Finite Group Databasebeta

← #30604875

Subgroups — #30604875

Order 15616.

442 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 68 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#21461566C211yesmaximal, characteristic
2#21461057C211yesmaximal, characteristic
3#21460815C212yesmaximal, complemented
4#21460799C212yesmaximal, complemented
5#21451692C211yesmaximal, characteristic
6C2×#14519835—21no—
7C2×C22⋊#92420—21no—
8C4×C22⋊#6923—21no—
9#16840640C2212yes—
10#16840608—21no—
11#16840606C2212yes—
12C2×#14519768C411yescharacteristic, complemented
13C2×#14519737C2211yescharacteristic
14#16839699C2211yescharacteristic
15#16839698C411yescharacteristic, complemented
16#16839696—21no—
17#16839688C2211yescharacteristic
18#16839677—21no—
19#16839637—21no—
20#16839636—21no—
21#16839629C411yescharacteristic, complemented
22C61×#397C411yescharacteristic, complemented
23C22×#92420—41no—
24C2×C22⋊#6923—22no—
25C2×C4×#6923—41no—
26#14519835—24no—
27C22⋊#92420—22no—
28C22⋊#92420—41no—
29(C2×#6923)⋊C4—41no—
30C22×C22⋊D61D411yescharacteristic
31C23×Dic61D411yescharacteristic
32C22×C4×D61D411yescharacteristic
33#14519768—41no—
34#14519768—41no—
35#14519737—41no—
36#14519737—41no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758196D411yescharacteristic
43C2×C22⋊Dic61—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198C2311yescharacteristic
47C2×C4×Dic61C2×C411yescharacteristic
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684C2×C411yescharacteristic
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61C2×C411yescharacteristic
55#95×C61—21no—
56#95×C61C2×C411yescharacteristic
57C24×C61C22⋊C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C2×C22⋊D61—22no—
65C2×C22⋊D61—22no—
66C2×C22⋊D61—41no—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—81no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—81no—
78C22⋊#6923—24no—
79C22⋊#6923—42no—
80C2×#92420—42no—
81C2×#92420—82no—
82C4×#6923—82no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C22⋊C4×C61—42no—
86C22⋊Dic61—81no—
87#11758198—41no—
88#11758198—42no—
89#11758198—81no—
90C4⋊Dic61—41no—
91C4⋊Dic61—41no—
92C4⋊Dic61—41no—
93C4⋊Dic61—81no—
94#11758196—41no—
95#11758196—42no—
96#11758196—81no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61#10412yes—
104C23×C61C2×C22⋊C411yescharacteristic
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61#13512yes—
110C2×#6923—44no—
111C2×#6923—82no—
112C2×#6923—82no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C22⋊D61—44no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—41no—
129C2×Dic61—42no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—81no—
138C2×Dic61#9911yescharacteristic
139C2×Dic61#9911yescharacteristic
140C4×D61—42no—
141C4×D61—81no—
142#92420—82no—
143#11732—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—42no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—81no—
151C22×C61C2×#9911yescharacteristic
152C22×C61#41611yescharacteristic
153C22×C61C2×#10411yescharacteristic
154C2×D61—22no—
155C2×D61—41no—
156C2×D61—41no—
157C2×D61—42no—
158C2×D61—81no—
159#6923—84no—
160C4×C61—41no—
161C4×C61—41no—
162C4×C61—81no—
163C4×C61—81no—
164Dic61—21no—
165Dic61—21no—
166Dic61—41no—
167Dic61—41no—
168Dic61—81no—
169Dic61—81no—
170#2633—611no—
171C4×#99—611no—
172#1459—611no—
173#1352—612no—
174#1344—612no—
175C2×C61—21no—
176C2×C61—21no—
177C2×C61—42no—
178C2×C61#150111yescharacteristic
179C2×C61#212311yescharacteristic
180C2×C61#212711yescharacteristic
181D61—42no—
182C2×#125—611no—
183C2×C4×D4—611no—
184C2×#99—1221no—
185C2×C22⋊C8—1221no—
186#399—611no—
187#399—611no—
188#398—1221no—
189#397—611no—
190#397#692311yescharacteristic, complemented
191#394—1221no—
192#385—1221no—
193#385—1221no—
194#383—611no—
195C4×C22⋊C4—1221no—
196#342—612no—
197#324—1221no—
198#323—612no—
199C61#1173211yescharacteristic, complemented, Sylow
200C22×D4—611no—
201C23×C4—611no—
202C23×C4—611no—
203C22×C8—2441no—
204#125—2441no—
205#125—2441no—
206C4×D4—2441no—
207C4×D4—2441no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4C2×#692311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—1222no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C22⋊C4—1221no—
221C2×C22⋊C4C2×#692311yescharacteristic
222C2×C42—611no—
223C2×C42—2441no—
224#99—1224no—
225C22⋊C8—1222no—
226C22⋊C8—2441no—
227#95—21no—
228#95—611no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—2441no—
233#95C2×#692311yescharacteristic
234C24—611no—
235C24C22⋊#692311yescharacteristic
236C2×D4—1222no—
237C2×D4—1222no—
238C2×D4—2441no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—4881no—
251C22×C4—4881no—
252C22×C4—2442no—
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22×#692311yescharacteristic
255C2×C8—4882no—
256C2×C8—2442no—
257C4⋊C4—41no—
258C4⋊C4—4881no—
259C4⋊C4—4881no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C4⋊C4—2441no—
263C4⋊C4—2441no—
264C4⋊C4—2442no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—1224no—
268C22⋊C4—4881no—
269C22⋊C4—4881no—
270C22⋊C4—2441no—
271C22⋊C4—2442no—
272C22⋊C4—2442no—
273C42—1221no—
274C42—1221no—
275C42—4882no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—612no—
281C23—1221no—
282C23—1221no—
283C23—2441no—
284C23—2441no—
285C23#1451984112yes—
286C23C2×C22⋊#692311yescharacteristic
287D4—2444no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—611no—
299C2×C4—611no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—4881no—
305C2×C4—4881no—
306C2×C4—4882no—
307C2×C4—4882no—
308C2×C4—2442no—
309C2×C4—2441no—
310C2×C4—2442no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2444no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2441no—
319C8—4882no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—1222no—
328C22—4881no—
329C22—2442no—
330C22—2441no—
331C22—2441no—
332C22D61⋊#9911yescharacteristic
333C22#1684075211yescharacteristic
334C22C2×#1451984111yescharacteristic
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—1221no—
340C4—1221no—
341C4—4881no—
342C4—4884no—
343C4—4881no—
344C4—2441no—
345C4—2441no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—2442no—
350C2#2146119511yescharacteristic
351C2#2146208211yescharacteristic
352C2#2146209711yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#397C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758198C23
hypercenter#397#6923
hypocenterC61#11732
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #397

Complements

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

NH ≅ G/Ntype
3. #21460815C2semidirect
4. #21460799C2semidirect
12. C2×#14519768C4semidirect
15. #16839698C4semidirect
21. #16839629C4semidirect
22. C61×#397C4semidirect
190. #397#6923semidirect
199. C61#11732semidirect

Maximal subgroups

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

subgroupindex
#214615662
#214610572
#214608152
#214607992
#214516922
#1173261