Finite Group Databasebeta

← #29711968

Subgroups — #29711968

Order 15616.

1318 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1318 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×#16839912C212yesmaximal
2#21448399C211yesmaximal, characteristic
3#21446962C212yesmaximal, complemented
4#21446902C212yesmaximal, complemented
5#21446174C214yesmaximal, complemented
6#21446132C212yesmaximal, complemented
7#21445890C211yesmaximal, characteristic, complemented
8C2×C61×#541C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C22×#11758201C2214yes—
11C2×#14519736C2211yescharacteristic
12#16839912—28no—
13C2×#14519701C2214yescomplemented
14C2×#14519697C2214yes—
15C2×C22⋊#92418C2212yes—
16#16839664C2212yes—
17#16839664C2212yes—
18#16839663—22no—
19#16839617—22no—
20#16839524C2214yescomplemented
21#16839517C2212yescomplemented
22#16839517C2212yescomplemented
23C61×#541—28no—
24C2×C4⋊Q8×C61C2211yescharacteristic, complemented
25C2×#126×C61—22no—
26C2×#125×C61—22no—
27C2×#125×C61C2211yescharacteristic, complemented
28C2×C22⋊Q8×C61C2212yes—
29C2×C4×Q8×C61C2212yescomplemented
30C2×#117×C61C2211yescharacteristic
31C24×D61—21no—
32C22×D244—21no—
33C22×D244—24no—
34C22×D244—24no—
35C22×D244—42no—
36C22×D244C2312yes—
37C2×#11758201—28no—
38C2×#11758201—28no—
39C2×#11758201—44no—
40C22×#92418C2314yes—
41#14519736—42no—
42C2×C4⋊Dic61C2312yes—
43C2×C22⋊Dic61—21no—
44C2×#11758198—24no—
45C2×#11758198—24no—
46C2×C4×Dic61—21no—
47#14519701—44no—
48#14519697—28no—
49C22⋊#92418—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
73C23×D61—22no—
74C23×D61—24no—
75C23×D61—42no—
76C23×D61—42no—
77C23×D61—84no—
78C2×Q8×C61—22no—
79C2×Q8×C61—24no—
80C2×Q8×C61—24no—
81C2×Q8×C61C2×D414yes—
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D414yes—
90C22×C4×C61C2411yescharacteristic
91C22×Dic61—22no—
92C2×D244—24no—
93C2×D244—24no—
94C2×D244—24no—
95C2×D244—22no—
96C2×D244—416no—
97C2×D244—42no—
98C2×D244—42no—
99C2×D244#4114yes—
100C4⋊C4×C61—28no—
101C4⋊C4×C61—28no—
102C4⋊C4×C61—28no—
103C4⋊C4×C61—28no—
104C4⋊C4×C61—24no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61C2×D414yes—
107C22⋊C4×C61—28no—
108C22⋊C4×C61—24no—
109C42×C61—28no—
110C42×C61—22no—
111C42×C61—22no—
112C22⋊Dic61—42no—
113#11758201—416no—
114#11758198—84no—
115#11758198—84no—
116C4⋊Dic61—24no—
117C4×Dic61—42no—
118C2×#92418—24no—
119C2×#92418—24no—
120C2×#92418—24no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—24no—
124C23×C61#12012yes—
125C23×C61C22×D411yescharacteristic
126C22×D61—24no—
127C22×D61—48no—
128C22×D61—42no—
129C22×D61—42no—
130C22×D61—44no—
131C22×D61—44no—
132C22×D61—88no—
133C22×D61—88no—
134C22×D61—88no—
135Q8×C61—28no—
136Q8×C61—44no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—28no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—24no—
143C2×C4×C61—24no—
144C2×C4×C61—24no—
145C2×C4×C61—24no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—44no—
149C2×C4×C61#12012yes—
150C2×C4×C61C22×D412yes—
151C2×C4×C61C2×#4112yes—
152C2×C4×C61H(5,𝔽2)12yes—
153C2×Dic61—42no—
154C2×Dic61—44no—
155D244—28no—
156D244—48no—
157D244—44no—
158#92418—44no—
159#23942—611nomaximal, Sylow
160C22×C61—22no—
161C22×C61—24no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61C2×#12011yescharacteristic
165C22×C61#52511yescharacteristic
166C22×C61#53511yescharacteristic
167C22×C61#57714yes—
168C2×D61—48no—
169C2×D61—44no—
170C2×D61—816no—
171C2×D61—82no—
172C2×D61—82no—
173C2×D61—84no—
174C4×C61—22no—
175C4×C61—44no—
176C4×C61—42no—
177C4×C61—42no—
178C4×C61—44no—
179C4×C61#54614yes—
180Dic61—82no—
181C2×#541D6111yescharacteristic, complemented
182C2×#450—612no—
183#2309—611no—
184#2025—612no—
185#2013—612no—
186#1880—614no—
187#1873—612no—
188#1818—611no—
189C2×C61—42no—
190C2×C61#240411yescharacteristic
191C2×C61#301012yes—
192C2×C61#312312yes—
193C2×C61#322212yes—
194D61—84no—
195C23×D4—611no—
196C22×SD16—614no—
197#541—28no—
198C2×C4⋊Q8C2×D6111yescharacteristic, complemented
199C2×#126—22no—
200C2×#125—22no—
201C2×#125C2×D6111yescharacteristic, complemented
202C2×C22⋊Q8C2×D6112yes—
203C2×C4×Q8C2×D6112yescomplemented
204C2×#117—611no—
205C2×#117C2×D6111yescharacteristic
206#450—1228no—
207C2×#103—614no—
208C2×#102—614no—
209C2×C22⋊C8—612no—
210#390—612no—
211#390—612no—
212#390—1222no—
213#379—1222no—
214#340—614no—
215#336—612no—
216#336—612no—
217C61#2394211yescharacteristic, complemented, Sylow
218C25—1221no—
219C22×Q8C22×D6112yes—
220C22×D4—612no—
221C22×D4—1224no—
222C22×D4—1221no—
223C22×D4—1224no—
224C22×D4—2442no—
225C23×C4C22×D6111yescharacteristic
226C2×SD16—1228no—
227C2×SD16—1228no—
228C2×SD16—2444no—
229C22×C8—614no—
230C4⋊Q8—24no—
231#126—216no—
232#125—216no—
233#125—24no—
234C22⋊Q8—24no—
235C22⋊Q8C22⋊D6118yes—
236C4×Q8—24no—
237C4×Q8—24no—
238#117—24no—
239#117—2442no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—612no—
245C2×C4⋊C4C22⋊D6112yes—
246C2×C4⋊C4C22×D6112yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—1224no—
249C2×C22⋊C4—1224no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4C22⋊D6112yes—
252C2×C42—22no—
253C2×C42—1221no—
254C2×C42C22×D6111yescharacteristic
255C2×C42C22×D6111yescharacteristic
256#103—2444no—
257#102—1228no—
258C22⋊C8—1224no—
259C24—1224no—
260C24—1222no—
261C24—4884no—
262C24—2442no—
263C24—2442no—
264C24C2×C22⋊D6111yescharacteristic
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8C2×C22⋊D6114yes—
269C2×D4—614no—
270C2×D4—1224no—
271C2×D4—1224no—
272C2×D4—1222no—
273C2×D4—1224no—
274C2×D4—24416no—
275C2×D4—2442no—
276C2×D4—2442no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—1222no—
282C22×C4C2×C22⋊D6112yes—
283C22×C4C2×C22⋊D6112yes—
284C22×C4C2×C22⋊D6114yes—
285C22×C4C2×C22⋊D6111yescharacteristic
286C22×C4C23×D6111yescharacteristic
287SD16—24416no—
288C2×C8—1224no—
289C2×C8—1224no—
290C2×C8—1224no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—1224no—
298C4⋊C4C2×C22⋊D6114yes—
299C22⋊C4—28no—
300C22⋊C4—24no—
301C22⋊C4—4884no—
302C22⋊C4—4884no—
303C22⋊C4—2442no—
304C42—28no—
305C42—22no—
306C42—22no—
307C42—2442no—
308C23—21no—
309C23—24no—
310C23—21no—
311C23—1224no—
312C23—4888no—
313C23—4888no—
314C23—4888no—
315C23—2448no—
316C23—2442no—
317C23—2444no—
318C23—2444no—
319C23—2442no—
320C23#1451974612yes—
321C23C22×C22⋊D6111yescharacteristic
322Q8—28no—
323Q8—44no—
324D4—1228no—
325D4—2448no—
326D4—2444no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—28no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—2444no—
340C2×C4—2442no—
341C2×C4#1451974612yes—
342C2×C4C22×C22⋊D6112yes—
343C2×C4#41×D6112yes—
344C2×C4#1451982312yes—
345C8—2444no—
346C22—24no—
347C22—22no—
348C22—42no—
349C22—42no—
350C22—48816no—
351C22—4884no—
352C22—4882no—
353C22—4882no—
354C22—2444no—
355C22—2448no—
356C22C2×#1451974611yescharacteristic
357C22#1684026911yescharacteristic
358C22#1684032611yescharacteristic
359C22#1684057014yes—
360C4—22no—
361C4—42no—
362C4—42no—
363C4—44no—
364C4—44no—
365C4—4882no—
366C4#1684040014yes—
367C2—42no—
368C2—4884no—
369C2#2144913411yescharacteristic
370C2#2145510512yes—
371C2#2145606612yes—
372C2#2145688512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#541C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenterC2×#541D61
hypocenterC61#23942
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. #21446962C2semidirect
4. #21446902C2semidirect
5. #21446174C2semidirect
6. #21446132C2semidirect
7. #21445890C2semidirect
8. C2×C61×#541C2semidirect
13. C2×#14519701C22semidirect
20. #16839524C22semidirect
21. #16839517C22semidirect
22. #16839517C22semidirect
24. C2×C4⋊Q8×C61C22semidirect
27. C2×#125×C61C22semidirect
29. C2×C4×Q8×C61C22semidirect
181. C2×#541D61semidirect
198. C2×C4⋊Q8C2×D61semidirect
201. C2×#125C2×D61semidirect
203. C2×C4×Q8C2×D61semidirect
217. C61#23942semidirect

Maximal subgroups

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

subgroupindex
C2×#168399122
#214483992
#214469622
#214469022
#214461742
#214461322
#214458902
C2×C61×#5412
#2394261