Finite Group Databasebeta

← #29572910

Subgroups — #29572910

Order 15616.

550 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#16839825C211yesmaximal, characteristic
2#21445846C212yesmaximal, complemented
3#21445804C212yesmaximal
4#21443687C211yesmaximal, characteristic, complemented
5C61×#1289C211yesmaximal, characteristic, complemented
6C23×#92418C2211yescharacteristic
7C22×C8×D61C411yescharacteristic, complemented
8C2×C4×C22⋊D61C2211yescharacteristic
9#16839825—22no—
10#16839825—22no—
11C2×#14519710—22no—
12C2×#14519706C411yescharacteristic, complemented
13C2×#14519691—22no—
14C2×#14519689C411yescharacteristic, complemented
15C2×C8×Dic61C411yescharacteristic, complemented
16#16839622C2212yes—
17#16839528—22no—
18#16839524—22no—
19C2×C22⋊C8×C61C2211yescharacteristic
20C61×#380C2212yes—
21C22×C22⋊D61D411yescharacteristic
22C22×C4×D61C2×C411yescharacteristic
23C22×#92418—42no—
24C22×#92418C2×C411yescharacteristic
25C22×#92418C2×C411yescharacteristic
26C22×#92418D412yes—
27C22×#92418D412yes—
28C2×C8×D61—21no—
29C2×C8×D61—21no—
30C2×C8×D61—42no—
31C4×C22⋊D61—22no—
32C4×C22⋊D61—22no—
33C2×C4⋊Dic61—22no—
34C2×#11758196Q811yescharacteristic
35C2×C22⋊Dic61—22no—
36C2×C22⋊Dic61D411yescharacteristic
37C2×#11758198Q811yescharacteristic
38C2×C4×Dic61C2×C411yescharacteristic
39#14519710—44no—
40#14519706—22no—
41#14519691—44no—
42#14519689—22no—
43C8×Dic61—21no—
44C8×Dic61—21no—
45#14519684—22no—
46C23×C4×C61C2311yescharacteristic
47C22×C8×C61C2×C411yescharacteristic
48C22×C8×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61—22no—
50C2×C22⋊C4×C61—22no—
51C22⋊C8×C61—22no—
52#95×C61—22no—
53C24×C61C2×D411yescharacteristic
54C23×D61C4⋊C411yescharacteristic
55C22×C4×C61—41no—
56C22×C4×C61—41no—
57C22×C4×C61—42no—
58C22×C4×C61C22×C411yescharacteristic
59C22×C4×C61C2×D412yes—
60C22×C4×C61C2×Q811yescharacteristic
61C22×C4×C61#4112yes—
62C2×C22⋊D61—22no—
63C2×C22⋊D61—22no—
64C2×C22⋊D61D812yes—
65C2×C22⋊D61Q1612yes—
66C22×Dic61—42no—
67C22×Dic61C4⋊C411yescharacteristic
68C22×Dic61C4⋊C411yescharacteristic
69C22×Dic61C4⋊C411yescharacteristic
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—41no—
73C2×C4×D61—41no—
74C2×C8×C61—21no—
75C2×C8×C61—21no—
76C2×C8×C61—41no—
77C2×C8×C61—41no—
78C4⋊C4×C61—42no—
79C4⋊C4×C61—42no—
80C22⋊C4×C61—42no—
81C22⋊C4×C61—42no—
82C22⋊Dic61—22no—
83C22⋊Dic61—42no—
84C22⋊Dic61—42no—
85#11758198—22no—
86C4⋊Dic61—42no—
87C4⋊Dic61—42no—
88#11758196—22no—
89C4×Dic61—21no—
90C4×Dic61—21no—
91C2×#92418—24no—
92C2×#92418—22no—
93C2×#92418—41no—
94C2×#92418—42no—
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C22⋊C412yes—
97C2×#92418C22⋊C411yescharacteristic
98C8×D61—42no—
99C8×D61—42no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61C2×C4⋊C411yescharacteristic
103C23×C61#12112yes—
104C23×C61C2×D811yescharacteristic
105C23×C61C2×Q1611yescharacteristic
106C23×C61#13512yes—
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—41no—
110C22×D61—41no—
111C22×D61#10712yescomplemented
112C2×C4×C61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—22no—
115C2×C4×C61—21no—
116C2×C4×C61—41no—
117C2×C4×C61—42no—
118C2×C4×C61—82no—
119C2×C4×C61—82no—
120C2×C4×C61—82no—
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C2×C4×C61#11711yescharacteristic
123C2×C4×C61C22⋊Q812yes—
124C22⋊D61—24no—
125C22⋊D61—24no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—82no—
133C2×Dic61—82no—
134C2×Dic61—82no—
135C2×Dic61#10712yescomplemented
136C4×D61—42no—
137C4×D61—41no—
138C4×D61—41no—
139C8×C61—41no—
140C8×C61—41no—
141#92418—21no—
142#92418—22no—
143#92418#9912yescomplemented
144#8609—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—21no—
148C22×C61—22no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61—42no—
152C22×C61#38011yescharacteristic
153C22×C61C2×#10711yescharacteristic
154C22×C61#42711yescharacteristic
155C22×C61#46612yes—
156C22×C61#46712yes—
157C2×D61—22no—
158C2×D61—42no—
159C2×D61—41no—
160C2×D61—41no—
161C2×D61—42no—
162C4×C61—21no—
163C4×C61—22no—
164C4×C61—162no—
165C4×C61C2×#9911yescharacteristic
166C4×C61#41011yescharacteristic
167Dic61—22no—
168Dic61—41no—
169Dic61—41no—
170Dic61—162no—
171C2×C8×D4—611no—
172#1808—612no—
173#1799—612no—
174#1289—611no—
175#1289D6111yescharacteristic, complemented
176C2×C61—22no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61—22no—
180C2×C61#179911yescharacteristic
181C2×C61#180811yescharacteristic
182C2×C61#181211yescharacteristic
183D61—42no—
184C23×C8—611no—
185C23×C8—611no—
186C2×C4×D4—611no—
187C8×D4—1222no—
188C8×D4—1222no—
189C2×#107—1222no—
190C2×C4⋊C8—611no—
191C2×#99—1222no—
192C2×C22⋊C8—611no—
193C2×C22⋊C8C2×D6111yescharacteristic
194C2×C4×C8—611no—
195#380—612no—
196#380C2×D6112yes—
197#342—1222no—
198#340—1222no—
199C61#860911yescharacteristic, complemented, Sylow
200C22×D4—611no—
201C23×C4—611no—
202C23×C4C22×D6111yescharacteristic
203C22×C8—611no—
204C22×C8—612no—
205C22×C8—611no—
206C22×C8—612no—
207C22×C8—1221no—
208C22×C8—1221no—
209C22×C8—2442no—
210C22×C8—2442no—
211C22×C8C4×D6111yescharacteristic
212C22×C8C4×D6111yescharacteristic
213C4×D4—1222no—
214C4×D4—1222no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—611no—
217C2×C4⋊C4—1222no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—611no—
220C2×C22⋊C4—611no—
221C2×C22⋊C4—1222no—
222C2×C42—611no—
223#107—2444no—
224C4⋊C8—1222no—
225#99—2444no—
226C22⋊C8—22no—
227C22⋊C8—1222no—
228C4×C8—1221no—
229C4×C8—1221no—
230#95—22no—
231#95—1222no—
232C24—611no—
233C24D4×D6111yescharacteristic
234C2×D4—612no—
235C2×D4—612no—
236C2×D4—1222no—
237C2×D4—1222no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—2441no—
247C22×C4—2441no—
248C22×C4—2442no—
249C22×C4C2×C4×D6111yescharacteristic
250C22×C4#1175822212yes—
251C22×C4D4×D6112yes—
252C22×C4Q8×D6111yescharacteristic
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—611no—
258C2×C8—611no—
259C2×C8—612no—
260C2×C8—1224no—
261C2×C8—1222no—
262C2×C8—2442no—
263C2×C8—2442no—
264C2×C8—2441no—
265C2×C8—2442no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—1222no—
269C4⋊C4—2442no—
270C4⋊C4—2442no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—1222no—
274C22⋊C4—1222no—
275C22⋊C4—2442no—
276C22⋊C4—2442no—
277C42—1221no—
278C42—1221no—
279C23—41no—
280C23—41no—
281C23—612no—
282C23—1221no—
283C23—1221no—
284C23—2441no—
285C23—2441no—
286C23C4⋊C4×D6111yescharacteristic
287C23#1451975012yes—
288C23D8×D6111yescharacteristic
289C23Q16×D6111yescharacteristic
290C23#1451979812yes—
291D4—1224no—
292D4—1224no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—612no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—1221no—
307C2×C4—4882no—
308C2×C4—4882no—
309C2×C4—4882no—
310C2×C4—2442no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4C22⋊C4×D6111yescharacteristic
316C2×C4#1451973411yescharacteristic
317C2×C4#1451975512yes—
318C8—41no—
319C8—41no—
320C8—612no—
321C8—1222no—
322C8—1221no—
323C22—22no—
324C22—21no—
325C22—22no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—1222no—
331C22—2442no—
332C22—2442no—
333C22—2441no—
334C22—2441no—
335C22#1683962111yescharacteristic
336C22#107×D6111yescharacteristic
337C22#1683979711yescharacteristic
338C22#1683999712yes—
339C22#1684000212yes—
340C4—21no—
341C4—22no—
342C4—162no—
343C4—1222no—
344C4—9762no—
345C4—2441no—
346C4—2441no—
347C4#99×D6111yescharacteristic
348C4#1683973411yescharacteristic
349C2—22no—
350C2—21no—
351C2—21no—
352C2—22no—
353C2—2442no—
354C2#2144580611yescharacteristic
355C2#2144584711yescharacteristic
356C2#2144586411yescharacteristic
subgroupG/H
centerC22#16839621
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1289C2
socleC22×C61#380
prime radicalC22×C22⋊D61D4
cosocleC23×C4×C61C23
hypercenter#1289D61
hypocenterC61#8609
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #21445846C2semidirect
4. #21443687C2semidirect
5. C61×#1289C2semidirect
7. C22×C8×D61C4semidirect
12. C2×#14519706C4semidirect
14. C2×#14519689C4semidirect
15. C2×C8×Dic61C4semidirect
111. C22×D61#107semidirect
135. C2×Dic61#107semidirect
143. #92418#99semidirect
175. #1289D61semidirect
199. C61#8609semidirect

Maximal subgroups

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

subgroupindex
C2×#168398252
#214458462
#214458042
#214436872
C61×#12892
#860961