Finite Group Databasebeta

← #30600468

Subgroups — #30600468

Order 15616.

436 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 80 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#21462720C211yesmaximal, characteristic
2#21461591C211yesmaximal, characteristic
3#21461292C211yesmaximal, characteristic
4#21461202C211yesmaximal, characteristic
5#21460712C211yesmaximal, characteristic
6#21460671C211yesmaximal, characteristic
7#21455948C211yesmaximal, characteristic
8#16840814—22no—
9#16840813—21no—
10#16840811—21no—
11#16840785—21no—
12#16840784C2211yescharacteristic
13#16840781C2211yescharacteristic
14#16840780—21no—
15C2×#14519840—21no—
16#16840721C2211yescharacteristic
17C4×C4⋊#6923C2211yescharacteristic
18#16840635—21no—
19#16840619—22no—
20#16840615—21no—
21#16840615—21no—
22#16840317C2211yescharacteristic
23C4×Q8×D61C2211yescharacteristic
24#16840091—21no—
25#16840048C411yescharacteristic, complemented
26#16840047C411yescharacteristic, complemented
27#16839969C411yescharacteristic, complemented
28#16839916—21no—
29#16839869—21no—
30#16839866—21no—
31#16839818C2211yescharacteristic
32C61×#462C411yescharacteristic, complemented
33C2×D61⋊C8—21no—
34C2×C4⋊#6923—21no—
35C2×C4×#6923—21no—
36#14519844—22no—
37#14519844—22no—
38C4⋊#92420—21no—
39C4⋊#92420—21no—
40#14519840—24no—
41#14519834—21no—
42#14519834D411yescharacteristic
43#14519834D411yescharacteristic
44#14519832—22no—
45#14519832—21no—
46#14519832—21no—
47#14519832D412yes—
48C4×#92420—22no—
49C4×#92420—21no—
50(C2×#6923)⋊C4—21no—
51C2×Q8×D61D411yescharacteristic
52C2×#11758202—21no—
53#14519777C2×C411yescharacteristic
54#14519768—21no—
55#14519757—21no—
56#14519753—21no—
57Q8×Dic61—21no—
58Q8×Dic61D411yescharacteristic
59C4×Dic122—21no—
60C4×Dic122C2×C411yescharacteristic
61#14519742—21no—
62#14519742—21no—
63#14519742D411yescharacteristic
64#14519735—21no—
65C4⋊C4×D61—21no—
66C4⋊C4×D61D411yescharacteristic
67C42×D61—21no—
68C42×D61C2311yescharacteristic
69#14519709—21no—
70#14519708—21no—
71C4⋊#92418C2×C411yescharacteristic
72#14519702—21no—
73#14519701—21no—
74#14519701—21no—
75#14519689—21no—
76#14519687—21no—
77C2×Q16×C61—21no—
78C4⋊Q8×C61C2×C411yescharacteristic
79C4×Q8×C61C2×C411yescharacteristic
80C4⋊C8×C61C2×C411yescharacteristic
81#103×C61—21no—
82C22×#6923—41no—
83C2×Q8×C61—21no—
84C2×Q8×C61C22⋊C411yescharacteristic
85Q8×D61—22no—
86Q8×D61—41no—
87C2×C4×D61—21no—
88C2×C4×D61—21no—
89C2×C4×D61C2×D411yescharacteristic
90C2×C4×D61C2×D411yescharacteristic
91C2×C4×D61#4111yescharacteristic
92C2×Dic122—21no—
93C2×Dic122—21no—
94C2×Dic122C22⋊C411yescharacteristic
95C2×#92420—21no—
96C2×#92420—21no—
97C2×#92420—22no—
98C2×#92420—41no—
99C4⋊#6923—22no—
100C4⋊#6923—41no—
101C4×#6923—81no—
102D61⋊C8—22no—
103D61⋊C8—41no—
104Q16×C61—42no—
105C2×C8×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61C22⋊C411yescharacteristic
110C42×C61—21no—
111C42×C61C22×C411yescharacteristic
112#11758202—42no—
113#11758198—41no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—41no—
118C4⋊Dic61C22⋊C411yescharacteristic
119#11758196—21no—
120#11758196—21no—
121#11758196—21no—
122#11758196—41no—
123C4×Dic61—21no—
124C4×Dic61—21no—
125C4×Dic61—21no—
126C4×Dic61—41no—
127C4×Dic61C2×D411yescharacteristic
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61#4111yescharacteristic
130C2×#92418—21no—
131C22×D61#12111yescharacteristic
132C2×#6923—42no—
133C2×#6923—81no—
134C2×#6923—81no—
135Q8×C61—21no—
136Q8×C61—21no—
137Q8×C61—41no—
138Q8×C61—41no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—41no—
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×C4×C61C4×D411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—41no—
148C2×Dic61#12011yescharacteristic
149C2×Dic61#12111yescharacteristic
150C2×Dic61#12311yescharacteristic
151C4×D61—21no—
152C4×D61—21no—
153C4×D61—21no—
154C4×D61—21no—
155C4×D61—41no—
156C4×D61—42no—
157C4×D61—41no—
158C4×D61—41no—
159Dic122—21no—
160Dic122—42no—
161Dic122—81no—
162Dic122#10412yes—
163#92420—42no—
164#92420—41no—
165#92420—41no—
166#92420—42no—
167C8×C61—41no—
168#92418—41no—
169#9166—611nomaximal, Sylow
170C22×C61#38611yescharacteristic
171C2×D61—21no—
172C2×D61—21no—
173C2×D61#47111yescharacteristic
174C2×D61#47311yescharacteristic
175#6923—82no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—41no—
179C4×C61—41no—
180C4×C61—81no—
181C4×C61C2×#10411yescharacteristic
182C4×C61#42111yescharacteristic
183Dic61—21no—
184Dic61—22no—
185Dic61—21no—
186Dic61—21no—
187Dic61—41no—
188Dic61—81no—
189#3113—611no—
190#2984—611no—
191#1763—611no—
192#1531—611no—
193#1503—611no—
194#1302—611no—
195#1285—611no—
196C2×C61#187111yescharacteristic
197C2×C61#190511yescharacteristic
198C2×C61#190811yescharacteristic
199D61—21no—
200D61—21no—
201#533—611no—
202C2×C4×Q8—611no—
203#483—1221no—
204#475—611no—
205#475—611no—
206#462—611no—
207#462#692311yescharacteristic, complemented
208#451—1221no—
209#446—1222no—
210#446—1221no—
211C8×Q8—1221no—
212#441—1221no—
213#441—1221no—
214#432—611no—
215#432—611no—
216#432—1221no—
217#431—611no—
218#431—1221no—
219C2×#103—1221no—
220C2×#97—611no—
221C4×C4⋊C4—611no—
222#340—1221no—
223#330—1222no—
224#328—1221no—
225#328—1221no—
226C61#916611yescharacteristic, complemented, Sylow
227C22×Q8—611no—
228C2×Q16—21no—
229C2×Q16—1221no—
230C22×C8—1221no—
231C4⋊Q8—611no—
232C4⋊Q8C2×#692311yescharacteristic
233#125—1221no—
234C22⋊Q8—1221no—
235C22⋊Q8—1221no—
236C4×Q8—611no—
237C4×Q8—611no—
238C4×Q8—611no—
239C4×Q8—1221no—
240C4×Q8—1221no—
241C4×Q8—1221no—
242C4×Q8—1221no—
243C4×Q8C2×#692311yescharacteristic
244#117—1221no—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C42—611no—
249C2×C42—1221no—
250C2×C42—1221no—
251#106—1221no—
252#106—1221no—
253C4⋊C8—611no—
254C4⋊C8—1221no—
255C4⋊C8—1221no—
256C4⋊C8—1222no—
257C4⋊C8—1222no—
258C4⋊C8C2×#692311yescharacteristic
259#103—21no—
260#103—1221no—
261#103—1221no—
262#103—1224no—
263#103—1221no—
264C22⋊C8—611no—
265C22⋊C8—611no—
266C22⋊C8—1221no—
267C22⋊C8—1221no—
268#97—612no—
269#97—1221no—
270#97—1221no—
271#97—1221no—
272#97—1222no—
273C4×C8—1221no—
274C4×C8—1222no—
275#95—1221no—
276C2×Q8—21no—
277C2×Q8—611no—
278C2×Q8—1221no—
279C2×Q8—1222no—
280C2×Q8—1221no—
281C2×Q8—2441no—
282C2×Q8C22⋊#692311yescharacteristic
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—611no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—2441no—
289Q16—42no—
290Q16—2442no—
291C2×C8—21no—
292C2×C8—1221no—
293C2×C8—1222no—
294C2×C8—1221no—
295C2×C8—1222no—
296C2×C8—1221no—
297C2×C8—2441no—
298C2×C8—2441no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—611no—
303C4⋊C4—1221no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4—1221no—
307C4⋊C4—1222no—
308C4⋊C4—1221no—
309C4⋊C4—1221no—
310C4⋊C4—2441no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C4⋊C4C22⋊#692311yescharacteristic
314C22⋊C4—2441no—
315C42—21no—
316C42—611no—
317C42—611no—
318C42—611no—
319C42—1221no—
320C42—1221no—
321C42—1221no—
322C42—4881no—
323C42—2441no—
324C42C22×#692311yescharacteristic
325C23—611no—
326Q8—21no—
327Q8—21no—
328Q8—41no—
329Q8—41no—
330Q8—612no—
331Q8—1221no—
332Q8—4881no—
333Q8—2442no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—611no—
338C2×C4—611no—
339C2×C4—611no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—2441no—
349C2×C4—2442no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2442no—
353C2×C4—2441no—
354C2×C4C2×C22⋊#692311yescharacteristic
355C2×C4D4×#692311yescharacteristic
356C2×C4D4×#692311yescharacteristic
357C8—41no—
358C8—2441no—
359C8—2442no—
360C8—2441no—
361C8—2442no—
362C8—2441no—
363C22—611no—
364C22—611no—
365C22—1221no—
366C22—1221no—
367C22#1684071011yescharacteristic
368C4—21no—
369C4—21no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—1221no—
374C4—1222no—
375C4—1221no—
376C4—1221no—
377C4—4882no—
378C4—4881no—
379C4—2441no—
380C4D61⋊#10411yescharacteristic
381C4#1684076011yescharacteristic
382C2—1221no—
383C2—1221no—
384C2#2146180311yescharacteristic
385C2#2146185411yescharacteristic
386C2#2146185911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#462C4
socleC22×C61#386
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#462#6923
hypocenterC61#9166
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
25. #16840048C4semidirect
26. #16840047C4semidirect
27. #16839969C4semidirect
32. C61×#462C4semidirect
207. #462#6923semidirect
226. C61#9166semidirect

Maximal subgroups

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

subgroupindex
#214627202
#214615912
#214612922
#214612022
#214607122
#214606712
#214559482
#916661