Finite Group Databasebeta

← #30612910

Subgroups — #30612910

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462542C212yesmaximal, complemented
2#21461452C211yesmaximal, characteristic, complemented
3#21461451C211yesmaximal, characteristic, complemented
4#21461450C212yesmaximal, complemented
5#21461400C211yesmaximal, characteristic, complemented
6#21461399C212yesmaximal, complemented
7D8×#92420C211yesmaximal, characteristic, complemented
8#21461376C211yesmaximal, characteristic, complemented
9#21461164C212yesmaximal, complemented
10C2×C8×#92420C211yesmaximal, characteristic, complemented
11#21456393C211yesmaximal, characteristic
12C2×#14519864—22no—
13#16840797C2212yes—
14#16840796C2212yes—
15#16840795—22no—
16#16840790C2212yes—
17#16840789—22no—
18D4×#92420C2212yes—
19#16840765C2212yes—
20#16840735—22no—
21C2×C4×#92420C2211yescharacteristic
22C2×C4×#92420C2211yescharacteristic
23#16840626C2212yes—
24#16840625C2211yescharacteristic
25#16840625C2211yescharacteristic
26#16840610C2212yes—
27#16840610C2212yes—
28D4⋊#92420C2212yes—
29D4⋊#92420C2212yes—
30C8×#92420C2211yescharacteristic
31C8×#92420C2211yescharacteristic
32C8×#92420C2211yescharacteristic
33C8×#92420C2211yescharacteristic
34#16840307C2212yes—
35#16840166C411yescharacteristic
36#16840137C411yescharacteristic
37#16840107C412yes—
38#16839998C411yescharacteristic
39#16839996C412yes—
40D8×Dic61C2211yescharacteristic
41#16839844C2211yescharacteristic
42#16839842C2212yes—
43C2×C8×Dic61C2211yescharacteristic
44C61×#466C411yescharacteristic
45#14519864—48no—
46C2×#11758213—22no—
47C2×#11758217—22no—
48C2×#11758217—22no—
49C2×D61⋊C8—22no—
50C22×#92420—21no—
51C22×#92420—22no—
52C22×#92420D411yescharacteristic
53C22×#92420D411yescharacteristic
54#14519844—22no—
55C4⋊#92420C2312yes—
56C4⋊#92420C2312yes—
57#14519834—22no—
58#14519834—22no—
59C22⋊#92420—22no—
60C22⋊#92420—22no—
61C4×#92420—21no—
62C4×#92420D411yescharacteristic
63C4×#92420D411yescharacteristic
64C4×#92420C2311yescharacteristic
65C4×#92420C2311yescharacteristic
66C4×#92420C2311yescharacteristic
67C4×#92420C2311yescharacteristic
68C2×#11758224—22no—
69C2×D488C2×C411yescharacteristic
70C2×#11758199C2×C412yes—
71C22×#92418C2×C411yescharacteristic
72#14519776C2×C412yes—
73#14519773C2×C412yes—
74#14519763—22no—
75#14519751C2×C412yes—
76#14519749—22no—
77D4×Dic61C2312yes—
78#14519739C2312yes—
79#14519737—22no—
80C2×C4×Dic61C2311yescharacteristic
81#14519711C2×C411yescharacteristic
82#14519710C2×C412yes—
83#14519699C2×C412yes—
84#14519698C2×C412yes—
85#14519697C2×C412yes—
86C8×Dic61C2×C411yescharacteristic
87C8×Dic61C2×C411yescharacteristic
88C8×Dic61C2311yescharacteristic
89C8×Dic61C2311yescharacteristic
90C2×D8×C61C2×C411yescharacteristic
91C22×C8×C61C2×C411yescharacteristic
92#121×C61C2×C412yes—
93#107×C61C2×C411yescharacteristic
94#102×C61C2×C412yes—
95C2×D4×C61—22no—
96C2×D4×C61C22×C412yes—
97C22×C4×C61C22×C411yescharacteristic
98C2×C22⋊D61—22no—
99C2×C22⋊D61—22no—
100C2×C22⋊D61—22no—
101C22×Dic61—22no—
102C22×Dic61C2×D411yescharacteristic
103C22×Dic61#4111yescharacteristic
104#11758224—44no—
105#11758224—44no—
106C2×D244C22×C412yes—
107C2×C4×D61—22no—
108C2×Dic122—22no—
109#11758217—44no—
110#11758217—44no—
111C2×#92420—24no—
112C2×#92420—22no—
113C2×#92420—21no—
114C2×#92420—21no—
115C2×#92420—22no—
116C2×#92420—22no—
117C2×#92420—44no—
118C2×#92420D814yescomplemented
119C2×#92420C2×D411yescharacteristic
120C2×#92420C2×D411yescharacteristic
121C2×#92420C2×D411yescharacteristic
122C2×#92420C2×D411yescharacteristic
123#11758213—44no—
124D61⋊C8—44no—
125D8×C61—22no—
126C2×C8×C61—22no—
127C2×C8×C61C22×C411yescharacteristic
128C2×C8×C61C22×C411yescharacteristic
129C4⋊C4×C61C22×C412yes—
130C22⋊C4×C61—22no—
131C22⋊Dic61—22no—
132#11758199—24no—
133#11758198—22no—
134#11758198—22no—
135C4⋊Dic61C22×C412yes—
136#11758196—22no—
137C4×Dic61C22×C411yescharacteristic
138C4×Dic61C2×D411yescharacteristic
139C4×Dic61#4111yescharacteristic
140C4×Dic61C2411yescharacteristic
141C2×#92418—22no—
142C2×#92418C22×C411yescharacteristic
143C2×#92418C22×C411yescharacteristic
144D488—22no—
145C23×C61—22no—
146C23×C61C4×D411yescharacteristic
147C22×D61—22no—
148D4×C61—22no—
149D4×C61—22no—
150D4×C61—44no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61C23×C411yescharacteristic
156C22⋊D61—44no—
157C22⋊D61—44no—
158C22⋊D61—44no—
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61C4×D411yescharacteristic
164C2×Dic61C4×D411yescharacteristic
165C2×Dic61C2×D812yes—
166C2×Dic61#13512yes—
167C2×Dic61C22×D411yescharacteristic
168C2×Dic61C2×#4111yescharacteristic
169D244—22no—
170D244—22no—
171C4×D61—42no—
172C4×D61—42no—
173Dic122—44no—
174#92420—22no—
175#92420—22no—
176#92420—21no—
177#92420—21no—
178#92420—42no—
179#92420—42no—
180#92420C2×D812yes—
181#92420C2×D812yes—
182C8×C61—21no—
183C8×C61#13112yescomplemented
184#92418—21no—
185#92418#13112yescomplemented
186#17956—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—42no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61C4×D812yes—
193C22×C61C2×C4×D411yescharacteristic
194C2×D61—42no—
195C2×D61—42no—
196C2×D61—42no—
197C4×C61—21no—
198C4×C61—42no—
199C4×C61C2×#13111yescharacteristic
200C4×C61C2×#13111yescharacteristic
201Dic61—21no—
202Dic61—42no—
203Dic61C4×D812yes—
204Dic61#44312yes—
205Dic61C22×D811yescharacteristic
206Dic61C2×#13511yescharacteristic
207#3169—611no—
208#2931—612no—
209#1589—612no—
210#1589—611no—
211#1589—611no—
212C8×D8—611no—
213C8×D8—612no—
214C8×D8—611no—
215#1567—611no—
216#1492—612no—
217C2×C82—611no—
218C2×C61—22no—
219C2×C61—82no—
220C2×C61#293111yescharacteristic
221C2×C61C2×C4×D811yescharacteristic
222C2×C61C2×#44311yescharacteristic
223D61—82no—
224C2×#131—1222no—
225#532—612no—
226#500—611no—
227#493—611no—
228#486—612no—
229#466—611no—
230#466—612no—
231#466#692311yescharacteristic
232C4×D8—611no—
233C4×D8—611no—
234C4×D8—612no—
235#436—612no—
236#436—612no—
237#436—1222no—
238C8×D4—612no—
239C8×D4—612no—
240C8×D4—1222no—
241#423—612no—
242#408—1222no—
243C2×C4×C8—611no—
244C2×C4×C8—611no—
245C2×C4×C8—611no—
246#335—611no—
247#335—611no—
248#335—612no—
249D4⋊C8—612no—
250D4⋊C8—612no—
251D4⋊C8—612no—
252D4⋊C8—612no—
253C82—611no—
254C82—611no—
255C82—611no—
256C82—611no—
257C61#1795611yescharacteristic, complemented, Sylow
258C2×#41—1222no—
259C2×D8—612no—
260C2×D8—611no—
261C2×D8C2×#692311yescharacteristic
262#131—2448no—
263C2×M16—1222no—
264C2×M16—1222no—
265C2×M16—1222no—
266C22×C8—611no—
267C22×C8—611no—
268C22×C8—611no—
269C22×C8—1222no—
270C22×C8—1222no—
271C22×C8—1221no—
272C22×C8C2×#692311yescharacteristic
273C4⋊Q8—612no—
274#127—612no—
275#124—1222no—
276#121—612no—
277#121—1222no—
278#121C2×#692312yes—
279C4×D4—612no—
280C4×D4—612no—
281C4×D4—1222no—
282C2×C42—611no—
283#107—611no—
284#107—612no—
285#107C2×#692311yescharacteristic
286C4⋊C8—612no—
287C4⋊C8—612no—
288C4⋊C8—1222no—
289#102—612no—
290#102—612no—
291#102—612no—
292#102C2×#692312yes—
293C22⋊C8—1222no—
294C22⋊C8—1222no—
295C22⋊C8—1222no—
296C22⋊C8—1222no—
297C4×C8—611no—
298C4×C8—611no—
299C4×C8—611no—
300C4×C8—611no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8—611no—
304C4×C8—611no—
305C4×C8—611no—
306C4×C8—611no—
307C4×C8—1221no—
308#41—2444no—
309#41—2444no—
310C2×Q8—1222no—
311C2×D4—22no—
312C2×D4—612no—
313C2×D4—1222no—
314C2×D4—1222no—
315C2×D4—1222no—
316C2×D4C22×#692312yes—
317C22×C4—611no—
318C22×C4—611no—
319C22×C4—1222no—
320C22×C4—1222no—
321C22×C4C22×#692311yescharacteristic
322D8—22no—
323D8—1224no—
324D8—1222no—
325M16—2444no—
326M16—2444no—
327M16—2444no—
328C2×C8—22no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—614no—
332C2×C8—611no—
333C2×C8—611no—
334C2×C8—611no—
335C2×C8—611no—
336C2×C8—1222no—
337C2×C8—1221no—
338C2×C8—1221no—
339C2×C8—1222no—
340C2×C8—1224no—
341C2×C8—1222no—
342C2×C8—1222no—
343C2×C8—2444no—
344C2×C8—2444no—
345C2×C8C22×#692311yescharacteristic
346C2×C8C22×#692311yescharacteristic
347C4⋊C4—612no—
348C4⋊C4—1222no—
349C4⋊C4C22×#692312yes—
350C22⋊C4—22no—
351C22⋊C4—1222no—
352C22⋊C4—1222no—
353C22⋊C4—1222no—
354C42—611no—
355C42—611no—
356C42—611no—
357C42—611no—
358C23—22no—
359C23—1222no—
360C23D4×#692311yescharacteristic
361Q8—2444no—
362D4—22no—
363D4—22no—
364D4—44no—
365D4—1222no—
366D4—1222no—
367D4—2444no—
368D4—2444no—
369D4—2444no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—612no—
374C2×C4—612no—
375C2×C4—611no—
376C2×C4—611no—
377C2×C4—611no—
378C2×C4—611no—
379C2×C4—1222no—
380C2×C4—1222no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4D4×#692311yescharacteristic
386C2×C4C23×#692311yescharacteristic
387C8—21no—
388C8—612no—
389C8—612no—
390C8—612no—
391C8—1221no—
392C8—1221no—
393C8—1222no—
394C8—1222no—
395C8—1221no—
396C8—2442no—
397C8—2442no—
398C8#1451986412yescomplemented
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—2442no—
405C22—2442no—
406C22—2442no—
407C22D8×#692312yes—
408C22C2×D4×#692311yescharacteristic
409C4—21no—
410C4—42no—
411C4—611no—
412C4—611no—
413C4—612no—
414C4—612no—
415C4—1221no—
416C4—2442no—
417C4C2×#1451986411yescharacteristic
418C4D61⋊#13111yescharacteristic
419C2—22no—
420C2—82no—
421C2—4882no—
422C2#2146254211yescharacteristic
423C2C2×D8×#692311yescharacteristic
424C2D61⋊#44311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#466C4
socleC22×C61C2×C4×D4
prime radical#21456393C2
cosocleC4×Dic61C24
hypercenter#466#6923
hypocenterC61#17956
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #21462542C2semidirect
2. #21461452C2semidirect
3. #21461451C2semidirect
4. #21461450C2semidirect
5. #21461400C2semidirect
6. #21461399C2semidirect
7. D8×#92420C2semidirect
8. #21461376C2semidirect
9. #21461164C2semidirect
10. C2×C8×#92420C2semidirect
118. C2×#92420D8semidirect
183. C8×C61#131semidirect
185. #92418#131semidirect
257. C61#17956semidirect
398. C8#14519864semidirect

Maximal subgroups

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

subgroupindex
#214625422
#214614522
#214614512
#214614502
#214614002
#214613992
D8×#924202
#214613762
#214611642
C2×C8×#924202
#214563932
#1795661