Finite Group Databasebeta

← #33057211

Subgroups — #33057211

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619982C211yesmaximal, characteristic
2#22619090C211yesmaximal, characteristic, complemented
3#22618963C211yesmaximal, characteristic, complemented
4#22608843C211yesmaximal, characteristic, complemented
5#22608834C211yesmaximal, characteristic, complemented
6#22608770C211yesmaximal, characteristic, complemented
7#22608763C211yesmaximal, characteristic, complemented
8#22608762C211yesmaximal, characteristic, complemented
9#22608754C211yesmaximal, characteristic, complemented
10#22608741C211yesmaximal, characteristic, complemented
11#22608713C211yesmaximal, characteristic, complemented
12#22608390C211yesmaximal, characteristic, complemented
13D4⋊C8×D73C211yesmaximal, characteristic, complemented
14#22608222C211yesmaximal, characteristic, complemented
15C73×#1583C211yesmaximal, characteristic, complemented
16#17989124C2211yescharacteristic
17C4×D4×D73C2211yescharacteristic
18#17988921C411yescharacteristic
19#17988907C411yescharacteristic
20#17988886C411yescharacteristic
21#17988885C411yescharacteristic
22#17988805C411yescharacteristic
23#17988804C411yescharacteristic
24#17988736C411yescharacteristic
25#17988722C411yescharacteristic
26#17988718C2211yescharacteristic, complemented
27C4×#11924015C2211yescharacteristic
28C4×D584C2211yescharacteristic
29C4×#11924013C2211yescharacteristic
30#17988666C2211yescharacteristic, complemented
31#17988665C2211yescharacteristic
32#17988661—21no—
33#17988657—21no—
34#17988656—21no—
35C8×D292—21no—
36C8×D292C2211yescharacteristic
37#17988652C2211yescharacteristic
38#17988652C2211yescharacteristic, complemented
39#17988646—21no—
40#17988643C2211yescharacteristic
41#17988638C2211yescharacteristic
42#17988638C2211yescharacteristic
43#17988612—21no—
44C4⋊C8×D73C2211yescharacteristic
45#17988550—21no—
46C22⋊C8×D73—21no—
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic, complemented
49#17988339C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51Q8⋊#10596831C2211yescharacteristic, complemented
52#17988316C2211yescharacteristic
53#17988316C2211yescharacteristic, complemented
54D4⋊#10596831C2211yescharacteristic
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988309C2211yescharacteristic, complemented
59#17988308C2211yescharacteristic, complemented
60C8×Q8×C73C2211yescharacteristic, complemented
61C4×SD16×C73C2211yescharacteristic
62#436×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic, complemented
64Q8⋊C8×C73C2211yescharacteristic, complemented
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic, complemented
67C2×D4×D73C2×C411yescharacteristic
68C22×C4×D73—21no—
69C2×#11924015C2×C411yescharacteristic
70C2×D584C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72C2×#11924012—21no—
73C2×#11924011—21no—
74C2×C8×D73—21no—
75C2×C8×D73—21no—
76C2×C8×D73—21no—
77C2×C8×D73—21no—
78C2×C8×D73—21no—
79#14556586—21no—
80#14556582C2×C411yescharacteristic
81#14556580C2×C411yescharacteristic
82#14556576—21no—
83#14556567C2×C411yescharacteristic
84#14556566C2×C411yescharacteristic
85C4×D292—21no—
86C4×D292C2311yescharacteristic
87C4×D292C2311yescharacteristic
88D4×Dic73C2×C411yescharacteristic
89#14556553C2×C411yescharacteristic
90C4×C22⋊D73—21no—
91#14556551—21no—
92#14556548—21no—
93C4⋊C4×D73C2×C411yescharacteristic
94C22⋊C4×D73—21no—
95C42×D73C2311yescharacteristic
96#14556525C2×C411yescharacteristic
97#14556524C2×C411yescharacteristic
98#14556522C2×C411yescharacteristic
99C4⋊#10596831—21no—
100C4⋊#10596831C2311yescharacteristic
101C4⋊#10596831C2311yescharacteristic
102#14556520—21no—
103#14556520C2×C411yescharacteristic
104#14556520C2×C411yescharacteristic
105#14556520D411yescharacteristic
106#14556520D411yescharacteristic
107#14556515C2×C411yescharacteristic
108#14556513C2×C411yescharacteristic
109#14556513C2×C411yescharacteristic
110#14556512C2×C411yescharacteristic
111#14556511C2×C411yescharacteristic
112#14556511C2×C411yescharacteristic
113C22⋊#10596831—21no—
114#14556503—21no—
115#14556503—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503C2×C411yescharacteristic
120#14556503C2×C411yescharacteristic
121#14556503D411yescharacteristic
122#14556503D411yescharacteristic
123#14556502—21no—
124#14556501—21no—
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C8×Dic73—21no—
129C2×SD16×C73C2×C411yescharacteristic
130C2×M16×C73—21no—
131C4×Q8×C73C2311yescharacteristic
132C4×D4×C73C2311yescharacteristic
133#106×C73C2×C411yescharacteristic
134C4⋊C8×C73—21no—
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#103×C73C2×C411yescharacteristic
138#102×C73C2×C411yescharacteristic
139C22⋊C8×C73—21no—
140C4×C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C23×D73—21no—
145C2×D292—21no—
146C2×D292C22×C411yescharacteristic
147C2×D292C22×C411yescharacteristic
148C2×C22⋊D73—21no—
149D4×D73—22no—
150D4×D73—41no—
151C22×Dic73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—81no—
157C2×C4×D73—81no—
158C2×C4×D73C22×C411yescharacteristic
159C2×C4×D73C2×D411yescharacteristic
160C2×C4×D73#4111yescharacteristic
161#11924015—22no—
162D584—22no—
163#11924013—41no—
164#11924012—81no—
165#11924011—81no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C2×D411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C8×D73—42no—
174C8×D73—81no—
175C8×D73—81no—
176C8×D73—81no—
177C8×D73—81no—
178C4⋊Dic73—21no—
179C4⋊Dic73C22×C411yescharacteristic
180C4⋊Dic73C22×C411yescharacteristic
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184C22⋊Dic73—21no—
185#11924005—21no—
186#11924005—21no—
187#11924005—21no—
188#11924005—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192C4×Dic73C22×C411yescharacteristic
193C4×Dic73C2×D411yescharacteristic
194C4×Dic73#4111yescharacteristic
195C2×Q8×C73C22×C411yescharacteristic
196C2×D4×C73C22×C411yescharacteristic
197C22×C4×C73—21no—
198SD16×C73—41no—
199M16×C73—81no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73C22×C411yescharacteristic
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C2×D411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C4⋊C4×C73—21no—
208C4⋊C4×C73C22×C411yescharacteristic
209C4⋊C4×C73C22×C411yescharacteristic
210C22⋊C4×C73—21no—
211C42×C73—21no—
212C42×C73C2411yescharacteristic
213C23×C73—21no—
214C22×D73—21no—
215C22×D73—21no—
216C22×D73—42no—
217C22×D73—42no—
218C22×D73—81no—
219C22×D73C4×D411yescharacteristic
220Q8×C73—21no—
221Q8×C73—21no—
222D4×C73—21no—
223D4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73C23×C411yescharacteristic
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C22⋊D73—42no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—81no—
236C2×Dic73C4×D411yescharacteristic
237C2×Dic73C4×D411yescharacteristic
238C2×Dic73C4×D411yescharacteristic
239D292—21no—
240D292—21no—
241D292—21no—
242D292—42no—
243D292#13112yes—
244C4×D73—22no—
245C4×D73—21no—
246C4×D73—21no—
247C4×D73—21no—
248C4×D73—21no—
249C4×D73—41no—
250C4×D73—41no—
251C4×D73—41no—
252C4×D73—41no—
253C4×D73—81no—
254C4×D73—81no—
255C8×C73—21no—
256C8×C73—21no—
257C8×C73—21no—
258C8×C73—21no—
259C8×C73—41no—
260C8×C73—41no—
261C8×C73—41no—
262#10596831—21no—
263#10596831—21no—
264#10596831—21no—
265#10596831—21no—
266#10596831—41no—
267#10596831—41no—
268#10596831—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73C2×C4×D411yescharacteristic
273C2×D73—21no—
274C2×D73—21no—
275C2×D73—22no—
276C2×D73—41no—
277C2×D73—41no—
278C2×D73—41no—
279C2×D73—41no—
280C2×D73—41no—
281C2×D73—84no—
282C2×D73#44412yes—
283C4×C73—21no—
284C4×C73—41no—
285C4×C73—41no—
286C4×C73—41no—
287C4×C73C2×#13111yescharacteristic
288C4×C73#56811yescharacteristic
289C4×C73#57611yescharacteristic
290C4×C73#57711yescharacteristic
291Dic73—21no—
292Dic73—21no—
293Dic73—21no—
294Dic73—41no—
295Dic73—41no—
296#18020—731nomaximal, Sylow
297C2×C73—81no—
298C2×C73#293311yescharacteristic
299C2×C73#294611yescharacteristic
300C2×C73C2×#44411yescharacteristic
301D73—22no—
302D73—42no—
303D73—81no—
304#3105—731no—
305#2994—731no—
306#2974—731no—
307#1594—731no—
308#1592—731no—
309#1583D7311yescharacteristic, complemented
310#1582—731no—
311#1581—731no—
312#1581—731no—
313#1580—731no—
314SD16⋊C8—731no—
315#1573—731no—
316#1494—731no—
317C2×D4⋊C8—731no—
318#1452—731no—
319C73#1802011yescharacteristic, complemented, Sylow
320#529—731no—
321C2×C4×D4—731no—
322#484—731no—
323#481—731no—
324#478—731no—
325#478—731no—
326#463—731no—
327#463—731no—
328#450—731no—
329#447—731no—
330#446—731no—
331C8×Q8C2×D7311yescharacteristic, complemented
332C4×SD16—731no—
333C4×SD16C2×D7311yescharacteristic
334C4×D8—731no—
335C4×D8—731no—
336#436—731no—
337#436—731no—
338#436C2×D7311yescharacteristic
339#435—1461no—
340#435—1461no—
341#435—1461no—
342C8×D4—731no—
343C8×D4—731no—
344C8×D4—731no—
345C8×D4—1461no—
346#433—1461no—
347#432—731no—
348#431—731no—
349#431—731no—
350#424—1461no—
351C2×C4⋊C8—731no—
352#407—1461no—
353C2×C22⋊C8—1461no—
354C2×C4×C8—731no—
355#335—731no—
356#335—731no—
357#334—731no—
358#334C2×D7311yescharacteristic, complemented
359Q8⋊C8—731no—
360Q8⋊C8C2×D7311yescharacteristic, complemented
361D4⋊C8—731no—
362D4⋊C8—731no—
363D4⋊C8—731no—
364D4⋊C8—731no—
365D4⋊C8—731no—
366D4⋊C8C2×D7311yescharacteristic
367#322—731no—
368#322—731no—
369#322—731no—
370#322C2×D7311yescharacteristic, complemented
371C22×D4—731no—
372C23×C4—1461no—
373C2×SD16—731no—
374C2×SD16C4×D7311yescharacteristic
375C2×D8—731no—
376C2×D8—731no—
377C2×M16—21no—
378C2×M16—1461no—
379C2×M16—1461no—
380C22×C8—1461no—
381C22×C8—1461no—
382C22×C8—1461no—
383C22×C8—1461no—
384C22×C8—1461no—
385#126—1461no—
386#125—731no—
387#124—731no—
388#123—1461no—
389C22⋊Q8—731no—
390#121—731no—
391C4×Q8C22×D7311yescharacteristic
392C4×D4—731no—
393C4×D4—731no—
394C4×D4—731no—
395C4×D4—731no—
396C4×D4—1461no—
397C4×D4—1461no—
398C4×D4—1461no—
399C4×D4C22×D7311yescharacteristic
400#117—1461no—
401C2×C4⋊C4—731no—
402C2×C22⋊C4—1461no—
403C2×C42—731no—
404#107—731no—
405#107—731no—
406#106—731no—
407#106C4×D7311yescharacteristic
408C4⋊C8—21no—
409C4⋊C8—731no—
410C4⋊C8—731no—
411C4⋊C8—731no—
412C4⋊C8—731no—
413C4⋊C8—731no—
414C4⋊C8—731no—
415C4⋊C8—1461no—
416C4⋊C8—1461no—
417C4⋊C8C22×D7311yescharacteristic
418C4⋊C8C22×D7311yescharacteristic
419#103—731no—
420#103C4×D7311yescharacteristic
421#102—731no—
422#102—731no—
423#102—731no—
424#102—731no—
425#102—731no—
426#102C4×D7311yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—731no—
429C22⋊C8—731no—
430C22⋊C8—731no—
431C22⋊C8—731no—
432C22⋊C8—1461no—
433C22⋊C8—1461no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8—1461no—
437C22⋊C8—1461no—
438#97—1461no—
439#97—1461no—
440C4×C8—21no—
441C4×C8—731no—
442C4×C8—731no—
443C4×C8—731no—
444C4×C8—1461no—
445C4×C8C22×D7311yescharacteristic
446C4×C8C22×D7311yescharacteristic
447C4×C8C22×D7311yescharacteristic
448C24—1461no—
449C2×Q8C2×C4×D7311yescharacteristic
450C2×D4—2921no—
451C2×D4—731no—
452C2×D4—731no—
453C2×D4—1461no—
454C2×D4—1461no—
455C2×D4—1462no—
456C2×D4C2×C4×D7311yescharacteristic
457C22×C4—21no—
458C22×C4—5841no—
459C22×C4—5841no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468SD16—41no—
469SD16—1462no—
470D8—2921no—
471D8—1462no—
472M16—81no—
473M16—5841no—
474M16—5841no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—2922no—
479C2×C8—5841no—
480C2×C8—5841no—
481C2×C8—5841no—
482C2×C8—5841no—
483C2×C8—731no—
484C2×C8—731no—
485C2×C8—731no—
486C2×C8—731no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8C2×C4×D7311yescharacteristic
491C2×C8C2×C4×D7311yescharacteristic
492C2×C8D4×D7311yescharacteristic
493C2×C8D4×D7311yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—731no—
496C4⋊C4—731no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4C2×C4×D7311yescharacteristic
502C4⋊C4C2×C4×D7311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1461no—
505C22⋊C4—1461no—
506C22⋊C4—1461no—
507C22⋊C4—1461no—
508C22⋊C4—1461no—
509C22⋊C4—1461no—
510C22⋊C4—1461no—
511C22⋊C4—1461no—
512C42—21no—
513C42—731no—
514C42—731no—
515C42—731no—
516C42C23×D7311yescharacteristic
517C23—21no—
518C23—2922no—
519C23—2922no—
520C23—5841no—
521C23—731no—
522C23—1461no—
523C23—1461no—
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—2922no—
529D4—2922no—
530D4—732no—
531D4—1461no—
532D4—1461no—
533D4—1461no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—5841no—
544C2×C4—5841no—
545C2×C4—5841no—
546C2×C4—731no—
547C2×C4—731no—
548C2×C4—731no—
549C2×C4—1462no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4C22×C4×D7311yescharacteristic
557C2×C4C2×D4×D7311yescharacteristic
558C2×C4#41×D7311yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—2921no—
567C8—2921no—
568C8—2921no—
569C8—1461no—
570C8—1461no—
571C8—1461no—
572C8—1461no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—2921no—
577C22—2921no—
578C22—2921no—
579C22—2921no—
580C22—2921no—
581C22—5844no—
582C22—732no—
583C22—1462no—
584C22—1461no—
585C22—1461no—
586C22C4×D4×D7311yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—2921no—
592C4—2921no—
593C4—1461no—
594C4—1461no—
595C4—1461no—
596C4#131×D7311yescharacteristic
597C4#1798935311yescharacteristic
598C4#1798938911yescharacteristic
599C4#1798939311yescharacteristic
600C2—81no—
601C2—2922no—
602C2—5841no—
603C2—1462no—
604C2#2261865411yescharacteristic
605C2#2261875811yescharacteristic
606C2#444×D7311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1583C2
socleC22×C73C2×C4×D4
prime radical#17988722C4
cosocleC42×C73C24
hypercenter#1583D73
hypocenterC73#18020
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
2. #22619090C2semidirect
3. #22618963C2semidirect
4. #22608843C2semidirect
5. #22608834C2semidirect
6. #22608770C2semidirect
7. #22608763C2semidirect
8. #22608762C2semidirect
9. #22608754C2semidirect
10. #22608741C2semidirect
11. #22608713C2semidirect
12. #22608390C2semidirect
13. D4⋊C8×D73C2semidirect
14. #22608222C2semidirect
15. C73×#1583C2semidirect
26. #17988718C22semidirect
30. #17988666C22semidirect
38. #17988652C22semidirect
48. #17988340C22semidirect
49. #17988339C22semidirect
50. #17988337C22semidirect
51. Q8⋊#10596831C22semidirect
53. #17988316C22semidirect
56. #17988314C22semidirect
57. #17988309C22semidirect
58. #17988309C22semidirect
59. #17988308C22semidirect
60. C8×Q8×C73C22semidirect
63. #334×C73C22semidirect
64. Q8⋊C8×C73C22semidirect
66. #322×C73C22semidirect
309. #1583D73semidirect
319. C73#18020semidirect
331. C8×Q8C2×D73semidirect
358. #334C2×D73semidirect
360. Q8⋊C8C2×D73semidirect
370. #322C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199822
#226190902
#226189632
#226088432
#226088342
#226087702
#226087632
#226087622
#226087542
#226087412
#226087132
#226083902
D4⋊C8×D732
#226082222
C73×#15832
#1802073