Finite Group Databasebeta

← #30624291

Subgroups — #30624291

Order 15616.

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#21462581C211yesmaximal, characteristic, complemented
2#21462578C211yesmaximal, characteristic, complemented
3#21462217C212yesmaximal, complemented
4#21462215C212yesmaximal, complemented
5SD32×#6923C212yesmaximal, complemented
6D16×#6923C212yesmaximal, complemented
7#21462171C211yesmaximal, characteristic, complemented
8#21462126C211yesmaximal, characteristic, complemented
9C2×#16840674C211yesmaximal, characteristic, complemented
10#21462031C211yesmaximal, characteristic, complemented
11#509×D61C211yesmaximal, characteristic
12#16840831—21no—
13#16840829—21no—
14C2×D61⋊C16C2211yescharacteristic
15#16840803—22no—
16#16840801—22no—
17Q16×#6923C2212yescomplemented
18D8×#6923C2212yes—
19D8×#6923C2212yes—
20D8×#6923C2212yescomplemented
21#16840772C2211yescharacteristic
22C2×#14519846C2211yescharacteristic
23#16840754—21no—
24#16840753—21no—
25#16840725C2211yescharacteristic
26#16840689C2212yescomplemented
27#16840688C2212yescomplemented
28#16840675C2211yescharacteristic, complemented
29#16840675C2211yescharacteristic, complemented
30#16840674C2212yes—
31#16840674C2212yes—
32#16840674C2211yescharacteristic, complemented
33#16840674C2211yescharacteristic, complemented
34#16840649C2212yescomplemented
35C16×#6923C2212yescomplemented
36#135×D61C2211yescharacteristic
37C2×D8×D61C2211yescharacteristic
38#16840200C411yescharacteristic, complemented
39#16840199C412yescomplemented
40#16840198C411yescharacteristic, complemented
41#16840197C411yescharacteristic, complemented
42#16840196C412yescomplemented
43SD32×D61C2212yes—
44D16×D61C2212yes—
45M32×D61C2211yescharacteristic
46C61×#509C411yescharacteristic, complemented
47D61⋊M16D411yescharacteristic
48Q8×#6923—22no—
49D4×#6923—22no—
50D4×#6923—22no—
51D4×#6923—22no—
52D4×#6923—42no—
53D4×#6923—41no—
54D4×#6923—41no—
55#14519856—21no—
56#14519856—41no—
57#14519856D411yescharacteristic
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923—41no—
60C2×C22⋊#6923—42no—
61D61⋊C16—21no—
62D61⋊C16C2311yescharacteristic
63D61⋊C16C2311yescharacteristic
64C2×#11758187—21no—
65#14519846—21no—
66#14519846C2312yes—
67#14519846C2311yescharacteristic
68#14519846C2311yescharacteristic
69#14519845—21no—
70#14519840—22no—
71#14519839—22no—
72#14519839—22no—
73#14519839—22no—
74#14519831D412yes—
75C8×#6923C2312yes—
76#41×D61—21no—
77C2×D4×D61—21no—
78#14519800C2×C411yescharacteristic, complemented
79#14519799C2×C411yescharacteristic, complemented
80#14519798—21no—
81#14519797—21no—
82#14519796C2×C411yescharacteristic, complemented
83Q16×D61C2311yescharacteristic
84SD16×D61—21no—
85C2×D488C2×C411yescharacteristic, complemented
86C2×#11758199—21no—
87D8×D61—21no—
88D8×D61—21no—
89D8×D61C2×C412yescomplemented
90D8×D61C2312yes—
91D8×D61C2311yescharacteristic
92C2×C8×D61C2311yescharacteristic
93#14519722C2×C412yescomplemented
94#14519721C2×C412yescomplemented
95#14519720C2×C412yescomplemented
96D976C2×C412yescomplemented
97#14519718C2×C412yescomplemented
98#14519718C2×C412yescomplemented
99#14519717C2×C411yescharacteristic, complemented
100#14519716C2×C412yescomplemented
101C16×D61C2312yes—
102#135×C61C2×C411yescharacteristic, complemented
103C2×D8×C61C2×C411yescharacteristic, complemented
104SD32×C61C2×C412yescomplemented
105D16×C61C2×C412yescomplemented
106M32×C61C2×C411yescharacteristic, complemented
107C23×D61—41no—
108C22×#6923—41no—
109C22×#6923—42no—
110C22×#6923—42no—
111C22×#6923—42no—
112#41×C61—21no—
113C2×D4×C61—21no—
114C2×C22⋊D61—41no—
115#11758226—21no—
116Q8×D61—21no—
117#11758224—21no—
118#11758224—41no—
119D4×D61—22no—
120D4×D61—22no—
121D4×D61—21no—
122D4×D61—41no—
123D4×D61—41no—
124D4×D61—41no—
125#11758222—21no—
126#11758222—41no—
127C2×D244—21no—
128C2×C4×D61—41no—
129C2×C4×D61C2×D411yescharacteristic
130C22⋊#6923—41no—
131C22⋊#6923—42no—
132C22⋊#6923—42no—
133C22⋊#6923—42no—
134C22⋊#6923D812yes—
135#11758217—21no—
136C4⋊#6923—21no—
137C4⋊#6923—21no—
138C4⋊#6923—22no—
139C4⋊#6923—42no—
140C4⋊#6923—41no—
141C4⋊#6923D811yescharacteristic
142C4⋊#6923D811yescharacteristic
143C4×#6923—42no—
144C4×#6923—41no—
145C4×#6923C2×D412yes—
146#11758213—21no—
147D61⋊C8C2×D412yes—
148Q16×C61C22×C411yescharacteristic
149SD16×C61—21no—
150D8×C61—21no—
151D8×C61C22×C411yescharacteristic
152D8×C61C22×C412yes—
153C2×C8×C61C22×C411yescharacteristic
154#11758202—21no—
155#11758201—21no—
156#11758200—21no—
157#11758199—22no—
158#11758199—22no—
159#11758199—21no—
160C2×#92418#4111yescharacteristic
161Dic244C22×C411yescharacteristic
162D488—21no—
163D488C22×C411yescharacteristic
164D488C22×C412yes—
165#11758190—21no—
166C8×D61C22×C411yescharacteristic
167C8×D61C2×D411yescharacteristic
168C8×D61#4111yescharacteristic
169C8×D61C2411yescharacteristic
170#11758187—21no—
171#11758187—21no—
172C16×C61C22×C412yes—
173#11758185C22×C412yes—
174C23×C61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—161no—
180C22×D61—161no—
181C22×D61C2×D811yescharacteristic
182C2×#6923—42no—
183C2×#6923—41no—
184C2×#6923—41no—
185C2×#6923—162no—
186C2×#6923—162no—
187C2×#6923—162no—
188C2×#6923—161no—
189C2×#6923C2×D812yes—
190C2×#6923C2×D812yes—
191Q8×C61—21no—
192D4×C61—22no—
193D4×C61—21no—
194D4×C61—41no—
195D4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61C4×D411yescharacteristic
198C22⋊D61—41no—
199C22⋊D61—41no—
200C22⋊D61—41no—
201C22⋊D61—42no—
202C22⋊D61—42no—
203C2×Dic61—41no—
204C2×Dic61#13511yescharacteristic
205D244—21no—
206D244—22no—
207D244—41no—
208D244—41no—
209C4×D61—41no—
210C4×D61—41no—
211C4×D61—41no—
212C4×D61C4×D411yescharacteristic
213C4×D61C2×D811yescharacteristic
214C4×D61#13511yescharacteristic
215C4×D61C22×D411yescharacteristic
216Dic122—21no—
217Dic122—41no—
218#92420—22no—
219C8×C61C4×D411yescharacteristic
220C8×C61C23×C411yescharacteristic
221#92418C4×D411yescharacteristic
222#92418C2×#4111yescharacteristic
223#33454—611nomaximal, Sylow
224C22×C61—41no—
225C22×C61—42no—
226C22×C61—161no—
227C22×C61C4×D811yescharacteristic
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—41no—
231C2×D61—42no—
232C2×D61—81no—
233C2×D61—82no—
234C2×D61—82no—
235C2×D61—81no—
236C2×D61—162no—
237C2×D61—161no—
238C2×D61C4×D811yescharacteristic
239C2×D61C22×D811yescharacteristic
240#6923—22no—
241#6923—22no—
242#6923—82no—
243#6923—81no—
244#6923—81no—
245C4×C61—41no—
246C4×C61C4×D811yescharacteristic
247C4×C61C2×C4×D411yescharacteristic
248Dic61—41no—
249Dic61C4×D811yescharacteristic
250Dic61C2×#13511yescharacteristic
251C2×#509—611no—
252#2951—611no—
253#2948—611no—
254#2177—612no—
255#2175—612no—
256C4×SD32—612no—
257C4×D16—612no—
258#2159—611no—
259#2138—611no—
260C2×#357—611no—
261#2108—611no—
262C2×C61—21no—
263C2×C61—82no—
264C2×C61—81no—
265C2×C61C2×C4×D811yescharacteristic
266D61—21no—
267D61—81no—
268D61—82no—
269D61#341511yescharacteristic
270D61#341611yescharacteristic
271C2×#135—611no—
272C22×D8—611no—
273#520—1221no—
274#518—1221no—
275#509—611no—
276#509—611no—
277#509—612no—
278#509—612no—
279#509—611no—
280#509#692311yescharacteristic, complemented
281C2×SD32—612no—
282C2×D16—612no—
283C2×M32—611no—
284C22×C16—611no—
285#442—1222no—
286#440—1222no—
287C4×Q16—612no—
288C4×D8—612no—
289C4×D8—612no—
290C4×D8—612no—
291#427—611no—
292C2×#107—611no—
293#418—1221no—
294#417—1221no—
295#405—611no—
296#367—612no—
297#366—612no—
298#358—611no—
299#358—611no—
300#357—612no—
301#357—612no—
302#357—611no—
303#357—611no—
304#346—612no—
305C4×C16—612no—
306C61#3345411yescharacteristic, complemented, Sylow
307C2×#41—1221no—
308C22×D4—1221no—
309#135—611no—
310#135—611no—
311#135—611no—
312#135—1221no—
313#135—1221no—
314#135C2×#692311yescharacteristic, complemented
315C2×Q16—611no—
316C2×SD16—1221no—
317C2×D8—612no—
318C2×D8—612no—
319C2×D8—611no—
320C2×D8—611no—
321C2×D8—1221no—
322C2×D8—1221no—
323C2×D8—1221no—
324C2×D8C2×#692311yescharacteristic, complemented
325C2×M16—611no—
326C22×C8—611no—
327C4×Q8—1222no—
328C4×D4—1222no—
329C4×D4—1222no—
330C4×D4—1222no—
331C4×D4—2442no—
332C4×D4—2441no—
333C4×D4—2441no—
334#117—611no—
335#117—1221no—
336#117—2441no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—2441no—
339C2×C22⋊C4—2442no—
340SD32—612no—
341SD32—612no—
342SD32—612no—
343SD32C2×#692312yescomplemented
344D16—612no—
345D16—612no—
346D16—612no—
347D16C2×#692312yescomplemented
348M32—611no—
349M32—612no—
350M32C2×#692311yescharacteristic, complemented
351C2×C16—612no—
352C2×C16—611no—
353C2×C16—611no—
354C2×C16—1221no—
355C2×C16—1221no—
356#107—612no—
357#107—611no—
358#107—611no—
359#107—1221no—
360#106—1221no—
361#103—1222no—
362#102—1222no—
363#102—1222no—
364#102—1222no—
365#97—612no—
366C4×C8—612no—
367C24—2441no—
368#41—21no—
369#41—1221no—
370#41—1221no—
371#41—1221no—
372#41—2441no—
373#41—2441no—
374C2×Q8—1221no—
375C2×D4—21no—
376C2×D4—1222no—
377C2×D4—1221no—
378C2×D4—1222no—
379C2×D4—1221no—
380C2×D4—2441no—
381C2×D4—2441no—
382C2×D4—2441no—
383C2×D4—2441no—
384C22×C4—611no—
385C22×C4—2441no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2441no—
390Q16—611no—
391Q16—1221no—
392Q16C22×#692311yescharacteristic
393SD16—21no—
394SD16—1221no—
395SD16—1221no—
396SD16—1221no—
397D8—21no—
398D8—611no—
399D8—612no—
400D8—1222no—
401D8—1222no—
402D8—1221no—
403D8—1221no—
404D8C22×#692311yescharacteristic
405D8C22×#692312yes—
406M16—1221no—
407M16—1221no—
408C2×C8—611no—
409C2×C8—611no—
410C2×C8—611no—
411C2×C8—611no—
412C2×C8—611no—
413C2×C8—612no—
414C2×C8C22×#692311yescharacteristic
415C4⋊C4—611no—
416C4⋊C4—611no—
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C4⋊C4—1222no—
420C4⋊C4—2441no—
421C4⋊C4—2442no—
422C22⋊C4—612no—
423C22⋊C4—2442no—
424C22⋊C4—2442no—
425C22⋊C4—2442no—
426C22⋊C4—2441no—
427C42—612no—
428C42—2441no—
429C42—2442no—
430C16—612no—
431C16—1221no—
432C16—1221no—
433C16C22×#692312yes—
434C23—41no—
435C23—611no—
436C23—9761no—
437C23—9761no—
438C23—2442no—
439C23—2441no—
440C23—2442no—
441C23—2441no—
442Q8—21no—
443Q8—1221no—
444Q8—2441no—
445D4—22no—
446D4—21no—
447D4—41no—
448D4—41no—
449D4—1222no—
450D4—1221no—
451D4—2441no—
452D4—2442no—
453D4—2442no—
454D4—2441no—
455D4—2441no—
456D4—2441no—
457D4—2441no—
458C2×C4—41no—
459C2×C4—611no—
460C2×C4—611no—
461C2×C4—611no—
462C2×C4—611no—
463C2×C4—611no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—9762no—
467C2×C4—9762no—
468C2×C4—9762no—
469C2×C4—9761no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4D4×#692311yescharacteristic
478C8—611no—
479C8—611no—
480C8—1222no—
481C8D4×#692311yescharacteristic
482C8C23×#692311yescharacteristic
483C22—41no—
484C22—42no—
485C22—161no—
486C22—611no—
487C22—611no—
488C22—1221no—
489C22—1221no—
490C22—9761no—
491C22—9762no—
492C22—4881no—
493C22—4882no—
494C22—4882no—
495C22—4881no—
496C22—2441no—
497C22—2442no—
498C22D8×#692311yescharacteristic
499C4—41no—
500C4—611no—
501C4—611no—
502C4—1222no—
503C4—1222no—
504C4—4881no—
505C4—4882no—
506C4—4881no—
507C4—2441no—
508C4D8×#692311yescharacteristic
509C4C2×D4×#692311yescharacteristic
510C2—21no—
511C2—82no—
512C2—81no—
513C2—611no—
514C2—611no—
515C2—1221no—
516C2—4882no—
517C2—4881no—
518C2C2×D8×#692311yescharacteristic
subgroupG/H
centerC2C2×D8×#6923
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#509C4
socleC2×C61C2×C4×D8
prime radical#509×D61C2
cosocleC8×D61C24
hypercenter#509#6923
hypocenterC61#33454
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #21462581C2semidirect
2. #21462578C2semidirect
3. #21462217C2semidirect
4. #21462215C2semidirect
5. SD32×#6923C2semidirect
6. D16×#6923C2semidirect
7. #21462171C2semidirect
8. #21462126C2semidirect
9. C2×#16840674C2semidirect
10. #21462031C2semidirect
17. Q16×#6923C22semidirect
20. D8×#6923C22semidirect
26. #16840689C22semidirect
27. #16840688C22semidirect
28. #16840675C22semidirect
29. #16840675C22semidirect
32. #16840674C22semidirect
33. #16840674C22semidirect
34. #16840649C22semidirect
35. C16×#6923C22semidirect
38. #16840200C4semidirect
39. #16840199C4semidirect
40. #16840198C4semidirect
41. #16840197C4semidirect
42. #16840196C4semidirect
46. C61×#509C4semidirect
78. #14519800C2×C4semidirect
79. #14519799C2×C4semidirect
82. #14519796C2×C4semidirect
85. C2×D488C2×C4semidirect
89. D8×D61C2×C4semidirect
93. #14519722C2×C4semidirect
94. #14519721C2×C4semidirect
95. #14519720C2×C4semidirect
96. D976C2×C4semidirect
97. #14519718C2×C4semidirect
98. #14519718C2×C4semidirect
99. #14519717C2×C4semidirect
100. #14519716C2×C4semidirect
102. #135×C61C2×C4semidirect
103. C2×D8×C61C2×C4semidirect
104. SD32×C61C2×C4semidirect
105. D16×C61C2×C4semidirect
106. M32×C61C2×C4semidirect
280. #509#6923semidirect
306. C61#33454semidirect
314. #135C2×#6923semidirect
324. C2×D8C2×#6923semidirect
343. SD32C2×#6923semidirect
347. D16C2×#6923semidirect
350. M32C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625812
#214625782
#214622172
#214622152
SD32×#69232
D16×#69232
#214621712
#214621262
C2×#168406742
#214620312
#509×D612
#3345461