Finite Group Databasebeta

← #33154103

Subgroups — #33154103

Order 18688.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614305C211yesmaximal, characteristic
2#22612934C211yesmaximal, characteristic
3#22611254C211yesmaximal, characteristic, complemented
4#22611135C211yesmaximal, characteristic
5#22611073C211yesmaximal, characteristic, complemented
6#22610941C211yesmaximal, characteristic
7#22610878C211yesmaximal, characteristic, complemented
8#22610607C211yesmaximal, characteristic, complemented
9#22610573C211yesmaximal, characteristic, complemented
10#22610524C211yesmaximal, characteristic, complemented
11#22610271C211yesmaximal, characteristic, complemented
12#22610236C211yesmaximal, characteristic
13#22610022C211yesmaximal, characteristic
14C4×#14556516C211yesmaximal, characteristic, complemented
15C73×#1923C211yesmaximal, characteristic
16C22×Dic292C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×#14556570C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21C22×C4⋊Dic73C2211yescharacteristic
22C2×#14556525C2211yescharacteristic
23C2×#14556516C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28C2×#14556516C2211yescharacteristic
29#17988529—21no—
30#17988499—21no—
31#17988493C2211yescharacteristic
32#17988493C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988491—21no—
35#17988485C2211yescharacteristic
36#17988484—21no—
37#17988480—21no—
38#17988474—21no—
39#17988459—21no—
40#17988447C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43C22⋊C4×Dic73—21no—
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C2×C4⋊C8×C73C2211yescharacteristic
53C2×C22⋊C8×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55C2×C4×C8×C73C2211yescharacteristic
56C4×C22⋊C4×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×Dic146C2311yescharacteristic
61C22×Dic146C2311yescharacteristic
62C23×Dic73—21no—
63C2×Dic292—22no—
64C2×Dic292—41no—
65C2×Dic292—41no—
66#14556591—22no—
67#14556591—41no—
68#14556570—22no—
69#14556570—22no—
70#14556570—22no—
71#14556570—41no—
72C4×Dic146—22no—
73C4×Dic146—41no—
74C2×C4⋊Dic73—22no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73—41no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94#14556525—22no—
95#14556516—22no—
96#14556516—22no—
97#14556516—22no—
98#14556516—22no—
99#14556516—41no—
100#14556516—41no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111C23×C4×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C2×C22⋊C4×C73D411yescharacteristic
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C4⋊C8×C73—41no—
121C22⋊C8×C73—41no—
122C22⋊C8×C73—41no—
123C4×C8×C73—41no—
124#95×C73D411yescharacteristic
125#95×C73D411yescharacteristic
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146—42no—
138C2×Dic146—42no—
139C2×Dic146—81no—
140C2×Dic146#4112yes—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73—21no—
149C22×Dic73—42no—
150C22×Dic73—42no—
151C22×Dic73—81no—
152Dic292—44no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—42no—
163C4⋊Dic73—42no—
164C4⋊Dic73#4112yes—
165C4⋊Dic73#4112yes—
166C4⋊Dic73#4112yes—
167#11924007—24no—
168#11924007—42no—
169#11924007—42no—
170#11924007—81no—
171C22⋊Dic73—24no—
172C22⋊Dic73—42no—
173C22⋊Dic73—81no—
174C4×Dic73—42no—
175C4×Dic73—81no—
176C4×Dic73—81no—
177C24×C73C2×D411yescharacteristic
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2411yescharacteristic
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C22⋊C4×C73—41no—
203C22⋊C4×C73Q1614yescomplemented
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73C2×Q1612yes—
215C23×C73C22×D411yescharacteristic
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73C2×Q1612yes—
234C2×C4×C73C2×Q1612yes—
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73H(5,𝔽2)11yescharacteristic
240C2×C4×C73H(5,𝔽2)11yescharacteristic
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—84no—
266Dic146—22no—
267Dic146—22no—
268Dic146—41no—
269Dic146—41no—
270Dic146—41no—
271Dic146—41no—
272Dic146—41no—
273Dic146—41no—
274Dic146—84no—
275C8×C73—41no—
276C8×C73—41no—
277C8×C73—41no—
278C8×C73—41no—
279C22×C73—22no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73#52511yescharacteristic
287C22×C73#54611yescharacteristic
288C22×C73#54611yescharacteristic
289C22×C73C22×Q1611yescharacteristic
290C22×C73#57511yescharacteristic
291C22×C73#57711yescharacteristic
292C22×C73#57711yescharacteristic
293C4×C73—22no—
294C4×C73—22no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73#52911yescharacteristic
299C4×C73#53811yescharacteristic
300C4×C73#55511yescharacteristic
301C4×C73#55911yescharacteristic
302Dic73—42no—
303Dic73—42no—
304Dic73—42no—
305Dic73—42no—
306Dic73—81no—
307Dic73—81no—
308Dic73—81no—
309Dic73—81no—
310#26726—731nomaximal, Sylow
311C2×C73—42no—
312C2×C73#245911yescharacteristic
313C2×C73#310411yescharacteristic
314C2×C73#312511yescharacteristic
315C2×C73#329911yescharacteristic
316C2×C73#329911yescharacteristic
317C2×C73#331111yescharacteristic
318C2×C73#332311yescharacteristic
319#2492—731no—
320#2360—731no—
321#2056—731no—
322#2036—731no—
323#2024—731no—
324#2001—731no—
325#1987—731no—
326#1933—731no—
327#1928—731no—
328#1923D7311yescharacteristic
329#1921—731no—
330#1878—731no—
331#1872—731no—
332#1824—731no—
333C4×#103—731no—
334C73#2672611yescharacteristic, complemented, Sylow
335C22×Q16—731no—
336C2×C4⋊Q8—731no—
337C2×C22⋊Q8—731no—
338C2×C22⋊Q8—731no—
339C2×C4×Q8—731no—
340C22×C4⋊C4—731no—
341C2×#107—731no—
342C2×C4⋊C8C2×D7311yescharacteristic
343C2×#103—731no—
344C2×#103—731no—
345C2×#103—731no—
346C2×#103—731no—
347C2×#103—731no—
348C2×#103—731no—
349C2×C22⋊C8C2×D7311yescharacteristic
350C2×C22⋊C8C2×D7311yescharacteristic
351C2×C4×C8C2×D7311yescharacteristic
352#400—1461no—
353#393—1461no—
354#391—731no—
355#391—731no—
356#391—731no—
357#391—1461no—
358#389—731no—
359#389—1461no—
360#388—1461no—
361#387—1461no—
362#385—1461no—
363#380—731no—
364C4×C4⋊C4—731no—
365C4×C4⋊C4—731no—
366C4×C22⋊C4—1461no—
367C4×C22⋊C4C2×D7311yescharacteristic
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#336C2×D7311yescharacteristic
377#336C2×D7311yescharacteristic
378C22×Q8—731no—
379C22×Q8—731no—
380C22×Q8—1461no—
381C23×C4—1461no—
382C23×C4C22×D7311yescharacteristic
383C2×Q16—2921no—
384C2×Q16—2921no—
385C2×Q16—1462no—
386C22×C8C22×D7311yescharacteristic
387C22×C8C22×D7311yescharacteristic
388C22×C8C22×D7311yescharacteristic
389C22×C8C22×D7311yescharacteristic
390C4⋊Q8—2921no—
391C4⋊Q8—1462no—
392C22⋊Q8—2921no—
393C22⋊Q8—1462no—
394C22⋊Q8—1462no—
395C22⋊Q8—1462no—
396C4×Q8—2921no—
397C4×Q8—1462no—
398C2×C4⋊C4—2921no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1462no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1461no—
412C2×C22⋊C4—1461no—
413C2×C22⋊C4—1461no—
414C2×C22⋊C4—1461no—
415C2×C22⋊C4D29211yescharacteristic
416C2×C22⋊C4D29211yescharacteristic
417C2×C42—1461no—
418C2×C42—1461no—
419C2×C42—1461no—
420C2×C42C22×D7311yescharacteristic
421C2×C42C22×D7311yescharacteristic
422#107—1462no—
423C4⋊C8—41no—
424#103—2921no—
425#103—2921no—
426#103—1462no—
427#103—1462no—
428#103—1462no—
429#103—1462no—
430C22⋊C8—41no—
431C22⋊C8—41no—
432C4×C8—41no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443#95D29211yescharacteristic
444#95D29211yescharacteristic
445C24C2×D29211yescharacteristic
446C2×Q8—2922no—
447C2×Q8—2922no—
448C2×Q8—5841no—
449C2×Q8—732no—
450C2×Q8—1461no—
451C2×Q8—1461no—
452C2×Q8—1461no—
453C2×Q8—1461no—
454C2×Q8—1461no—
455C2×Q8—1461no—
456C2×Q8—1461no—
457C2×Q8—1461no—
458C2×Q8—1461no—
459C2×Q8—1461no—
460C2×Q8—1461no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—2922no—
468C22×C4—2922no—
469C22×C4—5841no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4C2×D29211yescharacteristic
479C22×C4C2×D29211yescharacteristic
480C22×C4C2×D29211yescharacteristic
481C22×C4C2×D29211yescharacteristic
482C22×C4C2×D29211yescharacteristic
483C22×C4C23×D7311yescharacteristic
484Q16—2924no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C4⋊C4—2922no—
498C4⋊C4—2922no—
499C4⋊C4—2922no—
500C4⋊C4—2922no—
501C4⋊C4—5841no—
502C4⋊C4—732no—
503C4⋊C4—732no—
504C4⋊C4—732no—
505C4⋊C4—1464no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C22⋊C4—41no—
516C22⋊C4—2922no—
517C22⋊C4—5841no—
518C22⋊C4—1464no—
519C22⋊C4Dic29214yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—2922no—
525C42—5841no—
526C42—5841no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×Dic29212yes—
534C23C22×D29211yescharacteristic
535Q8—2921no—
536Q8—2921no—
537Q8—2921no—
538Q8—2921no—
539Q8—2921no—
540Q8—2921no—
541Q8—5844no—
542Q8—1462no—
543Q8—1462no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—5844no—
582C2×C4—1462no—
583C2×C4—1462no—
584C2×C4—1462no—
585C2×C4—1462no—
586C2×C4C2×Dic29212yes—
587C2×C4C2×Dic29212yes—
588C2×C4#41×D7311yescharacteristic
589C2×C4#41×D7311yescharacteristic
590C2×C4#41×D7311yescharacteristic
591C2×C4#41×D7311yescharacteristic
592C2×C4#1455663611yescharacteristic
593C2×C4#1455663611yescharacteristic
594C8—41no—
595C8—41no—
596C8—41no—
597C8—41no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1798909911yescharacteristic
606C22#1798923211yescharacteristic
607C22#1798923211yescharacteristic
608C22C22×Dic29211yescharacteristic
609C22#1798938711yescharacteristic
610C22#1798939911yescharacteristic
611C22#1798939911yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—2922no—
618C4—2922no—
619C4—2922no—
620C4—2922no—
621C4—5841no—
622C4—5841no—
623C4—5841no—
624C4—5841no—
625C4#1798912011yescharacteristic
626C4#1798916911yescharacteristic
627C4#1798929911yescharacteristic
628C4#1798932111yescharacteristic
629C2—42no—
630C2#2261386611yescharacteristic
631C2#2261997111yescharacteristic
632C2#2262021011yescharacteristic
633C2#2262180811yescharacteristic
634C2#2262180811yescharacteristic
635C2#2262196911yescharacteristic
636C2#2262214911yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1923C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1923D73
hypocenterC73#26726
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
3. #22611254C2semidirect
5. #22611073C2semidirect
7. #22610878C2semidirect
8. #22610607C2semidirect
9. #22610573C2semidirect
10. #22610524C2semidirect
11. #22610271C2semidirect
14. C4×#14556516C2semidirect
203. C22⋊C4×C73Q16semidirect
334. C73#26726semidirect
519. C22⋊C4Dic292semidirect

Maximal subgroups

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

subgroupindex
#226143052
#226129342
#226112542
#226111352
#226110732
#226109412
#226108782
#226106072
#226105732
#226105242
#226102712
#226102362
#226100222
C4×#145565162
C73×#19232
#2672673