Finite Group Databasebeta

← #34030262

Subgroups — #34030262

Order 18688.

696 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626693C211yesmaximal, characteristic, complemented
2#22626690C211yesmaximal, characteristic, complemented
3#22626329C212yesmaximal, complemented
4#22626327C212yesmaximal, complemented
5SD32×#64408C212yesmaximal, complemented
6D16×#64408C212yesmaximal, complemented
7#22626283C211yesmaximal, characteristic, complemented
8#22626238C211yesmaximal, characteristic, complemented
9C2×#17989499C211yesmaximal, characteristic, complemented
10#22626143C211yesmaximal, characteristic, complemented
11#509×D73C211yesmaximal, characteristic
12#17989656—21no—
13#17989654—21no—
14C2×#14556664C2211yescharacteristic
15#17989628—22no—
16#17989626—22no—
17Q16×#64408C2212yescomplemented
18D8×#64408C2212yes—
19D8×#64408C2212yes—
20D8×#64408C2212yescomplemented
21#17989597C2211yescharacteristic
22C2×#14556660C2211yescharacteristic
23#17989579—21no—
24#17989578—21no—
25#17989550C2211yescharacteristic
26#17989514C2212yescomplemented
27#17989513C2212yescomplemented
28#17989500C2211yescharacteristic, complemented
29#17989500C2211yescharacteristic, complemented
30#17989499C2212yes—
31#17989499C2212yes—
32#17989499C2211yescharacteristic, complemented
33#17989499C2211yescharacteristic, complemented
34#17989474C2212yescomplemented
35C16×#64408C2212yescomplemented
36#135×D73C2211yescharacteristic
37C2×D8×D73C2211yescharacteristic
38#17989025C411yescharacteristic, complemented
39#17989024C412yescomplemented
40#17989023C411yescharacteristic, complemented
41#17989022C411yescharacteristic, complemented
42#17989021C412yescomplemented
43SD32×D73C2212yes—
44D16×D73C2212yes—
45M32×D73C2211yescharacteristic
46#509×C73C411yescharacteristic, complemented
47D73⋊M16D411yescharacteristic
48Q8×#64408—22no—
49D4×#64408—22no—
50D4×#64408—22no—
51D4×#64408—22no—
52D4×#64408—41no—
53D4×#64408—41no—
54D4×#64408—42no—
55#14556670—21no—
56#14556670—41no—
57#14556670D411yescharacteristic
58C2×C4⋊#64408—21no—
59C2×C4⋊#64408—41no—
60C2×C22⋊#64408—42no—
61#14556664—21no—
62#14556664C2311yescharacteristic
63#14556664C2311yescharacteristic
64C2×#11924031—21no—
65#14556660—21no—
66#14556660C2312yes—
67#14556660C2311yescharacteristic
68#14556660C2311yescharacteristic
69#14556659—21no—
70#14556654—22no—
71#14556653—22no—
72#14556653—22no—
73#14556653—22no—
74#14556645D412yes—
75C8×#64408C2312yes—
76#41×D73—21no—
77C2×D4×D73—21no—
78#14556614C2×C411yescharacteristic, complemented
79#14556613C2×C411yescharacteristic, complemented
80#14556612—21no—
81#14556611—21no—
82#14556610C2×C411yescharacteristic, complemented
83Q16×D73C2311yescharacteristic
84SD16×D73—21no—
85C2×D584C2×C411yescharacteristic, complemented
86C2×#11924013—21no—
87D8×D73—21no—
88D8×D73—21no—
89D8×D73C2×C412yescomplemented
90D8×D73C2312yes—
91D8×D73C2311yescharacteristic
92C2×C8×D73C2311yescharacteristic
93#14556536C2×C412yescomplemented
94#14556535C2×C412yescomplemented
95#14556534C2×C412yescomplemented
96D1168C2×C412yescomplemented
97#14556532C2×C412yescomplemented
98#14556532C2×C412yescomplemented
99#14556531C2×C411yescharacteristic, complemented
100#14556530C2×C412yescomplemented
101C16×D73C2312yes—
102#135×C73C2×C411yescharacteristic, complemented
103C2×D8×C73C2×C411yescharacteristic, complemented
104SD32×C73C2×C412yescomplemented
105D16×C73C2×C412yescomplemented
106M32×C73C2×C411yescharacteristic, complemented
107C22×#64408—41no—
108C22×#64408—42no—
109C22×#64408—42no—
110C22×#64408—42no—
111#11924038—21no—
112#11924037—21no—
113#11924036C2×D412yes—
114C4⋊#64408—22no—
115C4⋊#64408—21no—
116C4⋊#64408—21no—
117C4⋊#64408—42no—
118C4⋊#64408—41no—
119C4⋊#64408D811yescharacteristic
120C4⋊#64408D811yescharacteristic
121C22⋊#64408—41no—
122C22⋊#64408—42no—
123C22⋊#64408—42no—
124C22⋊#64408—42no—
125C22⋊#64408D812yes—
126C4×#64408—42no—
127C4×#64408—41no—
128C4×#64408C2×D412yes—
129#11924031—21no—
130#11924031—21no—
131C23×D73—41no—
132#11924029—21no—
133#11924028—21no—
134#11924028—41no—
135#11924027—21no—
136#11924027—41no—
137Q8×D73—21no—
138C2×D292—21no—
139C2×C22⋊D73—41no—
140D4×D73—22no—
141D4×D73—22no—
142D4×D73—21no—
143D4×D73—41no—
144D4×D73—41no—
145D4×D73—41no—
146C2×C4×D73—41no—
147C2×C4×D73C2×D411yescharacteristic
148Dic292C22×C411yescharacteristic
149#11924018—21no—
150#11924017—21no—
151#11924016—21no—
152#11924015—21no—
153D584—21no—
154D584C22×C411yescharacteristic
155D584C22×C412yes—
156#11924013—22no—
157#11924013—22no—
158#11924013—21no—
159C2×#10596831#4111yescharacteristic
160C8×D73C22×C411yescharacteristic
161C8×D73C2×D411yescharacteristic
162C8×D73#4111yescharacteristic
163C8×D73C2411yescharacteristic
164#11924003C22×C412yes—
165#41×C73—21no—
166C2×D4×C73—21no—
167Q16×C73C22×C411yescharacteristic
168SD16×C73—21no—
169D8×C73—21no—
170D8×C73C22×C411yescharacteristic
171D8×C73C22×C412yes—
172C2×C8×C73C22×C411yescharacteristic
173C16×C73C22×C412yes—
174C23×C73—41no—
175C22×D73—41no—
176C22×D73—41no—
177C22×D73—42no—
178C22×D73—42no—
179C22×D73—161no—
180C22×D73—161no—
181C22×D73C2×D811yescharacteristic
182C2×#64408—41no—
183C2×#64408—41no—
184C2×#64408—42no—
185C2×#64408—162no—
186C2×#64408—162no—
187C2×#64408—162no—
188C2×#64408—161no—
189C2×#64408C2×D812yes—
190C2×#64408C2×D812yes—
191Q8×C73—21no—
192D4×C73—22no—
193D4×C73—21no—
194D4×C73—41no—
195D4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73C4×D411yescharacteristic
198C22⋊D73—41no—
199C22⋊D73—41no—
200C22⋊D73—42no—
201C22⋊D73—41no—
202C22⋊D73—42no—
203C2×Dic73—41no—
204C2×Dic73#13511yescharacteristic
205D292—22no—
206D292—21no—
207D292—41no—
208D292—41no—
209C4×D73—41no—
210C4×D73—41no—
211C4×D73—41no—
212C4×D73C4×D411yescharacteristic
213C4×D73C2×D811yescharacteristic
214C4×D73#13511yescharacteristic
215C4×D73C22×D411yescharacteristic
216Dic146—21no—
217Dic146—41no—
218#10596833—22no—
219C8×C73C4×D411yescharacteristic
220C8×C73C23×C411yescharacteristic
221#10596831C4×D411yescharacteristic
222#10596831C2×#4111yescharacteristic
223C22×C73—41no—
224C22×C73—42no—
225C22×C73—161no—
226C22×C73C4×D811yescharacteristic
227C2×D73—21no—
228C2×D73—21no—
229C2×D73—41no—
230C2×D73—42no—
231C2×D73—81no—
232C2×D73—82no—
233C2×D73—82no—
234C2×D73—81no—
235C2×D73—162no—
236C2×D73—161no—
237C2×D73C4×D811yescharacteristic
238C2×D73C22×D811yescharacteristic
239#64408—22no—
240#64408—22no—
241#64408—81no—
242#64408—82no—
243#64408—81no—
244C4×C73—41no—
245C4×C73C4×D811yescharacteristic
246C4×C73C2×C4×D411yescharacteristic
247Dic73—41no—
248Dic73C4×D811yescharacteristic
249Dic73C2×#13511yescharacteristic
250#33454—731nomaximal, Sylow
251C2×C73—21no—
252C2×C73—82no—
253C2×C73—81no—
254C2×C73C2×C4×D811yescharacteristic
255D73—21no—
256D73—81no—
257D73—82no—
258D73#341511yescharacteristic
259D73#341611yescharacteristic
260C2×#509—731no—
261#2951—731no—
262#2948—731no—
263#2177—732no—
264#2175—732no—
265C4×SD32—732no—
266C4×D16—732no—
267#2159—731no—
268#2138—731no—
269C2×#357—731no—
270#2108—731no—
271C73#3345411yescharacteristic, complemented, Sylow
272C2×#135—731no—
273C22×D8—731no—
274#520—1461no—
275#518—1461no—
276#509—731no—
277#509—731no—
278#509—732no—
279#509—732no—
280#509—731no—
281#509#6440811yescharacteristic, complemented
282C2×SD32—732no—
283C2×D16—732no—
284C2×M32—731no—
285C22×C16—731no—
286#442—1462no—
287#440—1462no—
288C4×Q16—732no—
289C4×D8—732no—
290C4×D8—732no—
291C4×D8—732no—
292#427—731no—
293C2×#107—731no—
294#418—1461no—
295#417—1461no—
296#405—731no—
297#367—732no—
298#366—732no—
299#358—731no—
300#358—731no—
301#357—732no—
302#357—732no—
303#357—731no—
304#357—731no—
305#346—732no—
306C4×C16—732no—
307C2×#41—1461no—
308C22×D4—1461no—
309#135—731no—
310#135—731no—
311#135—731no—
312#135—1461no—
313#135—1461no—
314#135C2×#6440811yescharacteristic, complemented
315C2×Q16—731no—
316C2×SD16—1461no—
317C2×D8—732no—
318C2×D8—732no—
319C2×D8—731no—
320C2×D8—731no—
321C2×D8—1461no—
322C2×D8—1461no—
323C2×D8—1461no—
324C2×D8C2×#6440811yescharacteristic, complemented
325C2×M16—731no—
326C22×C8—731no—
327C4×Q8—1462no—
328C4×D4—2921no—
329C4×D4—2921no—
330C4×D4—2922no—
331C4×D4—1462no—
332C4×D4—1462no—
333C4×D4—1462no—
334#117—2921no—
335#117—731no—
336#117—1461no—
337C2×C4⋊C4—2921no—
338C2×C4⋊C4—1461no—
339C2×C22⋊C4—2922no—
340SD32—732no—
341SD32—732no—
342SD32—732no—
343SD32C2×#6440812yescomplemented
344D16—732no—
345D16—732no—
346D16—732no—
347D16C2×#6440812yescomplemented
348M32—731no—
349M32—732no—
350M32C2×#6440811yescharacteristic, complemented
351C2×C16—732no—
352C2×C16—731no—
353C2×C16—731no—
354C2×C16—1461no—
355C2×C16—1461no—
356#107—732no—
357#107—731no—
358#107—731no—
359#107—1461no—
360#106—1461no—
361#103—1462no—
362#102—1462no—
363#102—1462no—
364#102—1462no—
365#97—732no—
366C4×C8—732no—
367C24—2921no—
368#41—21no—
369#41—2921no—
370#41—2921no—
371#41—1461no—
372#41—1461no—
373#41—1461no—
374C2×Q8—1461no—
375C2×D4—21no—
376C2×D4—2921no—
377C2×D4—2921no—
378C2×D4—2921no—
379C2×D4—2921no—
380C2×D4—1462no—
381C2×D4—1461no—
382C2×D4—1462no—
383C2×D4—1461no—
384C22×C4—2921no—
385C22×C4—2921no—
386C22×C4—2922no—
387C22×C4—2922no—
388C22×C4—2922no—
389C22×C4—731no—
390Q16—731no—
391Q16—1461no—
392Q16C22×#6440811yescharacteristic
393SD16—21no—
394SD16—1461no—
395SD16—1461no—
396SD16—1461no—
397D8—21no—
398D8—731no—
399D8—732no—
400D8—1462no—
401D8—1462no—
402D8—1461no—
403D8—1461no—
404D8C22×#6440811yescharacteristic
405D8C22×#6440812yes—
406M16—1461no—
407M16—1461no—
408C2×C8—731no—
409C2×C8—731no—
410C2×C8—731no—
411C2×C8—731no—
412C2×C8—731no—
413C2×C8—732no—
414C2×C8C22×#6440811yescharacteristic
415C4⋊C4—2921no—
416C4⋊C4—2922no—
417C4⋊C4—731no—
418C4⋊C4—731no—
419C4⋊C4—1461no—
420C4⋊C4—1461no—
421C4⋊C4—1462no—
422C22⋊C4—2921no—
423C22⋊C4—2922no—
424C22⋊C4—2922no—
425C22⋊C4—2922no—
426C22⋊C4—732no—
427C42—2922no—
428C42—2921no—
429C42—732no—
430C16—732no—
431C16—1461no—
432C16—1461no—
433C16C22×#6440812yes—
434C23—41no—
435C23—2921no—
436C23—2922no—
437C23—2922no—
438C23—2921no—
439C23—731no—
440C23—11681no—
441C23—11681no—
442Q8—21no—
443Q8—2921no—
444Q8—1461no—
445D4—22no—
446D4—21no—
447D4—41no—
448D4—41no—
449D4—2921no—
450D4—2921no—
451D4—2921no—
452D4—2922no—
453D4—2922no—
454D4—2921no—
455D4—2921no—
456D4—1462no—
457D4—1461no—
458C2×C4—41no—
459C2×C4—2921no—
460C2×C4—2921no—
461C2×C4—2921no—
462C2×C4—2921no—
463C2×C4—2921no—
464C2×C4—2921no—
465C2×C4—2922no—
466C2×C4—731no—
467C2×C4—731no—
468C2×C4—731no—
469C2×C4—731no—
470C2×C4—731no—
471C2×C4—732no—
472C2×C4—732no—
473C2×C4—11681no—
474C2×C4—11682no—
475C2×C4—11682no—
476C2×C4—11682no—
477C2×C4D4×#6440811yescharacteristic
478C8—731no—
479C8—731no—
480C8—1462no—
481C8D4×#6440811yescharacteristic
482C8C23×#6440811yescharacteristic
483C22—41no—
484C22—42no—
485C22—161no—
486C22—2921no—
487C22—2922no—
488C22—5841no—
489C22—5842no—
490C22—5842no—
491C22—5841no—
492C22—731no—
493C22—731no—
494C22—11681no—
495C22—11682no—
496C22—1461no—
497C22—1461no—
498C22D8×#6440811yescharacteristic
499C4—41no—
500C4—2921no—
501C4—5841no—
502C4—5842no—
503C4—5841no—
504C4—731no—
505C4—731no—
506C4—1462no—
507C4—1462no—
508C4D8×#6440811yescharacteristic
509C4C2×D4×#6440811yescharacteristic
510C2—21no—
511C2—82no—
512C2—81no—
513C2—5842no—
514C2—5841no—
515C2—731no—
516C2—731no—
517C2—1461no—
518C2C2×D8×#6440811yescharacteristic
subgroupG/H
centerC2C2×D8×#64408
commutatorC8×C73C23×C4
FrattiniC8C23×#64408
Fitting#509×C73C4
socleC2×C73C2×C4×D8
prime radical#509×D73C2
cosocleC8×D73C24
hypercenter#509#64408
hypocenterC73#33454
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22626693C2semidirect
2. #22626690C2semidirect
3. #22626329C2semidirect
4. #22626327C2semidirect
5. SD32×#64408C2semidirect
6. D16×#64408C2semidirect
7. #22626283C2semidirect
8. #22626238C2semidirect
9. C2×#17989499C2semidirect
10. #22626143C2semidirect
17. Q16×#64408C22semidirect
20. D8×#64408C22semidirect
26. #17989514C22semidirect
27. #17989513C22semidirect
28. #17989500C22semidirect
29. #17989500C22semidirect
32. #17989499C22semidirect
33. #17989499C22semidirect
34. #17989474C22semidirect
35. C16×#64408C22semidirect
38. #17989025C4semidirect
39. #17989024C4semidirect
40. #17989023C4semidirect
41. #17989022C4semidirect
42. #17989021C4semidirect
46. #509×C73C4semidirect
78. #14556614C2×C4semidirect
79. #14556613C2×C4semidirect
82. #14556610C2×C4semidirect
85. C2×D584C2×C4semidirect
89. D8×D73C2×C4semidirect
93. #14556536C2×C4semidirect
94. #14556535C2×C4semidirect
95. #14556534C2×C4semidirect
96. D1168C2×C4semidirect
97. #14556532C2×C4semidirect
98. #14556532C2×C4semidirect
99. #14556531C2×C4semidirect
100. #14556530C2×C4semidirect
102. #135×C73C2×C4semidirect
103. C2×D8×C73C2×C4semidirect
104. SD32×C73C2×C4semidirect
105. D16×C73C2×C4semidirect
106. M32×C73C2×C4semidirect
271. C73#33454semidirect
281. #509#64408semidirect
314. #135C2×#64408semidirect
324. C2×D8C2×#64408semidirect
343. SD32C2×#64408semidirect
347. D16C2×#64408semidirect
350. M32C2×#64408semidirect

Maximal subgroups

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

subgroupindex
#226266932
#226266902
#226263292
#226263272
SD32×#644082
D16×#644082
#226262832
#226262382
C2×#179894992
#226261432
#509×D732
#3345473