Finite Group Databasebeta

← #33136342

Subgroups — #33136342

Order 18688.

832 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988895C211yesmaximal, characteristic, complemented
2#22615097C211yesmaximal, characteristic
3#22614853C211yesmaximal, characteristic
4#22611364C211yesmaximal, characteristic, complemented
5#22611359C211yesmaximal, characteristic, complemented
6#22611128C211yesmaximal, characteristic, complemented
7#22611128C211yesmaximal, characteristic, complemented
8#22610900C211yesmaximal, characteristic, complemented
9#22610695C211yesmaximal, characteristic, complemented
10#22610364C211yesmaximal, characteristic, complemented
11#22610354C211yesmaximal, characteristic, complemented
12#22609964C211yesmaximal, characteristic, complemented
13#22609956C211yesmaximal, characteristic, complemented
14#22609762C211yesmaximal, characteristic, complemented
15C2×#435×C73C211yesmaximal, characteristic
16C2×#14556583C2211yescharacteristic
17C2×#14556570C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20#17988895—24no—
21C2×#14556525C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28C2×#14556516C2211yescharacteristic
29#17988525—21no—
30#17988510—21no—
31#17988510—21no—
32#17988501—21no—
33#17988490—21no—
34#17988483—21no—
35#17988481—21no—
36#17988481—21no—
37#17988457C2211yescharacteristic
38#17988447C2211yescharacteristic
39#17988447C2211yescharacteristic
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988447C2211yescharacteristic
43#17988447C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C22×M16×C73C2211yescharacteristic
54C23×C8×C73C2211yescharacteristic
55C2×C4×D4×C73C2211yescharacteristic
56#435×C73—24no—
57C2×C4⋊C8×C73C2211yescharacteristic
58C2×C22⋊C8×C73C2211yescharacteristic
59C2×C22⋊C8×C73C2211yescharacteristic
60C2×#97×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62#14556583—22no—
63#14556583—22no—
64#14556570—22no—
65#14556570—22no—
66#14556570—41no—
67#14556570—41no—
68C4×Dic146—22no—
69C4×Dic146—22no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C22⋊Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94#14556525—22no—
95#14556523—22no—
96#14556523—41no—
97#14556523—41no—
98#14556516—22no—
99#14556516—22no—
100#14556516—22no—
101#14556516—22no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109C22×D4×C73D411yescharacteristic
110C23×C4×C73C2311yescharacteristic
111C23×C4×C73C2311yescharacteristic
112C2×M16×C73—22no—
113C2×M16×C73—22no—
114C2×M16×C73—41no—
115C22×C8×C73—22no—
116C22×C8×C73—22no—
117C22×C8×C73—41no—
118C22×C8×C73C2311yescharacteristic
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C22×C8×C73C2311yescharacteristic
122C4×D4×C73—22no—
123C4×D4×C73—22no—
124C2×C4⋊C4×C73D411yescharacteristic
125C2×C22⋊C4×C73D411yescharacteristic
126C2×C22⋊C4×C73D411yescharacteristic
127C2×C42×C73C2311yescharacteristic
128C4⋊C8×C73D414yes—
129C22⋊C8×C73—22no—
130C22⋊C8×C73—22no—
131#97×C73—22no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146#4112yes—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—42no—
161C4⋊Dic73—81no—
162#11924007—42no—
163#11924007—42no—
164#11924007—42no—
165#11924007—42no—
166#11924007—42no—
167C22⋊Dic73—42no—
168C22⋊Dic73—42no—
169C22⋊Dic73—42no—
170C22⋊Dic73—42no—
171C22⋊Dic73—81no—
172C22⋊Dic73—81no—
173C22⋊Dic73—81no—
174C22⋊Dic73—81no—
175C4×Dic73—42no—
176C4×Dic73—81no—
177C24×C73C2×D411yescharacteristic
178C24×C73C2×D411yescharacteristic
179C2×D4×C73—22no—
180C2×D4×C73—22no—
181C2×D4×C73—41no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2411yescharacteristic
199M16×C73—44no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—44no—
210C2×C8×C73C2×D412yes—
211C2×C8×C73C2×D412yes—
212C2×C8×C73#4112yes—
213C4⋊C4×C73—22no—
214C22⋊C4×C73—22no—
215C22⋊C4×C73—22no—
216C42×C73C2×D412yes—
217C42×C73#4112yes—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73#13712yes—
232C23×C73C22×D411yescharacteristic
233D4×C73—44no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73#12112yes—
253C2×C4×C73#12112yes—
254C2×C4×C73C22×D411yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73H(5,𝔽2)11yescharacteristic
259C2×C4×C732−1+411yescharacteristic
260C2×Dic73—22no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
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—
284Dic146—22no—
285Dic146—22no—
286Dic146—41no—
287Dic146—41no—
288C8×C73—22no—
289C8×C73—22no—
290C8×C73—22no—
291C8×C73—41no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73—41no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73C2×#12111yescharacteristic
307C22×C73#54611yescharacteristic
308C22×C73#54711yescharacteristic
309C22×C73C2×#13711yescharacteristic
310C22×C73#57511yescharacteristic
311C22×C73#57711yescharacteristic
312C22×C73#57711yescharacteristic
313C4×C73—22no—
314C4×C73—22no—
315C4×C73—41no—
316C4×C73—41no—
317C4×C73#52911yescharacteristic
318C4×C73#54211yescharacteristic
319C4×C73#54611yescharacteristic
320C4×C73#54811yescharacteristic
321Dic73—42no—
322Dic73—81no—
323Dic73—81no—
324Dic73—81no—
325Dic73—81no—
326Dic73—81no—
327Dic73—81no—
328Dic73—81no—
329#25357—731nomaximal, Sylow
330C2×C73—42no—
331C2×C73—81no—
332C2×C73#242711yescharacteristic
333C2×C73#305611yescharacteristic
334C2×C73#306011yescharacteristic
335C2×C73#331711yescharacteristic
336C2×C73#332011yescharacteristic
337C2×C73#334411yescharacteristic
338C2×C73#335011yescharacteristic
339C2×#479—731no—
340C2×#435D7311yescharacteristic
341#2562—731no—
342#2538—731no—
343#2077—731no—
344#2076—731no—
345#2035—731no—
346#2035—731no—
347#1992—731no—
348#1948—731no—
349#1894—731no—
350#1892—731no—
351#1809—731no—
352#1807—731no—
353#1763—731no—
354C73#2535711yescharacteristic, complemented, Sylow
355C22×M16C2×D7311yescharacteristic
356C23×C8C2×D7311yescharacteristic
357C2×#125—731no—
358C2×C22⋊Q8—731no—
359C2×C22⋊Q8—731no—
360C2×C4×Q8—731no—
361C2×C4×D4C2×D7311yescharacteristic
362#479—1464no—
363#435—24no—
364C2×#107—731no—
365C2×#106—731no—
366C2×#106—731no—
367C2×#106—731no—
368C2×C4⋊C8C2×D7311yescharacteristic
369C2×#103—731no—
370C2×#103—731no—
371C2×#103—731no—
372C2×#103—731no—
373C2×C22⋊C8C2×D7311yescharacteristic
374C2×C22⋊C8C2×D7311yescharacteristic
375C2×#97C2×D7311yescharacteristic
376#400—1461no—
377#396—1461no—
378#396—1461no—
379#394—1461no—
380#390—1461no—
381#389—1461no—
382#388—1461no—
383#388—1461no—
384#384—731no—
385#380—731no—
386#380—731no—
387#380—731no—
388#380—731no—
389#380—731no—
390#380—731no—
391C4×C4⋊C4—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#340—731no—
398#340—731no—
399#340—731no—
400C22×Q8—731no—
401C22×D4D29211yescharacteristic
402C23×C4C22×D7311yescharacteristic
403C23×C4C22×D7311yescharacteristic
404C2×M16—22no—
405C2×M16—22no—
406C2×M16—41no—
407C22×C8—22no—
408C22×C8—22no—
409C22×C8—41no—
410C22×C8C22×D7311yescharacteristic
411C22×C8C22×D7311yescharacteristic
412C22×C8C22×D7311yescharacteristic
413C22×C8C22×D7311yescharacteristic
414#125—1462no—
415#125—1462no—
416C22⋊Q8—2921no—
417C22⋊Q8—2921no—
418C22⋊Q8—1462no—
419C22⋊Q8—1462no—
420C4×Q8—1462no—
421C4×Q8—1462no—
422C4×D4—22no—
423C4×D4—22no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1461no—
434C2×C4⋊C4—1461no—
435C2×C4⋊C4—1461no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4D29211yescharacteristic
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C22⋊C4—1461no—
442C2×C22⋊C4—1461no—
443C2×C22⋊C4—1461no—
444C2×C22⋊C4—1461no—
445C2×C22⋊C4—1461no—
446C2×C22⋊C4—1461no—
447C2×C22⋊C4D29211yescharacteristic
448C2×C22⋊C4D29211yescharacteristic
449C2×C42—1461no—
450C2×C42—1461no—
451C2×C42C22×D7311yescharacteristic
452#107—1462no—
453#106—2921no—
454#106—2921no—
455#106—1462no—
456C4⋊C8C22⋊D7314yes—
457#103—1462no—
458#103—1462no—
459#103—1462no—
460#103—1462no—
461C22⋊C8—22no—
462C22⋊C8—22no—
463#97—22no—
464#95—1461no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471C24C2×D29211yescharacteristic
472C24C2×D29211yescharacteristic
473C2×Q8—732no—
474C2×Q8—1461no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—41no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4C2×D29211yescharacteristic
503C22×C4C2×D29211yescharacteristic
504C22×C4C2×D29211yescharacteristic
505C22×C4C2×D29211yescharacteristic
506C22×C4C23×D7311yescharacteristic
507M16—44no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—44no—
518C2×C8C2×C22⋊D7312yes—
519C2×C8C2×C22⋊D7312yes—
520C2×C8#1192402712yes—
521C4⋊C4—22no—
522C4⋊C4—2922no—
523C4⋊C4—2922no—
524C4⋊C4—2922no—
525C4⋊C4—2922no—
526C4⋊C4—2922no—
527C4⋊C4—2922no—
528C4⋊C4—5841no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4—1461no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—2922no—
546C22⋊C4—2922no—
547C22⋊C4—2922no—
548C22⋊C4—2922no—
549C22⋊C4—5841no—
550C22⋊C4—5841no—
551C22⋊C4—5841no—
552C22⋊C4—5841no—
553C42—2922no—
554C42—5841no—
555C42C2×C22⋊D7312yes—
556C42#1192402812yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#1455662412yes—
571C23C22×D29211yescharacteristic
572Q8—2921no—
573Q8—2921no—
574Q8—1462no—
575Q8—1462no—
576D4—44no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
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—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—2921no—
618C2×C4—1462no—
619C2×C4#1455656512yes—
620C2×C4#1455656512yes—
621C2×C4C22×C22⋊D7311yescharacteristic
622C2×C4#41×D7311yescharacteristic
623C2×C4C2×#1192402711yescharacteristic
624C2×C4C2×#1192402811yescharacteristic
625C2×C4#1455663611yescharacteristic
626C2×C4#1455663811yescharacteristic
627C8—22no—
628C8—22no—
629C8—22no—
630C8—41no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22C2×#1455656511yescharacteristic
646C22#1798923211yescharacteristic
647C22#1798924111yescharacteristic
648C22C2×#1455662411yescharacteristic
649C22#1798938711yescharacteristic
650C22#1798939911yescharacteristic
651C22#1798939911yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—2922no—
657C4—5841no—
658C4—5841no—
659C4—5841no—
660C4—5841no—
661C4—5841no—
662C4—5841no—
663C4—5841no—
664C4#1798911911yescharacteristic
665C4#1798920211yescharacteristic
666C4#1798923111yescharacteristic
667C4#1798924411yescharacteristic
668C2—42no—
669C2—81no—
670C2#2261350211yescharacteristic
671C2#2261956111yescharacteristic
672C2#2261959311yescharacteristic
673C2#2262205911yescharacteristic
674C2#2262210411yescharacteristic
675C2#2262238811yescharacteristic
676C2#2262247811yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#435×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenterC2×#435D73
hypocenterC73#25357
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
1. C2×#17988895C2semidirect
4. #22611364C2semidirect
5. #22611359C2semidirect
6. #22611128C2semidirect
7. #22611128C2semidirect
8. #22610900C2semidirect
9. #22610695C2semidirect
10. #22610364C2semidirect
11. #22610354C2semidirect
12. #22609964C2semidirect
13. #22609956C2semidirect
14. #22609762C2semidirect
354. C73#25357semidirect

Maximal subgroups

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

subgroupindex
C2×#179888952
#226150972
#226148532
#226113642
#226113592
#226111282
#226109002
#226106952
#226103642
#226103542
#226099642
#226099562
#226097622
C2×#435×C732
#2535773