Finite Group Databasebeta

← #30596940

Subgroups — #30596940

Order 15616.

788 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462498C211yesmaximal, characteristic
2C4×#14519856C211yesmaximal, characteristic
3#21460700C214yesmaximal
4#21448158C211yesmaximal, characteristic
5C2×#14519856—21no—
6#16840785—21no—
7#16840782—21no—
8#16840780—21no—
9#16840779—21no—
10#16840767C412yescomplemented
11#16840733C2212yes—
12C4×#11758213C412yescomplemented
13C4×C4⋊#6923—21no—
14C4×C4⋊#6923C412yescomplemented
15C4×C22⋊#6923—21no—
16C4×C22⋊#6923C2212yes—
17#16840698C412yescomplemented
18C42×#6923—21no—
19C2×#14519734—21no—
20C2×C42×D61C2211yescharacteristic
21#16839613—21no—
22#16839613C412yescomplemented
23#16839609—21no—
24#16839609C2212yes—
25#16839608C411yescharacteristic, complemented
26C42×Dic61—21no—
27C61×#376C411yescharacteristic, complemented
28C2×#11758213C2×C414yes—
29#14519856—28no—
30C2×C4⋊#6923—22no—
31C2×C22⋊#6923—22no—
32C2×C4×#6923—22no—
33C2×C4×#6923—22no—
34C2×C4×#6923—22no—
35#14519844—22no—
36#14519844—22no—
37#14519834—22no—
38C22⋊#92420—22no—
39#14519832—22no—
40C4×#92420—22no—
41(C2×#6923)⋊C4—22no—
42(C2×#6923)⋊C4C2×C414yes—
43C22×C4×D61D411yescharacteristic
44C22×C4×D61D411yescharacteristic
45C22×C4×D61C2311yescharacteristic
46#14519734—28no—
47C2×#11758196—22no—
48C2×#11758198—22no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61—22no—
52C2×C4×Dic61C2×C411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C42×D61C2×C411yescharacteristic
56C42×D61C2×C411yescharacteristic
57C42×D61C2×C411yescharacteristic
58C42×D61C2×C411yescharacteristic
59C42×D61D411yescharacteristic
60C42×D61D411yescharacteristic
61C42×D61Q811yescharacteristic
62C42×D61Q811yescharacteristic
63#14519684—21no—
64#14519684C2×C412yes—
65C2×C42×C61—21no—
66C2×C42×C61C2×C411yescharacteristic
67#95×C61—21no—
68#95×C61C2×C412yes—
69C23×D61C2×D411yescharacteristic
70C22×#6923—24no—
71C22×C4×C61—22no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22×C411yescharacteristic
75C22×Dic61—22no—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C2×C4×D61—24no—
80C2×C4×D61—24no—
81C2×C4×D61—42no—
82C2×C4×D61C22⋊C411yescharacteristic
83C2×C4×D61C22⋊C411yescharacteristic
84C2×C4×D61C22⋊C411yescharacteristic
85C2×C4×D61C22⋊C411yescharacteristic
86C2×C4×D61C4⋊C411yescharacteristic
87C2×C4×D61C4⋊C411yescharacteristic
88C2×C4×D61C4⋊C411yescharacteristic
89C2×C4×D61C4⋊C411yescharacteristic
90C2×C4×D61C22×C411yescharacteristic
91C2×C4×D61C22×C411yescharacteristic
92C2×C4×D61C2×D411yescharacteristic
93C2×C4×D61C2×Q811yescharacteristic
94C22⋊#6923—28no—
95C2×#92420—24no—
96C4⋊#6923—28no—
97C4×#6923—28no—
98C4×#6923—44no—
99C4×#6923—44no—
100#11758213—44no—
101C42×C61—24no—
102C42×C61C4211yescharacteristic
103C42×C61C4211yescharacteristic
104C42×C61C4⋊C411yescharacteristic
105C42×C61C4⋊C411yescharacteristic
106#11758198—28no—
107#11758196—28no—
108C4×Dic61—24no—
109C4×Dic61—44no—
110C4×Dic61—44no—
111C4×Dic61C4211yescharacteristic
112C4×Dic61C4211yescharacteristic
113C4×Dic61C22⋊C411yescharacteristic
114C4×Dic61C22⋊C411yescharacteristic
115C4×Dic61C22⋊C411yescharacteristic
116C4×Dic61C22⋊C411yescharacteristic
117C4×Dic61C4⋊C411yescharacteristic
118C4×Dic61C4⋊C411yescharacteristic
119C4×Dic61C4⋊C411yescharacteristic
120C4×Dic61C4⋊C411yescharacteristic
121C4×Dic61C4⋊C411yescharacteristic
122C4×Dic61C4⋊C411yescharacteristic
123C23×C61C2×C22⋊C411yescharacteristic
124C22×D61—22no—
125C22×D61—22no—
126C22×D61—41no—
127C22×D61C2×C22⋊C411yescharacteristic
128C22×D61C2×C4⋊C411yescharacteristic
129C2×#6923—28no—
130C2×#6923—44no—
131C2×#6923—44no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—24no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61#9511yescharacteristic
139C2×C4×C61#9511yescharacteristic
140C2×C4×C61#9511yescharacteristic
141C2×C4×C61#9511yescharacteristic
142C2×C4×C61C2×C4211yescharacteristic
143C2×C4×C61C2×C4⋊C411yescharacteristic
144C2×Dic61—22no—
145C2×Dic61—22no—
146C2×Dic61—24no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61#9511yescharacteristic
150C2×Dic61#9511yescharacteristic
151C2×Dic61#9511yescharacteristic
152C2×Dic61#9511yescharacteristic
153C2×Dic61#10414yes—
154C2×Dic61C2×C22⋊C411yescharacteristic
155C2×Dic61C2×C4⋊C411yescharacteristic
156C4×D61—22no—
157C4×D61—22no—
158C4×D61—22no—
159C4×D61—22no—
160C4×D61—22no—
161C4×D61—22no—
162C4×D61—22no—
163C4×D61—22no—
164C4×D61—22no—
165C4×D61—41no—
166C4×D61—41no—
167C4×D61#10414yes—
168#92420—44no—
169#8048—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61C2×#9511yescharacteristic
173C22×C61C2×#10412yes—
174C2×D61—21no—
175C2×D61—21no—
176C2×D61—21no—
177C2×D61—21no—
178C2×D61—21no—
179C2×D61—44no—
180C2×D61C2×#10412yes—
181#6923—48no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—44no—
188C4×C61#33912yes—
189Dic61—21no—
190Dic61—21no—
191Dic61—44no—
192Dic61#33912yes—
193Dic61#33912yes—
194Dic61#33912yes—
195Dic61#41012yes—
196#2920—611no—
197C4×#117—611no—
198C4×#117—611no—
199#1297—614no—
200C2×C61—22no—
201C2×C61C2×#33911yescharacteristic
202C2×C61C2×#33911yescharacteristic
203C2×C61#174011yescharacteristic
204D61—22no—
205D61—41no—
206C2×#117—1221no—
207C2×#117—1221no—
208C22×C42—611no—
209#432—1221no—
210#432—1221no—
211#431—1221no—
212#431—1221no—
213#423—612no—
214#407—612no—
215C4×M16—612no—
216C4×C4⋊C4—612no—
217C4×C4⋊C4—612no—
218C4×C4⋊C4—1221no—
219C4×C4⋊C4—1221no—
220C4×C22⋊C4—612no—
221C4×C22⋊C4—612no—
222C4×C22⋊C4—1221no—
223C4×C22⋊C4—1221no—
224#376—612no—
225#376—611no—
226#376#692311yescharacteristic, complemented
227C43—1221no—
228C43—1221no—
229C61#804811yescharacteristic, complemented, Sylow
230C23×C4—611no—
231C23×C4—611no—
232C23×C4—611no—
233C2×M16—614no—
234#117—1228no—
235#117—1228no—
236C2×C4⋊C4—1222no—
237C2×C4⋊C4—1222no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4—1222no—
240C2×C42—21no—
241C2×C42—611no—
242C2×C42—611no—
243C2×C42—611no—
244C2×C42—611no—
245C2×C42—611no—
246C2×C42—611no—
247C2×C42—611no—
248C2×C42—611no—
249C2×C42—611no—
250C2×C42—611no—
251C2×C42—611no—
252C2×C42—1222no—
253C2×C42—1221no—
254C2×C42—1222no—
255C2×C42—1222no—
256C2×C42—1222no—
257C2×C42—1222no—
258C2×C42C2×#692311yescharacteristic
259C4⋊C8—1222no—
260C4⋊C8—1222no—
261C22⋊C8—1222no—
262C22⋊C8—1222no—
263#97—1222no—
264C4×C8—1222no—
265#95—21no—
266#95—614no—
267#95—612no—
268#95—1222no—
269#95—1221no—
270#95C2×#692312yes—
271C24—611no—
272C22×C4—22no—
273C22×C4—611no—
274C22×C4—611no—
275C22×C4—611no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—611no—
279C22×C4—611no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—611no—
286C22×C4—611no—
287C22×C4—611no—
288C22×C4—1224no—
289C22×C4—1224no—
290C22×C4—1224no—
291C22×C4—1222no—
292C22×C4—2442no—
293C22×C4C22⋊#692311yescharacteristic
294C22×C4C22⋊#692311yescharacteristic
295C22×C4C22×#692311yescharacteristic
296M16—2444no—
297C2×C8—1224no—
298C4⋊C4—1228no—
299C4⋊C4—1228no—
300C22⋊C4—1228no—
301C22⋊C4—1228no—
302C42—24no—
303C42—611no—
304C42—611no—
305C42—611no—
306C42—611no—
307C42—611no—
308C42—611no—
309C42—611no—
310C42—611no—
311C42—611no—
312C42—611no—
313C42—611no—
314C42—611no—
315C42—1224no—
316C42—1228no—
317C42—2444no—
318C42—2444no—
319C42—2444no—
320C42—2444no—
321C42C4×#692311yescharacteristic
322C42C4×#692311yescharacteristic
323C42C4⋊#692311yescharacteristic
324C42C4⋊#692311yescharacteristic
325C23—611no—
326C23—611no—
327C23—1222no—
328C23—1222no—
329C23—2441no—
330C23C2×C22⋊#692311yescharacteristic
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—614no—
338C2×C4—611no—
339C2×C4—611no—
340C2×C4—611no—
341C2×C4—611no—
342C2×C4—614no—
343C2×C4—611no—
344C2×C4—611no—
345C2×C4—1222no—
346C2×C4—1222no—
347C2×C4—1222no—
348C2×C4—1222no—
349C2×C4—1222no—
350C2×C4—1222no—
351C2×C4—1222no—
352C2×C4—1222no—
353C2×C4—1222no—
354C2×C4—1222no—
355C2×C4—1222no—
356C2×C4—1224no—
357C2×C4—1228no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2442no—
361C2×C4—2444no—
362C2×C4—2444no—
363C2×C4—2442no—
364C2×C4(C2×#6923)⋊C411yescharacteristic
365C2×C4(C2×#6923)⋊C411yescharacteristic
366C2×C4(C2×#6923)⋊C411yescharacteristic
367C2×C4(C2×#6923)⋊C411yescharacteristic
368C2×C4C2×C4×#692311yescharacteristic
369C2×C4C2×C4⋊#692311yescharacteristic
370C8—2444no—
371C22—21no—
372C22—21no—
373C22—612no—
374C22—1221no—
375C22—1221no—
376C22—1221no—
377C22—1221no—
378C22—1221no—
379C22—2444no—
380C22C2×(C2×#6923)⋊C411yescharacteristic
381C22D61⋊#10412yes—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—44no—
388C4—612no—
389C4—612no—
390C4—612no—
391C4—612no—
392C4—1221no—
393C4—1221no—
394C4—2444no—
395C4—2448no—
396C4#1684063312yes—
397C2—22no—
398C2—1222no—
399C2—2441no—
400C2#2146149511yescharacteristic
401C2C2×#1684063311yescharacteristic
402C2#2146151711yescharacteristic
subgroupG/H
centerC2×C4(C2×#6923)⋊C4
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#376C4
socleC22×C61C2×#95
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#376#6923
hypocenterC61#8048
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
10. #16840767C4semidirect
12. C4×#11758213C4semidirect
14. C4×C4⋊#6923C4semidirect
17. #16840698C4semidirect
22. #16839613C4semidirect
25. #16839608C4semidirect
27. C61×#376C4semidirect
226. #376#6923semidirect
229. C61#8048semidirect

Maximal subgroups

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

subgroupindex
#214624982
C4×#145198562
#214607002
#214481582
#804861