Finite Group Databasebeta

← #29711976

Subgroups — #29711976

Order 15616.

1158 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 207 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×#16839917C212yesmaximal
2#21448403C211yesmaximal, characteristic
3#21446967C212yesmaximal, complemented
4#21446919C212yesmaximal, complemented
5#21446160C214yesmaximal, complemented
6#21446132C212yesmaximal
7#21445890C211yesmaximal, characteristic
8C2×C61×#541C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C22×#11758202C2214yes—
11C2×#14519736C2211yescharacteristic
12#16839917—28no—
13C2×#14519701C2214yes—
14C2×#14519700C2214yes—
15C2×C22⋊#92418C2212yes—
16#16839668C2212yes—
17#16839668C2212yes—
18#16839667—22no—
19#16839623—22no—
20#16839524C2214yes—
21#16839517C2212yes—
22#16839517C2212yes—
23C61×#541—28no—
24C2×C4⋊Q8×C61C2211yescharacteristic
25C2×#126×C61—22no—
26C2×#125×C61—22no—
27C2×#125×C61C2211yescharacteristic
28C2×C22⋊Q8×C61C2212yes—
29C2×C4×Q8×C61C2212yes—
30C2×#117×C61C2211yescharacteristic
31C22×Dic122—21no—
32C22×Dic122—24no—
33C22×Dic122—24no—
34C22×Dic122—42no—
35C22×Dic122C2312yes—
36C23×Dic61—21no—
37C2×#11758202—28no—
38C2×#11758202—28no—
39C2×#11758202—44no—
40C22×#92418C2314yes—
41#14519736—42no—
42C2×C4⋊Dic61C2312yes—
43C2×C22⋊Dic61—21no—
44C2×C4×Dic61—21no—
45#14519701—44no—
46#14519700—28no—
47C22⋊#92418—24no—
48#14519684—24no—
49#14519684—24no—
50C22×Q8×C61C2312yes—
51C23×C4×C61C2311yescharacteristic
52C4⋊Q8×C61—24no—
53#126×C61—216no—
54#125×C61—216no—
55#125×C61—24no—
56C22⋊Q8×C61—24no—
57C22⋊Q8×C61D418yes—
58C4×Q8×C61—24no—
59C4×Q8×C61—24no—
60#117×C61—24no—
61C2×C4⋊C4×C61—22no—
62C2×C4⋊C4×C61—22no—
63C2×C4⋊C4×C61—22no—
64C2×C4⋊C4×C61—22no—
65C2×C4⋊C4×C61D412yes—
66C2×C4⋊C4×C61C2312yes—
67C2×C22⋊C4×C61—22no—
68C2×C22⋊C4×C61D412yes—
69C2×C42×C61—22no—
70C2×C42×C61C2311yescharacteristic
71C2×C42×C61C2311yescharacteristic
72C24×C61C2×D411yescharacteristic
73C2×Q8×C61—22no—
74C2×Q8×C61—24no—
75C2×Q8×C61—24no—
76C2×Q8×C61C2×D414yes—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D414yes—
85C22×C4×C61C2411yescharacteristic
86C22×Dic61—24no—
87C22×Dic61—22no—
88C22×Dic61—22no—
89C22×Dic61—42no—
90C22×Dic61—42no—
91C2×Dic122—24no—
92C2×Dic122—24no—
93C2×Dic122—24no—
94C2×Dic122—22no—
95C2×Dic122—416no—
96C2×Dic122—42no—
97C2×Dic122—42no—
98C2×Dic122#4114yes—
99C4⋊C4×C61—28no—
100C4⋊C4×C61—28no—
101C4⋊C4×C61—28no—
102C4⋊C4×C61—28no—
103C4⋊C4×C61—24no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61C2×D414yes—
106C22⋊C4×C61—28no—
107C22⋊C4×C61—24no—
108C42×C61—28no—
109C42×C61—22no—
110C42×C61—22no—
111C22⋊Dic61—42no—
112#11758202—416no—
113C4⋊Dic61—24no—
114C4×Dic61—42no—
115C2×#92418—24no—
116C2×#92418—24no—
117C2×#92418—24no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—24no—
121C23×C61#12012yes—
122C23×C61C22×D411yescharacteristic
123Q8×C61—28no—
124Q8×C61—44no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—28no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—24no—
131C2×C4×C61—24no—
132C2×C4×C61—24no—
133C2×C4×C61—24no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—44no—
137C2×C4×C61#12012yes—
138C2×C4×C61C22×D412yes—
139C2×C4×C61C2×#4112yes—
140C2×C4×C61H(5,𝔽2)12yes—
141C2×Dic61—24no—
142C2×Dic61—48no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61—44no—
147Dic122—28no—
148Dic122—48no—
149Dic122—44no—
150#92418—44no—
151#23943—611nomaximal, Sylow
152C22×C61—22no—
153C22×C61—24no—
154C22×C61—42no—
155C22×C61—42no—
156C22×C61C2×#12011yescharacteristic
157C22×C61#52511yescharacteristic
158C22×C61#53511yescharacteristic
159C22×C61#57814yes—
160C4×C61—22no—
161C4×C61—44no—
162C4×C61—42no—
163C4×C61—42no—
164C4×C61—44no—
165C4×C61#54614yes—
166Dic61—44no—
167Dic61—82no—
168C2×#541D6111yescharacteristic
169C2×#451—612no—
170#2310—611no—
171#2026—612no—
172#2016—612no—
173#1878—614no—
174#1873—612no—
175#1818—611no—
176C2×C61—42no—
177C2×C61#240411yescharacteristic
178C2×C61#301112yes—
179C2×C61#312712yes—
180C2×C61#322312yes—
181C23×Q8—611no—
182C22×Q16—614no—
183#541—28no—
184C2×C4⋊Q8C2×D6111yescharacteristic
185C2×#126—22no—
186C2×#125—22no—
187C2×#125C2×D6111yescharacteristic
188C2×C22⋊Q8C2×D6112yes—
189C2×C4×Q8C2×D6112yes—
190C2×#117—611no—
191C2×#117C2×D6111yescharacteristic
192#451—1228no—
193C2×#103—614no—
194C2×#103—614no—
195C2×C22⋊C8—612no—
196#391—612no—
197#391—612no—
198#391—1222no—
199#381—1222no—
200#340—614no—
201#336—612no—
202#336—612no—
203C61#2394311yescharacteristic, complemented, Sylow
204C22×Q8—612no—
205C22×Q8—1224no—
206C22×Q8—1221no—
207C22×Q8—1224no—
208C22×Q8—2442no—
209C22×Q8C22×D6112yes—
210C23×C4—1221no—
211C23×C4C22×D6111yescharacteristic
212C2×Q16—1228no—
213C2×Q16—1228no—
214C2×Q16—2444no—
215C22×C8—614no—
216C4⋊Q8—24no—
217#126—216no—
218#125—216no—
219#125—24no—
220C22⋊Q8—24no—
221C22⋊Q8C22⋊D6118yes—
222C4×Q8—24no—
223C4×Q8—24no—
224#117—24no—
225#117—2442no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—612no—
231C2×C4⋊C4C22⋊D6112yes—
232C2×C4⋊C4C22×D6112yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—1221no—
235C2×C22⋊C4C22⋊D6112yes—
236C2×C42—22no—
237C2×C42—1221no—
238C2×C42C22×D6111yescharacteristic
239C2×C42C22×D6111yescharacteristic
240#103—1228no—
241#103—2444no—
242C22⋊C8—1224no—
243#95—1224no—
244#95—1224no—
245C24C2×C22⋊D6111yescharacteristic
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—614no—
250C2×Q8—1224no—
251C2×Q8—1222no—
252C2×Q8—1224no—
253C2×Q8—1224no—
254C2×Q8—24416no—
255C2×Q8—2442no—
256C2×Q8—2442no—
257C2×Q8C2×C22⋊D6114yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—1224no—
263C22×C4—1222no—
264C22×C4—1222no—
265C22×C4—2442no—
266C22×C4—2442no—
267C22×C4C2×C22⋊D6112yes—
268C22×C4C2×C22⋊D6112yes—
269C22×C4C2×C22⋊D6114yes—
270C22×C4C2×C22⋊D6111yescharacteristic
271C22×C4C23×D6111yescharacteristic
272Q16—24416no—
273C2×C8—1224no—
274C2×C8—1224no—
275C2×C8—1224no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—22no—
280C4⋊C4—28no—
281C4⋊C4—24no—
282C4⋊C4—1224no—
283C4⋊C4C2×C22⋊D6114yes—
284C22⋊C4—28no—
285C22⋊C4—24no—
286C22⋊C4—2442no—
287C42—28no—
288C42—22no—
289C42—22no—
290C42—2442no—
291C23—21no—
292C23—24no—
293C23—21no—
294C23#1451974612yes—
295C23C22×C22⋊D6111yescharacteristic
296Q8—28no—
297Q8—44no—
298Q8—1228no—
299Q8—2448no—
300Q8—2444no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—1224no—
314C2×C4—2444no—
315C2×C4—2442no—
316C2×C4—2448no—
317C2×C4—2442no—
318C2×C4—2442no—
319C2×C4#1451974612yes—
320C2×C4C22×C22⋊D6112yes—
321C2×C4#41×D6112yes—
322C2×C4#1451982312yes—
323C8—2444no—
324C22—24no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22C2×#1451974611yescharacteristic
329C22#1684026911yescharacteristic
330C22#1684032611yescharacteristic
331C22#1684057714yes—
332C4—22no—
333C4—42no—
334C4—42no—
335C4—44no—
336C4—44no—
337C4—4882no—
338C4—2444no—
339C4#1684040014yes—
340C2—42no—
341C2#2144913411yescharacteristic
342C2#2145511412yes—
343C2#2145611412yes—
344C2#2145689612yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#541C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#541D61
hypocenterC61#23943
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
3. #21446967C2semidirect
4. #21446919C2semidirect
5. #21446160C2semidirect
203. C61#23943semidirect

Maximal subgroups

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

subgroupindex
C2×#168399172
#214484032
#214469672
#214469192
#214461602
#214461322
#214458902
C2×C61×#5412
#2394361