Finite Group Databasebeta

← #34018668

Subgroups — #34018668

Order 18688.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626876C211yesmaximal, characteristic
2#22626824C211yesmaximal, characteristic, complemented
3#22625558C211yesmaximal, characteristic, complemented
4#22625556C211yesmaximal, characteristic, complemented
5#22625544C211yesmaximal, characteristic, complemented
6#22625538C211yesmaximal, characteristic, complemented
7#22625524C211yesmaximal, characteristic, complemented
8#22625521C211yesmaximal, characteristic, complemented
9Q16×#10596833C211yesmaximal, characteristic, complemented
10#22625515C211yesmaximal, characteristic, complemented
11#22625507C211yesmaximal, characteristic, complemented
12#22625259C211yesmaximal, characteristic, complemented
13#22625244C211yesmaximal, characteristic
14#22625152C211yesmaximal, characteristic, complemented
15#22620012C211yesmaximal, characteristic
16#17989638C2211yescharacteristic
17Q8×#10596833—21no—
18Q8×#10596833—21no—
19Q8×#10596833C2211yescharacteristic
20Q8×#10596833C2211yescharacteristic
21Q8×#10596833C2211yescharacteristic
22Q8×#10596833C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24#17989621C2211yescharacteristic
25#17989620—21no—
26#17989619—21no—
27#17989615C2211yescharacteristic
28#17989612—21no—
29#17989604C2211yescharacteristic
30#17989603—21no—
31#17989603—21no—
32#17989603C2211yescharacteristic
33#17989603C2211yescharacteristic
34#17989594—21no—
35#17989590C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40Q8⋊#10596833C2211yescharacteristic
41Q8⋊#10596833C2211yescharacteristic
42Q8⋊#10596833C2211yescharacteristic
43Q8⋊#10596833C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46#17989435C2211yescharacteristic
47#17989435C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989428C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#17989123C2211yescharacteristic
53C2×Q8×Dic73C2211yescharacteristic
54#17988915C411yescharacteristic
55#17988901C411yescharacteristic
56#17988896C411yescharacteristic
57#17988896C411yescharacteristic
58#17988811C411yescharacteristic
59#17988811C411yescharacteristic
60#17988726C411yescharacteristic
61Q16×Dic73C2211yescharacteristic
62Q16×Dic73C2211yescharacteristic
63#17988672C2211yescharacteristic
64#17988672C2211yescharacteristic
65#17988637C2211yescharacteristic
66#451×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924037—21no—
69C2×#11924036—21no—
70#14556658—21no—
71#14556658—21no—
72#14556658—21no—
73C4⋊#10596833—21no—
74C4⋊#10596833—21no—
75C4⋊#10596833—21no—
76C4⋊#10596833—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833D411yescharacteristic
79C4⋊#10596833D411yescharacteristic
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88C22⋊#10596833D411yescharacteristic
89C22⋊#10596833D411yescharacteristic
90#14556646—21no—
91C4×#10596833—21no—
92C4×#10596833—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×#11924015C2×C411yescharacteristic
102C2×#11924015C2×C411yescharacteristic
103#14556586—21no—
104#14556582C2×C411yescharacteristic
105#14556580C2×C411yescharacteristic
106#14556575—21no—
107#14556565C2×C411yescharacteristic
108Q8×Dic73—21no—
109Q8×Dic73—21no—
110Q8×Dic73C2×C411yescharacteristic
111Q8×Dic73C2×C411yescharacteristic
112Q8×Dic73C2311yescharacteristic
113Q8×Dic73C2311yescharacteristic
114Q8×Dic73C2311yescharacteristic
115#14556553C2311yescharacteristic
116#14556551—21no—
117#14556549—21no—
118C2×C4⋊Dic73—21no—
119C2×C4⋊Dic73C2×C411yescharacteristic
120C2×C4×Dic73—21no—
121C2×C4×Dic73C2311yescharacteristic
122#14556525C2×C411yescharacteristic
123#14556525C2×C411yescharacteristic
124#14556522C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556515C2×C411yescharacteristic
129#14556515C2×C411yescharacteristic
130#14556515C2×C411yescharacteristic
131#14556515C2×C411yescharacteristic
132#14556513C2×C411yescharacteristic
133#14556513C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C22×Q8×C73C2×C411yescharacteristic
138C2×Q16×C73C2×C411yescharacteristic
139C2×Q16×C73C2×C411yescharacteristic
140C22⋊Q8×C73C2×C411yescharacteristic
141#103×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924038—81no—
145#11924037—81no—
146#11924036—81no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833C2×D411yescharacteristic
154C2×#10596833C2×D411yescharacteristic
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×D292C22×C411yescharacteristic
158C2×C22⋊D73—21no—
159C22×Dic73—21no—
160C22×Dic73C2×D411yescharacteristic
161C22×Dic73#4111yescharacteristic
162C2×C4×D73—21no—
163#11924015—22no—
164#11924015—22no—
165C2×#10596831C22×C411yescharacteristic
166C2×#10596831C22×C411yescharacteristic
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73C22×C411yescharacteristic
175C4⋊Dic73C22×C411yescharacteristic
176C4⋊Dic73C22×C411yescharacteristic
177#11924007—21no—
178#11924005—21no—
179#11924005—21no—
180#11924005—21no—
181#11924005—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73C22×C411yescharacteristic
188C4×Dic73C2×D411yescharacteristic
189C4×Dic73#4111yescharacteristic
190C4×Dic73C2411yescharacteristic
191C2×Q8×C73—22no—
192C2×Q8×C73—22no—
193C2×Q8×C73—21no—
194C2×Q8×C73C22×C411yescharacteristic
195C2×Q8×C73C22×C411yescharacteristic
196C2×Q8×C73C22×C411yescharacteristic
197C22×C4×C73—21no—
198C22×C4×C73C22×C411yescharacteristic
199Q16×C73—22no—
200Q16×C73—22no—
201C2×C8×C73C22×C411yescharacteristic
202C2×C8×C73C22×C411yescharacteristic
203C4⋊C4×C73—21no—
204C4⋊C4×C73C22×C411yescharacteristic
205C22⋊C4×C73—21no—
206C23×C73C4×D411yescharacteristic
207C22×D73—21no—
208Q8×C73—24no—
209Q8×C73—21no—
210Q8×C73—21no—
211Q8×C73—21no—
212Q8×C73—21no—
213Q8×C73#13112yes—
214Q8×C73#13112yes—
215C2×C4×C73—24no—
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×C73C4×D411yescharacteristic
223C2×C4×C73C23×C411yescharacteristic
224C22⋊D73—81no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—41no—
231C2×Dic73—81no—
232C2×Dic73C4×D411yescharacteristic
233C2×Dic73C4×D411yescharacteristic
234C2×Dic73C22×D411yescharacteristic
235C2×Dic73C2×#4111yescharacteristic
236D292—21no—
237D292—21no—
238C4×D73—41no—
239C4×D73—41no—
240#10596833—21no—
241#10596833—21no—
242#10596833—21no—
243#10596833—21no—
244#10596833—41no—
245#10596833—41no—
246#10596833—41no—
247#10596833—41no—
248#10596833—41no—
249#10596833—41no—
250C8×C73—21no—
251C8×C73—21no—
252#10596831—21no—
253#10596831—21no—
254C22×C73—21no—
255C22×C73—21no—
256C22×C73—21no—
257C22×C73C2×C4×D411yescharacteristic
258C2×D73—41no—
259C2×D73—41no—
260C2×D73—41no—
261C4×C73—22no—
262C4×C73—22no—
263C4×C73—21no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73C2×#13111yescharacteristic
267C4×C73C2×#13111yescharacteristic
268Dic73—21no—
269Dic73—21no—
270Dic73—21no—
271Dic73—41no—
272Dic73—41no—
273Dic73—41no—
274Dic73#57711yescharacteristic
275Dic73#57811yescharacteristic
276#17919—731nomaximal, Sylow
277C2×C73—41no—
278C2×C73#292911yescharacteristic
279C2×C73#294711yescharacteristic
280C2×C73#295711yescharacteristic
281D73—81no—
282#3108—731no—
283#2993—731no—
284#2982—731no—
285#1588—731no—
286#1587—731no—
287#1584—731no—
288#1583—731no—
289#1579—731no—
290SD16⋊C8—731no—
291C8×Q16—731no—
292C8×SD16—731no—
293#1574—731no—
294#1487—731no—
295#1482—731no—
296#1454—731no—
297C73#1791911yescharacteristic, complemented, Sylow
298#529—731no—
299C2×C4×Q8—731no—
300#483—731no—
301#480—731no—
302#479—731no—
303#479—731no—
304#464—731no—
305#464—731no—
306#451#6440811yescharacteristic
307#448—731no—
308#446—731no—
309C8×Q8—731no—
310C8×Q8—731no—
311C8×Q8—731no—
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—1461no—
315C8×Q8—1461no—
316C4×Q16—731no—
317C4×Q16—731no—
318C4×SD16—731no—
319C4×SD16—731no—
320#436—731no—
321#436—1461no—
322#435—1461no—
323C8×D4—731no—
324#433—1461no—
325#431—731no—
326#431—731no—
327#431—731no—
328#431—731no—
329#431—1461no—
330#431—1461no—
331#424—1461no—
332#423—731no—
333#335—731no—
334#335—731no—
335#334—731no—
336#334—731no—
337Q8⋊C8—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343D4⋊C8—731no—
344D4⋊C8—731no—
345#322—731no—
346#322—731no—
347#322—731no—
348C82—731no—
349C22×Q8C2×#6440811yescharacteristic
350C2×Q16C2×#6440811yescharacteristic
351C2×Q16C2×#6440811yescharacteristic
352C2×SD16—731no—
353C2×SD16—731no—
354C2×M16—1461no—
355C2×M16—1461no—
356C22×C8—1461no—
357#126—1461no—
358#125—731no—
359#124—731no—
360#123—1461no—
361C22⋊Q8C2×#6440811yescharacteristic
362#121—731no—
363C4×Q8—731no—
364C4×Q8—731no—
365C4×Q8—731no—
366C4×Q8—731no—
367C4×Q8—731no—
368C4×Q8—1461no—
369C4×Q8—1461no—
370C4×D4—731no—
371C4×D4—1461no—
372#117—1461no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—1461no—
375C2×C42—731no—
376C2×C42—1461no—
377#107—731no—
378#107—731no—
379#106—731no—
380#106—731no—
381C4⋊C8—731no—
382C4⋊C8—731no—
383C4⋊C8—731no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—731no—
388C4⋊C8—731no—
389C4⋊C8—1461no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396C4⋊C8—1461no—
397#103—731no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103C2×#6440811yescharacteristic
402#103C2×#6440811yescharacteristic
403#102—731no—
404#102—731no—
405C22⋊C8—731no—
406C22⋊C8—731no—
407C22⋊C8—731no—
408C22⋊C8—1461no—
409C22⋊C8—1461no—
410C22⋊C8—1461no—
411C22⋊C8—1461no—
412C22⋊C8C2×#6440811yescharacteristic
413#97—1461no—
414C4×C8—731no—
415C4×C8—731no—
416C4×C8—731no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—1461no—
421C4×C8—1461no—
422C4×C8—1461no—
423C4×C8—1461no—
424C4×C8—1461no—
425C4×C8—1461no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8C22×#6440811yescharacteristic
430C2×Q8C22×#6440811yescharacteristic
431C2×Q8C22×#6440811yescharacteristic
432C2×D4—731no—
433C2×D4—1461no—
434C22×C4—21no—
435C22×C4—731no—
436C22×C4—731no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4C22×#6440811yescharacteristic
440Q16—22no—
441Q16—22no—
442SD16—1462no—
443SD16—1462no—
444M16—5841no—
445M16—5841no—
446C2×C8—5841no—
447C2×C8—731no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8C22×#6440811yescharacteristic
460C2×C8C22×#6440811yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—731no—
463C4⋊C4—731no—
464C4⋊C4—731no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4C22×#6440811yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1461no—
476C22⋊C4—1461no—
477C22⋊C4—1461no—
478C22⋊C4—1461no—
479C42—731no—
480C42—731no—
481C42—731no—
482C42—731no—
483C42—1461no—
484C42—1461no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C23—1461no—
489C23D4×#6440811yescharacteristic
490Q8—24no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8#1455667812yes—
496Q8#1455667812yes—
497D4—5841no—
498D4—1461no—
499D4—1461no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—2921no—
508C2×C4—2921no—
509C2×C4—2921no—
510C2×C4—5841no—
511C2×C4—731no—
512C2×C4—731no—
513C2×C4—731no—
514C2×C4—731no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4D4×#6440811yescharacteristic
521C2×C4C23×#6440811yescharacteristic
522C8—21no—
523C8—21no—
524C8—2921no—
525C8—2921no—
526C8—2921no—
527C8—2921no—
528C8—2921no—
529C8—2921no—
530C8—1461no—
531C8—1461no—
532C8—1461no—
533C8—1461no—
534C8—1461no—
535C8—1461no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—2921no—
540C22—2921no—
541C22—2921no—
542C22C2×D4×#6440811yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—2921no—
549C4—2921no—
550C4—2921no—
551C4—731no—
552C4—731no—
553C4—1461no—
554C4—1461no—
555C4—1461no—
556C4C2×#1455667811yescharacteristic
557C4C2×#1455667811yescharacteristic
558C2—41no—
559C2—5841no—
560C2#2262664611yescharacteristic
561C2#2262668911yescharacteristic
562C2#2262671111yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#451×C73C4
socleC22×C73C2×C4×D4
prime radical#14556565C2×C4
cosocleC4×Dic73C24
hypercenter#451#64408
hypocenterC73#17919
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #22626824C2semidirect
3. #22625558C2semidirect
4. #22625556C2semidirect
5. #22625544C2semidirect
6. #22625538C2semidirect
7. #22625524C2semidirect
8. #22625521C2semidirect
9. Q16×#10596833C2semidirect
10. #22625515C2semidirect
11. #22625507C2semidirect
12. #22625259C2semidirect
14. #22625152C2semidirect
297. C73#17919semidirect

Maximal subgroups

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

subgroupindex
#226268762
#226268242
#226255582
#226255562
#226255442
#226255382
#226255242
#226255212
Q16×#105968332
#226255152
#226255072
#226252592
#226252442
#226251522
#226200122
#1791973