Finite Group Databasebeta

← #30599051

Subgroups — #30599051

Order 15616.

456 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461699C211yesmaximal, characteristic, complemented
2#21461695C211yesmaximal, characteristic, complemented
3#21461234C212yesmaximal
4#21460719C212yesmaximal, complemented
5#21454511C211yesmaximal, characteristic
6C2×#14519848—21no—
7C2×#14519845—21no—
8#16840733C2212yes—
9#16840704C2212yes—
10#16840638—21no—
11#16840636—21no—
12#16840616—22no—
13#16840613—22no—
14C22×C8×D61C2211yescharacteristic
15#16840269C2211yescharacteristic
16#16839831—21no—
17#16839831—21no—
18C8×C22⋊D61—21no—
19C8×C22⋊D61—21no—
20#16839790C411yescharacteristic, complemented
21#16839729C411yescharacteristic, complemented
22#16839725C2211yescharacteristic
23#16839717C411yescharacteristic, complemented
24C2×#131×C61C411yescharacteristic, complemented
25C2×#11758213—22no—
26C2×C4⋊#6923—22no—
27C2×C22⋊#6923—22no—
28#14519848—42no—
29#14519845—42no—
30#14519834—22no—
31C22⋊#92420—22no—
32(C2×#6923)⋊C4—22no—
33C22×C4×D61C2311yescharacteristic
34C2×#11758194C2×C411yescharacteristic
35C22×#92418C2×C411yescharacteristic
36C2×C8×D61—42no—
37C2×C8×D61D411yescharacteristic
38C2×C8×D61D411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61D411yescharacteristic
41#14519758—21no—
42#14519753D411yescharacteristic
43#14519753Q811yescharacteristic
44#14519748D411yescharacteristic
45#14519748Q811yescharacteristic
46#14519745—21no—
47C4×C22⋊D61—21no—
48C4×C22⋊D61—21no—
49#14519736C2×C411yescharacteristic
50#14519706—21no—
51#14519706—21no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54#14519689—21no—
55#14519689—21no—
56#14519687—21no—
57C8×Dic61—21no—
58C2×#41×C61C2×C411yescharacteristic
59#131×C61—42no—
60C2×M16×C61—21no—
61C2×M16×C61C2×C411yescharacteristic
62C22×C8×C61—21no—
63C22×C8×C61C2×C411yescharacteristic
64C23×D61#4111yescharacteristic
65C22×#6923—42no—
66#41×C61—41no—
67#41×C61—41no—
68C2×Q8×C61C4⋊C411yescharacteristic
69C2×D4×C61—21no—
70C2×D4×C61C4⋊C411yescharacteristic
71C22×C4×C61—21no—
72C22×C4×C61C22×C411yescharacteristic
73C2×C22⋊D61—21no—
74C2×C22⋊D61—21no—
75C22×Dic61#4111yescharacteristic
76C2×C4×D61—41no—
77C2×C4×D61—41no—
78C2×C4×D61C2×D411yescharacteristic
79C2×C4×D61C2×D411yescharacteristic
80C2×C4×D61C2×D411yescharacteristic
81C2×C4×D61C2×Q811yescharacteristic
82C22⋊#6923—82no—
83C2×#92420—42no—
84C4⋊#6923—42no—
85C4⋊#6923—42no—
86#11758213—42no—
87#11758213—42no—
88M16×C61—42no—
89M16×C61—41no—
90C2×C8×C61—22no—
91C2×C8×C61—21no—
92C2×C8×C61—42no—
93C2×C8×C61C22⋊C411yescharacteristic
94C2×C8×C61C22⋊C411yescharacteristic
95C22⋊Dic61—21no—
96C22⋊Dic61—21no—
97C22⋊Dic61—21no—
98#11758198—21no—
99#11758198—21no—
100C4⋊Dic61C4⋊C411yescharacteristic
101C4⋊Dic61C4⋊C411yescharacteristic
102#11758196—21no—
103#11758196—21no—
104C4×Dic61—21no—
105#11758194—41no—
106C2×#92418—21no—
107C2×#92418—41no—
108C2×#92418C22⋊C411yescharacteristic
109C2×#92418C22⋊C411yescharacteristic
110C8×D61—22no—
111C8×D61—22no—
112C8×D61—41no—
113C8×D61—41no—
114C23×C61—21no—
115C23×C61#11711yescharacteristic
116C22×D61—22no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61#12111yescharacteristic
120C22×D61C22⋊Q811yescharacteristic
121C2×#6923—82no—
122C2×#6923—82no—
123C2×#6923—82no—
124Q8×C61—41no—
125D4×C61—41no—
126D4×C61—42no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61C2×C22⋊C411yescharacteristic
133C2×C4×C61C2×C4⋊C411yescharacteristic
134C22⋊D61—42no—
135C22⋊D61—42no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—41no—
139C2×Dic61#12111yescharacteristic
140C2×Dic61C22⋊Q811yescharacteristic
141C4×D61—21no—
142C4×D61—21no—
143C4×D61—21no—
144C4×D61—21no—
145C4×D61—41no—
146C4×D61—41no—
147#92420—82no—
148C8×C61—21no—
149C8×C61—41no—
150C8×C61#9912yes—
151#92418—21no—
152#92418—21no—
153#92418—41no—
154#8629—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—41no—
159C22×C61—41no—
160C22×C61—41no—
161C22×C61#38011yescharacteristic
162C2×D61—21no—
163C2×D61—21no—
164C2×D61—21no—
165C2×D61—21no—
166C2×D61—21no—
167C2×D61—44no—
168C2×D61#47112yes—
169#6923—162no—
170C4×C61—21no—
171C4×C61—41no—
172C4×C61C2×#9911yescharacteristic
173C4×C61#41011yescharacteristic
174Dic61—21no—
175Dic61—21no—
176Dic61—41no—
177Dic61—41no—
178#2929—611no—
179#1810—611no—
180#1807—611no—
181#1513—612no—
182#1305—612no—
183C2×C61—41no—
184C2×C61—81no—
185C2×C61#179911yescharacteristic
186C2×C61#181411yescharacteristic
187C2×C61#181511yescharacteristic
188D61—22no—
189D61—41no—
190C2×#131#692311yescharacteristic, complemented
191C23×C8—611no—
192#525—611no—
193#435—1221no—
194#435—1221no—
195C8×D4—1221no—
196C8×D4—1221no—
197C2×#108—1221no—
198C2×#106—1221no—
199#424—611no—
200#408—611no—
201#407—611no—
202#407—612no—
203#405—611no—
204#380—612no—
205#341—1221no—
206#340—1221no—
207#328—1222no—
208#327—1222no—
209C61#862911yescharacteristic, complemented, Sylow
210C2×#41C2×#692311yescharacteristic
211C23×C4—611no—
212#131—42no—
213C2×M16—21no—
214C2×M16—611no—
215C2×M16—1222no—
216C2×M16C2×#692311yescharacteristic
217C22×C8—21no—
218C22×C8—611no—
219C22×C8—611no—
220C22×C8—611no—
221C22×C8—611no—
222C22×C8—611no—
223C22×C8—2442no—
224C22×C8C2×#692311yescharacteristic
225#123—1221no—
226C22⋊Q8—611no—
227C22⋊Q8—611no—
228#121—611no—
229#121—611no—
230#120—1221no—
231C4×D4—1221no—
232C4×D4—1221no—
233#117—611no—
234C2×C4⋊C4—1222no—
235C2×C22⋊C4—1222no—
236#108—2442no—
237#106—2442no—
238C4⋊C8—1221no—
239C4⋊C8—1221no—
240C22⋊C8—1221no—
241C22⋊C8—1221no—
242C22⋊C8—1221no—
243C22⋊C8—1221no—
244C22⋊C8—1222no—
245C22⋊C8—1222no—
246#97—1221no—
247C4×C8—1221no—
248#95—1222no—
249C24—611no—
250#41—41no—
251#41—41no—
252C2×Q8C4⋊#692311yescharacteristic
253C2×D4—21no—
254C2×D4—1221no—
255C2×D4—1221no—
256C2×D4C4⋊#692311yescharacteristic
257C22×C4—21no—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—611no—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4—2442no—
266C22×C4C22×#692311yescharacteristic
267M16—42no—
268M16—41no—
269M16—2442no—
270M16—2442no—
271M16—2441no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—42no—
275C2×C8—611no—
276C2×C8—611no—
277C2×C8—1222no—
278C2×C8—1222no—
279C2×C8—1221no—
280C2×C8—2441no—
281C2×C8—2441no—
282C2×C8—2441no—
283C2×C8—2442no—
284C2×C8C22⋊#692311yescharacteristic
285C2×C8C22⋊#692311yescharacteristic
286C4⋊C4—611no—
287C4⋊C4—611no—
288C4⋊C4—1221no—
289C4⋊C4—1221no—
290C4⋊C4—2442no—
291C4⋊C4—2442no—
292C22⋊C4—1221no—
293C22⋊C4—1221no—
294C22⋊C4—1221no—
295C22⋊C4—1221no—
296C22⋊C4—1221no—
297C22⋊C4—4882no—
298C42—1221no—
299C23—21no—
300C23—611no—
301C23—611no—
302C23—1222no—
303C23—2441no—
304C23—2441no—
305C23#1451985611yescharacteristic
306Q8—41no—
307D4—41no—
308D4—42no—
309D4—2442no—
310D4—2442no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—611no—
317C2×C4—611no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—1221no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—4882no—
325C2×C4—4882no—
326C2×C4—4882no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4C2×C22⋊#692311yescharacteristic
331C2×C4C2×C4⋊#692311yescharacteristic
332C8—21no—
333C8—41no—
334C8—1221no—
335C8—1221no—
336C8—4882no—
337C8—2441no—
338C8#1451983512yes—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—612no—
346C22—1221no—
347C22—1221no—
348C22—1221no—
349C22—1221no—
350C22—1221no—
351C22—2444no—
352C22#1684070411yescharacteristic
353C4—21no—
354C4—41no—
355C4—1221no—
356C4—1221no—
357C4—9762no—
358C4—2441no—
359C4—2441no—
360C4C2×#1451983511yescharacteristic
361C4#1684074111yescharacteristic
362C2—41no—
363C2—81no—
364C2—1222no—
365C2—2441no—
366C2#2146167911yescharacteristic
367C2#2146170611yescharacteristic
368C2#2146170711yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#131×C61C4
socleC22×C61#380
prime radical#16840269C22
cosocleC22×C4×D61C23
hypercenterC2×#131#6923
hypocenterC61#8629
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. #21461699C2semidirect
2. #21461695C2semidirect
4. #21460719C2semidirect
20. #16839790C4semidirect
21. #16839729C4semidirect
23. #16839717C4semidirect
24. C2×#131×C61C4semidirect
190. C2×#131#6923semidirect
209. C61#8629semidirect

Maximal subgroups

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

subgroupindex
#214616992
#214616952
#214612342
#214607192
#214545112
#862961