Finite Group Databasebeta

← #30624292

Subgroups — #30624292

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462581C211yesmaximal, characteristic
2#21462580C211yesmaximal, characteristic, complemented
3#21462216C212yesmaximal, complemented
4#21462215C212yesmaximal, complemented
5Q32×#6923C212yesmaximal, complemented
6SD32×#6923C212yesmaximal, complemented
7#21462171C211yesmaximal, characteristic, complemented
8#21462126C211yesmaximal, characteristic
9C2×#16840675C211yesmaximal, characteristic, complemented
10#21462031C211yesmaximal, characteristic, complemented
11#510×D61C211yesmaximal, characteristic
12#16840831—21no—
13#16840830—21no—
14C2×D61⋊C16C2211yescharacteristic
15#16840802—22no—
16#16840801—22no—
17Q16×#6923C2212yes—
18Q16×#6923C2212yes—
19Q16×#6923C2212yes—
20D8×#6923C2212yes—
21#16840772C2211yescharacteristic
22C2×#14519846C2211yescharacteristic
23#16840755—21no—
24#16840753—21no—
25#16840725C2211yescharacteristic
26#16840689C2212yes—
27#16840688C2212yes—
28#16840675C2212yes—
29#16840675C2212yes—
30#16840675C2211yescharacteristic
31#16840675C2211yescharacteristic
32#16840674C2211yescharacteristic
33#16840674C2211yescharacteristic
34#16840649C2212yes—
35C16×#6923C2212yes—
36#135×D61C2211yescharacteristic
37C2×Q16×D61C2211yescharacteristic
38#16840205C411yescharacteristic, complemented
39#16840204C412yescomplemented
40#16840203C411yescharacteristic, complemented
41#16840202C411yescharacteristic, complemented
42#16840201C412yescomplemented
43Q32×D61C2212yes—
44SD32×D61C2212yes—
45M32×D61C2211yescharacteristic
46C61×#510C411yescharacteristic, complemented
47D61⋊M16D411yescharacteristic
48Q8×#6923—22no—
49Q8×#6923—22no—
50Q8×#6923—22no—
51Q8×#6923—41no—
52Q8×#6923—41no—
53D4×#6923—22no—
54D4×#6923—42no—
55#14519856—21no—
56#14519856—42no—
57#14519856—41no—
58#14519856D411yescharacteristic
59C2×C4⋊#6923—21no—
60C2×C4⋊#6923—41no—
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#14519840—22no—
72#14519840—22no—
73#14519839—22no—
74#14519831D412yes—
75C8×#6923C2312yes—
76#41×D61—21no—
77C2×Q8×D61—21no—
78#14519800C2×C411yescharacteristic
79#14519799C2×C411yescharacteristic
80#14519798—21no—
81#14519797—21no—
82#14519796C2×C411yescharacteristic
83C2×Dic244C2×C411yescharacteristic, complemented
84C2×#11758202—21no—
85Q16×D61—21no—
86Q16×D61—21no—
87Q16×D61C2×C412yescomplemented
88Q16×D61C2312yes—
89Q16×D61C2311yescharacteristic
90SD16×D61—21no—
91D8×D61C2311yescharacteristic
92C2×C8×D61C2311yescharacteristic
93Dic488C2×C412yescomplemented
94#14519723C2×C412yescomplemented
95#14519723C2×C412yescomplemented
96#14519722C2×C412yescomplemented
97#14519721C2×C412yescomplemented
98#14519720C2×C412yescomplemented
99#14519717C2×C411yescharacteristic, complemented
100#14519716C2×C412yescomplemented
101C16×D61C2312yes—
102#135×C61C2×C411yescharacteristic
103C2×Q16×C61C2×C411yescharacteristic, complemented
104Q32×C61C2×C412yescomplemented
105SD32×C61C2×C412yescomplemented
106M32×C61C2×C411yescharacteristic, complemented
107C22×#6923—41no—
108C22×#6923—42no—
109#41×C61—21no—
110C2×Q8×C61—21no—
111#11758226—21no—
112Q8×D61—22no—
113Q8×D61—22no—
114Q8×D61—21no—
115Q8×D61—41no—
116Q8×D61—41no—
117#11758224—21no—
118#11758224—41no—
119D4×D61—21no—
120D4×D61—41no—
121#11758222—21no—
122#11758222—41no—
123C2×C4×D61—41no—
124C2×C4×D61—41no—
125C2×C4×D61C2×D411yescharacteristic
126C2×Dic122—21no—
127C2×Dic122—41no—
128C22⋊#6923—41no—
129C22⋊#6923—42no—
130C22⋊#6923D812yes—
131#11758217—21no—
132C4⋊#6923—22no—
133C4⋊#6923—21no—
134C4⋊#6923—21no—
135C4⋊#6923—42no—
136C4⋊#6923—42no—
137C4⋊#6923—42no—
138C4⋊#6923—41no—
139C4⋊#6923D811yescharacteristic
140C4⋊#6923D811yescharacteristic
141C4×#6923—42no—
142C4×#6923—42no—
143C4×#6923—42no—
144C4×#6923—41no—
145C4×#6923C2×D412yes—
146#11758213—21no—
147D61⋊C8C2×D412yes—
148Q16×C61—21no—
149Q16×C61C22×C412yes—
150Q16×C61C22×C411yescharacteristic
151SD16×C61—21no—
152D8×C61C22×C411yescharacteristic
153C2×C8×C61C22×C411yescharacteristic
154#11758202—22no—
155#11758202—22no—
156#11758202—21no—
157#11758201—21no—
158#11758200—21no—
159#11758199—21no—
160C2×#92418#4111yescharacteristic
161Dic244—21no—
162Dic244C22×C412yes—
163Dic244C22×C411yescharacteristic
164D488C22×C411yescharacteristic
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—
174C22×D61—41no—
175C22×D61C2×D811yescharacteristic
176C2×#6923—41no—
177C2×#6923—41no—
178C2×#6923—42no—
179C2×#6923—162no—
180C2×#6923—161no—
181C2×#6923C2×D812yes—
182C2×#6923C2×D812yes—
183Q8×C61—21no—
184Q8×C61—22no—
185Q8×C61—41no—
186D4×C61—21no—
187D4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61C4×D411yescharacteristic
191C22⋊D61—41no—
192C22⋊D61—41no—
193C22⋊D61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61#13511yescharacteristic
197D244—21no—
198D244—41no—
199C4×D61—41no—
200C4×D61—42no—
201C4×D61—42no—
202C4×D61—41no—
203C4×D61—41no—
204C4×D61C4×D411yescharacteristic
205C4×D61C2×D811yescharacteristic
206C4×D61#13511yescharacteristic
207C4×D61C22×D411yescharacteristic
208Dic122—21no—
209Dic122—22no—
210Dic122—41no—
211Dic122—42no—
212Dic122—41no—
213Dic122—42no—
214#92420—22no—
215C8×C61C4×D411yescharacteristic
216C8×C61C23×C411yescharacteristic
217#92418C4×D411yescharacteristic
218#92418C2×#4111yescharacteristic
219#33455—611nomaximal, Sylow
220C22×C61—41no—
221C22×C61C4×D811yescharacteristic
222C2×D61—21no—
223C2×D61—21no—
224C2×D61—41no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61C4×D811yescharacteristic
228C2×D61C22×D811yescharacteristic
229#6923—22no—
230#6923—22no—
231#6923—81no—
232#6923—82no—
233#6923—81no—
234C4×C61—42no—
235C4×C61—41no—
236C4×C61C4×D811yescharacteristic
237C4×C61C2×C4×D411yescharacteristic
238Dic61—42no—
239Dic61—41no—
240Dic61C4×D811yescharacteristic
241Dic61C2×#13511yescharacteristic
242C2×#510—611no—
243#2951—611no—
244#2950—611no—
245#2176—612no—
246#2175—612no—
247C4×Q32—612no—
248C4×SD32—612no—
249#2159—611no—
250#2138—611no—
251C2×#358—611no—
252#2108—611no—
253C2×C61—21no—
254C2×C61—81no—
255C2×C61C2×C4×D811yescharacteristic
256D61—21no—
257D61—81no—
258D61#341611yescharacteristic
259D61#341711yescharacteristic
260C2×#135—611no—
261C22×Q16—611no—
262#520—1221no—
263#519—1221no—
264#510—611no—
265#510—611no—
266#510—612no—
267#510—612no—
268#510—611no—
269#510#692311yescharacteristic, complemented
270C2×Q32—612no—
271C2×SD32—612no—
272C2×M32—611no—
273C22×C16—611no—
274#441—1222no—
275#440—1222no—
276C4×Q16—612no—
277C4×Q16—612no—
278C4×Q16—612no—
279C4×D8—612no—
280#427—611no—
281C2×#107—611no—
282#419—1221no—
283#417—1221no—
284#405—611no—
285#367—612no—
286#366—612no—
287#358—612no—
288#358—612no—
289#358—611no—
290#358—611no—
291#357—611no—
292#357—611no—
293#346—612no—
294C4×C16—612no—
295C61#3345511yescharacteristic, complemented, Sylow
296C2×#41—1221no—
297C22×Q8—1221no—
298#135—611no—
299#135—611no—
300#135—611no—
301#135—1221no—
302#135—1221no—
303#135C2×#692311yescharacteristic
304C2×Q16—612no—
305C2×Q16—612no—
306C2×Q16—611no—
307C2×Q16—611no—
308C2×Q16—1221no—
309C2×Q16—1221no—
310C2×Q16—1221no—
311C2×Q16C2×#692311yescharacteristic, complemented
312C2×SD16—1221no—
313C2×D8—611no—
314C2×M16—611no—
315C22×C8—611no—
316C4×Q8—1222no—
317C4×Q8—1222no—
318C4×Q8—1222no—
319C4×Q8—2441no—
320C4×Q8—2441no—
321C4×D4—1222no—
322C4×D4—2442no—
323#117—611no—
324#117—1221no—
325#117—2442no—
326#117—2441no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—2441no—
329Q32—612no—
330Q32—612no—
331Q32—612no—
332Q32C2×#692312yescomplemented
333SD32—612no—
334SD32—612no—
335SD32—612no—
336SD32C2×#692312yescomplemented
337M32—611no—
338M32—612no—
339M32C2×#692311yescharacteristic, complemented
340C2×C16—612no—
341C2×C16—611no—
342C2×C16—611no—
343C2×C16—1221no—
344C2×C16—1221no—
345#107—611no—
346#107—611no—
347#107—612no—
348#107—1221no—
349#106—1221no—
350#103—1222no—
351#103—1222no—
352#103—1222no—
353#102—1222no—
354#97—612no—
355C4×C8—612no—
356#41—21no—
357#41—1221no—
358#41—1221no—
359#41—1221no—
360#41—2441no—
361#41—2441no—
362C2×Q8—21no—
363C2×Q8—1221no—
364C2×Q8—1222no—
365C2×Q8—1222no—
366C2×Q8—1221no—
367C2×Q8—2441no—
368C2×Q8—2441no—
369C2×Q8—2441no—
370C2×D4—1221no—
371C2×D4—2441no—
372C22×C4—611no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2442no—
376C22×C4—2441no—
377Q16—21no—
378Q16—612no—
379Q16—611no—
380Q16—1222no—
381Q16—1222no—
382Q16—1221no—
383Q16—1221no—
384Q16C22×#692312yes—
385Q16C22×#692311yescharacteristic
386SD16—21no—
387SD16—1221no—
388SD16—1221no—
389SD16—1221no—
390D8—611no—
391D8—1221no—
392D8C22×#692311yescharacteristic
393M16—1221no—
394M16—1221no—
395C2×C8—611no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—611no—
399C2×C8—611no—
400C2×C8—612no—
401C2×C8C22×#692311yescharacteristic
402C4⋊C4—611no—
403C4⋊C4—611no—
404C4⋊C4—1222no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—2441no—
408C4⋊C4—2442no—
409C4⋊C4—2442no—
410C4⋊C4—2442no—
411C22⋊C4—612no—
412C22⋊C4—2442no—
413C22⋊C4—2441no—
414C42—612no—
415C42—2441no—
416C42—2442no—
417C42—2442no—
418C42—2442no—
419C16—612no—
420C16—1221no—
421C16—1221no—
422C16C22×#692312yes—
423C23—611no—
424C23—2441no—
425Q8—21no—
426Q8—22no—
427Q8—41no—
428Q8—1221no—
429Q8—1222no—
430Q8—2442no—
431Q8—2442no—
432Q8—2441no—
433Q8—2441no—
434D4—21no—
435D4—41no—
436D4—1221no—
437D4—2441no—
438D4—2441no—
439D4—2441no—
440D4—2441no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—611no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—611no—
448C2×C4—612no—
449C2×C4—612no—
450C2×C4—9762no—
451C2×C4—9761no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2441no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4—2442no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4D4×#692311yescharacteristic
463C8—611no—
464C8—611no—
465C8—1222no—
466C8D4×#692311yescharacteristic
467C8C23×#692311yescharacteristic
468C22—41no—
469C22—611no—
470C22—611no—
471C22—1221no—
472C22—1221no—
473C22—4881no—
474C22—4881no—
475C22—2441no—
476C22D8×#692311yescharacteristic
477C4—41no—
478C4—42no—
479C4—611no—
480C4—611no—
481C4—1222no—
482C4—1222no—
483C4—4882no—
484C4—4881no—
485C4—4881no—
486C4—2441no—
487C4—2442no—
488C4D8×#692311yescharacteristic
489C4C2×D4×#692311yescharacteristic
490C2—21no—
491C2—81no—
492C2—611no—
493C2—611no—
494C2—1221no—
495C2—4881no—
496C2C2×D8×#692311yescharacteristic
subgroupG/H
centerC2C2×D8×#6923
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#510C4
socleC2×C61C2×C4×D8
prime radical#135×D61C22
cosocleC8×D61C24
hypercenter#510#6923
hypocenterC61#33455
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #21462580C2semidirect
3. #21462216C2semidirect
4. #21462215C2semidirect
5. Q32×#6923C2semidirect
6. SD32×#6923C2semidirect
7. #21462171C2semidirect
9. C2×#16840675C2semidirect
10. #21462031C2semidirect
38. #16840205C4semidirect
39. #16840204C4semidirect
40. #16840203C4semidirect
41. #16840202C4semidirect
42. #16840201C4semidirect
46. C61×#510C4semidirect
83. C2×Dic244C2×C4semidirect
87. Q16×D61C2×C4semidirect
93. Dic488C2×C4semidirect
94. #14519723C2×C4semidirect
95. #14519723C2×C4semidirect
96. #14519722C2×C4semidirect
97. #14519721C2×C4semidirect
98. #14519720C2×C4semidirect
99. #14519717C2×C4semidirect
100. #14519716C2×C4semidirect
103. C2×Q16×C61C2×C4semidirect
104. Q32×C61C2×C4semidirect
105. SD32×C61C2×C4semidirect
106. M32×C61C2×C4semidirect
269. #510#6923semidirect
295. C61#33455semidirect
311. C2×Q16C2×#6923semidirect
332. Q32C2×#6923semidirect
336. SD32C2×#6923semidirect
339. M32C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625812
#214625802
#214622162
#214622152
Q32×#69232
SD32×#69232
#214621712
#214621262
C2×#168406752
#214620312
#510×D612
#3345561