Finite Group Databasebeta

← #30601166

Subgroups — #30601166

Order 15616.

738 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462669C211yesmaximal, characteristic
2#21461682C211yesmaximal, characteristic
3#21461658C212yesmaximal, complemented
4#21460681C212yesmaximal, complemented
5#21449179C211yesmaximal, characteristic
6#16840790—24no—
7#16840778—21no—
8C2×#14519840—24no—
9C2×#14519836—21no—
10D61⋊(C22⋊C8)—21no—
11D61⋊(C22⋊C8)C2212yes—
12#16840704C2212yes—
13#16840640—21no—
14C22×Q8×D61C2211yescharacteristic
15C2×#14519755—21no—
16C2×#14519739C2211yescharacteristic
17#16839682C411yescharacteristic, complemented
18#16839675C411yescharacteristic, complemented
19#16839673C411yescharacteristic, complemented
20#16839666—21no—
21#16839640—21no—
22#16839633—21no—
23#16839621C2211yescharacteristic
24C61×#395C411yescharacteristic, complemented
25C2×D61⋊C8—22no—
26C2×D61⋊C8D414yes—
27C2×C4⋊#6923—22no—
28C2×C22⋊#6923—22no—
29C4⋊#92420—22no—
30#14519840—48no—
31#14519836—42no—
32#14519834—22no—
33#14519834—22no—
34C22⋊#92420—22no—
35C4×#92420—22no—
36(C2×#6923)⋊C4—22no—
37C22×Dic122—21no—
38C22×Dic122C2×C411yescharacteristic
39C2×Q8×D61—22no—
40C2×Q8×D61—22no—
41C2×Q8×D61—44no—
42C22×D244D411yescharacteristic
43C22×C4×D61—21no—
44C22×C4×D61D411yescharacteristic
45C22×C4×D61C2311yescharacteristic
46#14519755—44no—
47#14519739—22no—
48#14519739—22no—
49C2×#11758196—21no—
50C2×#11758196—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198C2×C411yescharacteristic
55C2×#11758198D411yescharacteristic
56C2×C4×Dic61D411yescharacteristic
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684C2×C411yescharacteristic
61C22×Q8×C61C2×C411yescharacteristic
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61C2×C411yescharacteristic
64#95×C61—21no—
65#95×C61C2×C411yescharacteristic
66C23×D61C22⋊C411yescharacteristic
67C23×D61C2×D411yescharacteristic
68C22×#6923—42no—
69C2×Q8×C61—22no—
70C2×Q8×C61—42no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22×C411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C2×D411yescharacteristic
80Q8×D61—28no—
81Q8×D61—44no—
82Q8×D61—44no—
83Q8×D61—44no—
84C2×D244—22no—
85C2×D244—22no—
86C2×D244SD1612yes—
87C2×D244Q1612yes—
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—42no—
91C2×C4×D61—42no—
92C2×C4×D61—42no—
93C2×C4×D61—42no—
94C2×C4×D61—42no—
95C2×C4×D61C2×D412yes—
96C2×C4×D61#4112yes—
97C2×Dic122—22no—
98C2×Dic122—24no—
99C2×Dic122—42no—
100C2×Dic122—42no—
101C2×Dic122—41no—
102C2×Dic122—41no—
103C22⋊#6923—42no—
104C22⋊#6923—42no—
105C2×#92420—24no—
106C4⋊#6923—44no—
107D61⋊C8—44no—
108D61⋊C8—82no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—42no—
111#11758198—22no—
112#11758198—22no—
113#11758198—42no—
114#11758198—42no—
115#11758198—42no—
116#11758196—42no—
117#11758196—42no—
118C4×Dic61—21no—
119C4×Dic61—21no—
120C23×C61C2×C22⋊C411yescharacteristic
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61#10312yescomplemented
128C22×D61#12112yes—
129C22×D61C2×SD1611yescharacteristic
130C22×D61C2×Q1611yescharacteristic
131C22×D61#13512yes—
132C2×#6923—82no—
133C2×#6923—82no—
134C2×#6923—82no—
135Q8×C61—24no—
136Q8×C61—42no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—22no—
140C2×C4×C61—21no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61#10312yescomplemented
149C2×C4×C61C4×D412yes—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—21no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61#12011yescharacteristic
160C2×Dic61#12311yescharacteristic
161D244—24no—
162D244—24no—
163C4×D61—24no—
164C4×D61—42no—
165C4×D61—48no—
166C4×D61—42no—
167C4×D61—81no—
168Dic122—24no—
169Dic122—44no—
170Dic122—42no—
171Dic122—44no—
172#92420—44no—
173#9467—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61#38611yescharacteristic
177C22×C61C2×#10311yescharacteristic
178C22×C61#41611yescharacteristic
179C2×D61—22no—
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—41no—
183C2×D61—41no—
184C2×D61—42no—
185C2×D61—41no—
186C2×D61—41no—
187C2×D61—42no—
188C2×D61—81no—
189C2×D61#46512yes—
190C2×D61#46712yes—
191#6923—162no—
192C4×C61—22no—
193C4×C61—22no—
194C4×C61—42no—
195C4×C61—41no—
196C4×C61—81no—
197Dic61—21no—
198Dic61—42no—
199Dic61—41no—
200Dic61—81no—
201Dic61B(4,𝔽2)12yes—
202#2972—611no—
203#2409—611no—
204#1799—611no—
205#1787—612no—
206#1289—612no—
207C2×C61—21no—
208C2×C61—21no—
209C2×C61#188911yescharacteristic
210C2×C61#189211yescharacteristic
211C2×C61#190911yescharacteristic
212D61—22no—
213D61—22no—
214D61—42no—
215C23×Q8—611no—
216C2×C22⋊Q8—1221no—
217C2×C4×D4—611no—
218C8×D4—1224no—
219#431—1221no—
220C2×#103—1224no—
221C2×#99—1221no—
222C2×C22⋊C8—612no—
223C2×C22⋊C8—1221no—
224#395—611no—
225#395#692311yescharacteristic, complemented
226#393—611no—
227#393—611no—
228#391—1221no—
229#386—1221no—
230#385—1221no—
231#380—611no—
232#380—612no—
233#342—1221no—
234C61#946711yescharacteristic, complemented, Sylow
235C22×Q8—611no—
236C22×Q8—1221no—
237C22×Q8—1222no—
238C22×Q8—1222no—
239C22×Q8—2444no—
240C22×Q8C2×#692311yescharacteristic
241C22×D4—611no—
242C23×C4—611no—
243C23×C4—611no—
244C23×C4—1221no—
245C22×C8—614no—
246C22×C8—1222no—
247C22⋊Q8—2444no—
248C4×D4—1222no—
249C4×D4—1222no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1222no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4C2×#692311yescharacteristic
255C2×C22⋊C4—611no—
256C2×C22⋊C4—611no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1222no—
261C2×C42—611no—
262C4⋊C8—1222no—
263#103—2448no—
264#99—2442no—
265C22⋊C8—1222no—
266C22⋊C8—1222no—
267C22⋊C8—1222no—
268C4×C8—1222no—
269#95—21no—
270#95—611no—
271#95—1221no—
272#95—1221no—
273#95—1222no—
274#95—1221no—
275#95C2×#692311yescharacteristic
276C24—611no—
277C24—611no—
278C2×Q8—22no—
279C2×Q8—42no—
280C2×Q8—1224no—
281C2×Q8—1228no—
282C2×Q8—1222no—
283C2×Q8—2441no—
284C2×Q8—2441no—
285C2×Q8—2442no—
286C2×Q8—2442no—
287C2×Q8—2444no—
288C2×Q8—2444no—
289C2×Q8—2444no—
290C2×D4—612no—
291C2×D4—612no—
292C2×D4—1222no—
293C2×D4—1222no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—611no—
297C22×C4—612no—
298C22×C4—612no—
299C22×C4—611no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—2442no—
305C22×C4—2442no—
306C22×C4—2442no—
307C22×C4—2442no—
308C22×C4—2442no—
309C22×C4—2442no—
310C22×C4C22⋊#692311yescharacteristic
311C22×C4C22⋊#692311yescharacteristic
312C22×C4C22×#692311yescharacteristic
313C2×C8—1224no—
314C2×C8—4882no—
315C2×C8—2444no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—2442no—
319C4⋊C4—2442no—
320C4⋊C4—2444no—
321C22⋊C4—1222no—
322C22⋊C4—1222no—
323C22⋊C4—2442no—
324C22⋊C4—2442no—
325C22⋊C4—2442no—
326C22⋊C4—2442no—
327C22⋊C4—2442no—
328C42—1221no—
329C42—1221no—
330C23—612no—
331C23—611no—
332C23—611no—
333C23—612no—
334C23—612no—
335C23—1221no—
336C23—1221no—
337C23—2441no—
338C23—2441no—
339C23—2441no—
340C23—2441no—
341C23C2×C22⋊#692311yescharacteristic
342Q8—24no—
343Q8—42no—
344Q8—1224no—
345Q8—2442no—
346Q8—2444no—
347Q8—2444no—
348D4—1224no—
349D4—1224no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—611no—
362C2×C4—611no—
363C2×C4—1224no—
364C2×C4—1222no—
365C2×C4—1222no—
366C2×C4—1221no—
367C2×C4—4881no—
368C2×C4—4882no—
369C2×C4—4882no—
370C2×C4—4882no—
371C2×C4—2442no—
372C2×C4—2442no—
373C2×C4—2448no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4#1451984012yescomplemented
381C2×C4D4×#692312yes—
382C8—2444no—
383C22—21no—
384C22—21no—
385C22—612no—
386C22—612no—
387C22—1222no—
388C22—1222no—
389C22—1222no—
390C22—4881no—
391C22—2442no—
392C22—2441no—
393C22—2441no—
394C22—2442no—
395C22—2441no—
396C22—2441no—
397C22#1684071011yescharacteristic
398C22C2×#1451984011yescharacteristic
399C22#1684075211yescharacteristic
400C4—22no—
401C4—22no—
402C4—42no—
403C4—41no—
404C4—81no—
405C4—612no—
406C4—1221no—
407C4—9762no—
408C4—4881no—
409C4—2442no—
410C4—2441no—
411C2—21no—
412C2—21no—
413C2—1222no—
414C2—1222no—
415C2—2442no—
416C2#2146182811yescharacteristic
417C2#2146183211yescharacteristic
418C2#2146186111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#395C4
socleC22×C61#386
prime radicalC22×D244D4
cosocleC22×C4×D61C23
hypercenter#395#6923
hypocenterC61#9467
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
3. #21461658C2semidirect
4. #21460681C2semidirect
17. #16839682C4semidirect
18. #16839675C4semidirect
19. #16839673C4semidirect
24. C61×#395C4semidirect
127. C22×D61#103semidirect
148. C2×C4×C61#103semidirect
225. #395#6923semidirect
234. C61#9467semidirect
380. C2×C4#14519840semidirect

Maximal subgroups

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

subgroupindex
#214626692
#214616822
#214616582
#214606812
#214491792
#946761