711yescharacteristic
308C22×C73#57711yescharacteristic
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—42no—
312C4×C73—41no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—42no—
316C4×C73—81no—
317C4×C73#52511yescharacteristic
318C4×C73#53911yescharacteristic
319C4×C73D4×Q811yescharacteristic
320C4×C73#55411yescharacteristic
321Dic73—42no—
322Dic73—81no—
323Dic73—81no—
324Dic73—81no—
325#26366—731nomaximal, Sylow
326C2×C73—42no—
327C2×C73—82no—
328C2×C73#244911yescharacteristic
329C2×C73#307711yescharacteristic
330C2×C73#308411yescharacteristic
331C2×C73#327111yescharacteristic
332C2×C73#327211yescharacteristic
333C2×C73#334511yescharacteristic
334C2×C73#335111yescharacteristic
335C2×#478—731no—
336C2×#428—731no—
337#2441—731no—
338#2361D7311yescharacteristic
339#2094—731no—
340#2088—731no—
341#2067—731no—
342#2059—731no—
343#1948—731no—
344#1918—731no—
345#1895—731no—
346#1875—731no—
347#1864—731no—
348#1853—731no—
349#1851—731no—
350C73#2636611yescharacteristic, complemented, Sylow
351C22×M16—731no—
352C2×#125—731no—
353C2×#121C2×D7311yescharacteristic
354C2×C4×D4—21no—
355C2×C4×D4C2×D7311yescharacteristic
356C2×#117—731no—
357C22×C4⋊C4C2×D7311yescharacteristic
358#478—1464no—
359#428—1464no—
360C2×#107—731no—
361C2×#107—731no—
362C2×#106—731no—
363C2×#106—731no—
364C2×#106—731no—
365C2×#106—731no—
366C2×C4⋊C8—731no—
367C2×C4⋊C8—731no—
368C2×#102—731no—
369C2×#102—731no—
370C2×#102—731no—
371C2×#102—731no—
372C2×C22⋊C8—731no—
373#399—1461no—
374#396—1461no—
375#390C2×D7311yescharacteristic
376#389—731no—
377#389—731no—
378#389—1461no—
379#389C2×D7311yescharacteristic
380#388—21no—
381#387—1461no—
382#386—21no—
383#385—21no—
384#384—731no—
385#380—731no—
386C4×C4⋊C4C2×D7311yescharacteristic
387C4×C22⋊C4C2×D7311yescharacteristic
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#336—731no—
395#336—731no—
396C22×D4—21no—
397C22×D4C22×D7311yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4C22×D7311yescharacteristic
401C2×M16—1462no—
402C2×M16—1462no—
403C2×M16—1462no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408#125—2921no—
409#125—1462no—
410#121—22no—
411#121Dic14614yes—
412C4×D4—22no—
413C4×D4—41no—
414C4×D4—44no—
415#117—2921no—
416#117—1462no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—41no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4C22⋊D7311yescharacteristic
441C2×C22⋊C4C22⋊D7311yescharacteristic
442C2×C42—21no—
443C2×C42—1461no—
444C2×C42C22×D7311yescharacteristic
445C2×C42C22×D7311yescharacteristic
446#107—734no—
447#107—1462no—
448#106—2921no—
449#106—1462no—
450#106—1462no—
451#106—1462no—
452C4⋊C8—1462no—
453C4⋊C8—1462no—
454#102—2921no—
455#102—1462no—
456#102—1462no—
457#102—1462no—
458C22⋊C8—1462no—
459#95—21no—
460#95—21no—
461#95—1461no—
462#95—1461no—
463#95—1461no—
464#95C22⋊D7311yescharacteristic
465#95C22⋊D7311yescharacteristic
466C24—21no—
467C24C2×C22⋊D7311yescharacteristic
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4C2×Dic14612yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4C2×C22⋊D7311yescharacteristic
495C22×C4C2×C22⋊D7311yescharacteristic
496C22×C4C2×C22⋊D7311yescharacteristic
497C22×C4C2×C22⋊D7311yescharacteristic
498C22×C4C2×C22⋊D7311yescharacteristic
499C22×C4C2×Dic14612yes—
500C22×C4#1192402812yes—
501C22×C4C23×D7311yescharacteristic
502M16—1468no—
503C2×C8—732no—
504C2×C8—732no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—2922no—
522C4⋊C4—2922no—
523C4⋊C4—2922no—
524C4⋊C4—2922no—
525C4⋊C4—5841no—
526C4⋊C4—5841no—
527C4⋊C4—732no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4D4×D7312yes—
536C4⋊C4C2×Dic14612yes—
537C22⋊C4—24no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—2922no—
544C22⋊C4—5841no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C42—2922no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23#1455657012yes—
565C23C22×C22⋊D7311yescharacteristic
566D4—22no—
567D4—22no—
568D4—48no—
569D4—41no—
570D4—41no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—1462no—
611C2×C4#1455657012yes—
612C2×C4#1455661712yes—
613C2×C4C2×D4×D7311yescharacteristic
614C2×C4C22×Dic14611yescharacteristic
615C2×C4#41×D7311yescharacteristic
616C2×C4C2×#1192402811yescharacteristic
617C2×C4#1455663611yescharacteristic
618C2×C4#1455663811yescharacteristic
619C8—2921no—
620C8—2921no—
621C8—1462no—
622C8—1462no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C2×#1455657011yescharacteristic
638C22D4×C22⋊D7311yescharacteristic
639C22#1798923611yescharacteristic
640C22C2×#1455661711yescharacteristic
641C22#1798938511yescharacteristic
642C22#1798939511yescharacteristic
643C22#1798939811yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—42no—
651C4—81no—
652C4—2922no—
653C4—5841no—
654C4—5841no—
655C4—5841no—
656C4#1798909611yescharacteristic
657C4#1798917911yescharacteristic
658C4D4×Dic14611yescharacteristic
659C4#1798929611yescharacteristic
660C2—42no—
661C2—82no—
662C2#2261373411yescharacteristic
663C2#2261972811yescharacteristic
664C2#2261979311yescharacteristic
665C2#2262151711yescharacteristic
666C2#2262152711yescharacteristic
667C2#2262240111yescharacteristic
668C2#2262248511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2361C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2361D73
hypocenterC73#26366
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. C2×#17988887C2semidirect
2. C2×#17988627C2semidirect
3. #22613644C2semidirect
4. #22611464C2semidirect
5. #22611433C2semidirect
6. #22611319C2semidirect
7. #22611269C2semidirect
8. #22610694C2semidirect
9. #22610502C2semidirect
12. #22610192C2semidirect
13. #22610124C2semidirect
14. #22610116C2semidirect
86. #14556524D4semidirect
350. C73#26366semidirect

Maximal subgroups

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

subgroupindex
C2×#179888872
C2×#179886272
#226136442
#226114642
#226114332
#226113192
#226112692
#226106942
#226105022
#226103702
#226102542
#226101922
#226101242
#226101162
C73×#23612
#2636673