Finite Group Databasebeta

← #33103254

Subgroups — #33103254

Order 18688.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988596C211yesmaximal, characteristic
2#22612822C211yesmaximal, characteristic
3#22610624C211yesmaximal, characteristic, complemented
4#22610610C211yesmaximal, characteristic, complemented
5#22610569C211yesmaximal, characteristic, complemented
6#22610562C211yesmaximal, characteristic, complemented
7#22610199C211yesmaximal, characteristic, complemented
8#22610185C211yesmaximal, characteristic, complemented
9#22610023C211yesmaximal, characteristic
10#22609962C211yesmaximal, characteristic
11#22609866C211yesmaximal, characteristic
12#22609761C211yesmaximal, characteristic, complemented
13C4×#14556515C211yesmaximal, characteristic, complemented
14#22609715C211yesmaximal, characteristic
15Q8×C22⋊C4×C73C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×#14556550C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20#17988596C418yescomplemented
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×#14556502C2211yescharacteristic
31C2×C4×#10596831C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988481—21no—
35#17988481—21no—
36#17988454—21no—
37#17988447C2211yescharacteristic
38C4×C4⋊Dic73C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40#17988438—21no—
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C23×Q8×C73C2211yescharacteristic
50C2×C4×Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52#391×C73—21no—
53#391×C73—21no—
54#391×C73C2211yescharacteristic
55#391×C73C2211yescharacteristic
56#380×C73—21no—
57#380×C73C2211yescharacteristic
58C4×C22⋊C4×C73—21no—
59C4×C22⋊C4×C73C2211yescharacteristic
60C2×#11924012—41no—
61C2×#11924012—41no—
62C2×#11924012C2×C414yes—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556550—22no—
68#14556550C2×C414yes—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77C2×C4×Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73—21no—
80#14556524—41no—
81#14556522—41no—
82#14556515—22no—
83#14556515—22no—
84#14556515—41no—
85#14556515—41no—
86#14556515C2×C414yes—
87#14556515C2×C414yes—
88#14556515C2×C414yes—
89#14556515C2×C414yes—
90C22⋊#10596831—22no—
91#14556502—41no—
92C4×#10596831—41no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98C22×Q8×C73—21no—
99C22×Q8×C73—22no—
100C22×Q8×C73—22no—
101C22×Q8×C73—22no—
102C22×Q8×C73—41no—
103C22×Q8×C73—41no—
104C22×Q8×C73C2×C414yes—
105C22×Q8×C73C2311yescharacteristic
106C22×Q8×C73C2311yescharacteristic
107C23×C4×C73—21no—
108C23×C4×C73C2311yescharacteristic
109C4×Q8×C73—22no—
110C4×Q8×C73—22no—
111C4×Q8×C73—41no—
112C4×Q8×C73—41no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C42×C73—21no—
121C2×C42×C73—21no—
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#95×C73—21no—
125#95×C73—21no—
126#95×C73—21no—
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—44no—
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×#10596831C22×C412yes—
147C2×#10596831C22×C412yes—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73C22×C412yes—
155C4⋊Dic73C22×C412yes—
156#11924007—42no—
157#11924007—42no—
158C22⋊Dic73—24no—
159C22⋊Dic73—81no—
160C4×Dic73—24no—
161C4×Dic73—81no—
162C4×Dic73—81no—
163C24×C73C2×D411yescharacteristic
164C2×Q8×C73—24no—
165C2×Q8×C73—21no—
166C2×Q8×C73—21no—
167C2×Q8×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73—28no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—42no—
175C2×Q8×C73—42no—
176C2×Q8×C73—44no—
177C2×Q8×C73—44no—
178C2×Q8×C73—44no—
179C2×Q8×C73C22×C412yes—
180C2×Q8×C73C22×C412yes—
181C2×Q8×C73#4112yes—
182C2×Q8×C73#4112yes—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—24no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—42no—
193C22×C4×C73—42no—
194C22×C4×C73C22×C412yes—
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73#4112yes—
201C22×C4×C73C2411yescharacteristic
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—42no—
207C4⋊C4×C73—42no—
208C22⋊C4×C73—22no—
209C22⋊C4×C73—41no—
210C22⋊C4×C73—42no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—42no—
216C42×C73—42no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73C4×D412yes—
224C23×C73C22×D411yescharacteristic
225Q8×C73—22no—
226Q8×C73—22no—
227Q8×C73—22no—
228Q8×C73—22no—
229Q8×C73—22no—
230Q8×C73—48no—
231Q8×C73—41no—
232Q8×C73—41no—
233Q8×C73#11714yes—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
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—48no—
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×C73C4×D412yes—
265C2×C4×C73#13712yes—
266C2×C4×C73C23×C411yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C2×C4×C73H(5,𝔽2)11yescharacteristic
271C2×C4×C73H(5,𝔽2)11yescharacteristic
272C2×Dic73—22no—
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—
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×C73C2×C4×D411yescharacteristic
296C22×C73#54611yescharacteristic
297C22×C73#54611yescharacteristic
298C22×C73C2×#13711yescharacteristic
299C22×C73#57511yescharacteristic
300C22×C73#57711yescharacteristic
301C22×C73#57811yescharacteristic
302C4×C73—22no—
303C4×C73—24no—
304C4×C73—22no—
305C4×C73—41no—
306C4×C73—42no—
307C4×C73—41no—
308C4×C73—41no—
309C4×C73—81no—
310C4×C73—81no—
311C4×C73C2×#11711yescharacteristic
312C4×C73#51811yescharacteristic
313C4×C73#55111yescharacteristic
314C4×C73#55111yescharacteristic
315Dic73—42no—
316Dic73—42no—
317Dic73—81no—
318Dic73—81no—
319#22581—731nomaximal, Sylow
320C2×C73—42no—
321C2×C73#234211yescharacteristic
322C2×C73C4×#13711yescharacteristic
323C2×C73#294711yescharacteristic
324C2×C73#331311yescharacteristic
325C2×C73#332011yescharacteristic
326C2×C73#332611yescharacteristic
327C2×C73#332811yescharacteristic
328C2×#419—731no—
329#2347—731no—
330Q8×C22⋊C4D7311yescharacteristic
331#1935—731no—
332#1933—731no—
333#1928—731no—
334#1927—731no—
335#1865—731no—
336#1863—731no—
337#1825—731no—
338#1809—731no—
339#1787—731no—
340#1763—731no—
341C4×#103—731no—
342#1753—731no—
343C73#2258111yescharacteristic, complemented, Sylow
344C23×Q8C2×D7311yescharacteristic
345C22×M16—731no—
346C2×C4×Q8C2×D7311yescharacteristic
347C2×C4×Q8C2×D7311yescharacteristic
348C2×#117—731no—
349C2×#107—731no—
350C2×#106—731no—
351#419—738no—
352C2×#103—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#103—731no—
356C2×#103—731no—
357C2×#103—731no—
358C2×#103—731no—
359C2×#103—731no—
360C2×C22⋊C8—731no—
361C2×#97—731no—
362C2×C4×C8—731no—
363#391—21no—
364#391—21no—
365#391C2×D7311yescharacteristic
366#391C2×D7311yescharacteristic
367#389—731no—
368#389—731no—
369#388—1461no—
370#388—1461no—
371#383—1461no—
372#380—21no—
373#380—731no—
374#380C2×D7311yescharacteristic
375C4×C4⋊C4—731no—
376C4×C4⋊C4—731no—
377C4×C4⋊C4—1461no—
378C4×C22⋊C4—21no—
379C4×C22⋊C4C2×D7311yescharacteristic
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340—731no—
386#336—731no—
387#336—731no—
388C22×Q8—21no—
389C22×Q8—22no—
390C22×Q8—22no—
391C22×Q8—22no—
392C22×Q8—41no—
393C22×Q8—41no—
394C22×Q8C4×D7314yes—
395C22×Q8C22×D7311yescharacteristic
396C22×Q8C22×D7311yescharacteristic
397C23×C4—21no—
398C23×C4C22×D7311yescharacteristic
399C2×M16—2921no—
400C2×M16—2921no—
401C2×M16—734no—
402C22×C8—731no—
403C22×C8—731no—
404C22×C8—731no—
405C22×C8—731no—
406C4×Q8—22no—
407C4×Q8—22no—
408C4×Q8—41no—
409C4×Q8—41no—
410#117—734no—
411#117—1462no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4C22×D7311yescharacteristic
421C2×C4⋊C4C22×D7311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4C22⋊D7311yescharacteristic
426C2×C22⋊C4C22⋊D7311yescharacteristic
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—1461no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42C22×D7311yescharacteristic
433C2×C42C22×D7311yescharacteristic
434#107—2921no—
435#106—2921no—
436#103—2921no—
437#103—2921no—
438#103—734no—
439#103—734no—
440#103—734no—
441#103—734no—
442#103—1462no—
443#103—1462no—
444C22⋊C8—1462no—
445#97—2921no—
446C4×C8—2921no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95C22⋊D7311yescharacteristic
458#95C22⋊D7311yescharacteristic
459C24C2×C22⋊D7311yescharacteristic
460C2×Q8—24no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—28no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8C2×C4×D7312yes—
476C2×Q8C2×C4×D7312yes—
477C2×Q8#1192402712yes—
478C2×Q8#1192402712yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4C2×C4×D7312yes—
495C22×C4C2×C22⋊D7311yescharacteristic
496C22×C4C2×C22⋊D7311yescharacteristic
497C22×C4C2×C22⋊D7311yescharacteristic
498C22×C4C2×C22⋊D7311yescharacteristic
499C22×C4C2×C22⋊D7311yescharacteristic
500C22×C4#1192402812yes—
501C22×C4C23×D7311yescharacteristic
502M16—2924no—
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—732no—
524C4⋊C4—732no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C22⋊C4—22no—
532C22⋊C4—41no—
533C22⋊C4—42no—
534C22⋊C4—5841no—
535C22⋊C4—1464no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—42no—
542C42—5841no—
543C42—5841no—
544C42—1464no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C4×C22⋊D7312yes—
552C23C22×C22⋊D7311yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#1455654714yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—1462no—
603C2×C4—1462no—
604C2×C4C4×C22⋊D7312yes—
605C2×C4#1455662212yes—
606C2×C4C22×C4×D7311yescharacteristic
607C2×C4C2×#1192402711yescharacteristic
608C2×C4C2×#1192402711yescharacteristic
609C2×C4C2×#1192402811yescharacteristic
610C2×C4#1455663611yescharacteristic
611C2×C4#1455663611yescharacteristic
612C8—2921no—
613C8—2921no—
614C8—1462no—
615C8—1462no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D7311yescharacteristic
624C22#1798923111yescharacteristic
625C22#1798923111yescharacteristic
626C22C2×#1455662211yescharacteristic
627C22#1798938511yescharacteristic
628C22#1798939511yescharacteristic
629C22#1798940211yescharacteristic
630C4—22no—
631C4—24no—
632C4—22no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—2922no—
640C4—2922no—
641C4—5841no—
642C4—5841no—
643C4C2×#1455654711yescharacteristic
644C4#1798906011yescharacteristic
645C4#1798926311yescharacteristic
646C4#1798926311yescharacteristic
647C2—42no—
648C2#2261277611yescharacteristic
649C2C4×#1455662211yescharacteristic
650C2#2261877411yescharacteristic
651C2#2262199711yescharacteristic
652C2#2262210211yescharacteristic
653C2#2262218411yescharacteristic
654C2#2262221411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingQ8×C22⋊C4×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenterQ8×C22⋊C4D73
hypocenterC73#22581
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #22610624C2semidirect
4. #22610610C2semidirect
5. #22610569C2semidirect
6. #22610562C2semidirect
7. #22610199C2semidirect
8. #22610185C2semidirect
12. #22609761C2semidirect
13. C4×#14556515C2semidirect
20. #17988596C4semidirect
343. C73#22581semidirect

Maximal subgroups

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

subgroupindex
C2×#179885962
#226128222
#226106242
#226106102
#226105692
#226105622
#226101992
#226101852
#226100232
#226099622
#226098662
#226097612
C4×#145565152
#226097152
Q8×C22⋊C4×C732
#2258173