Finite Group Databasebeta

← #29733636

Subgroups — #29733636

Order 15616.

1052 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 191 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×#16840375C211yesmaximal, characteristic
2C2×#16840141C211yesmaximal, characteristic, complemented
3#21451586C211yesmaximal, characteristic
4#21446828C214yesmaximal
5#21446785C211yesmaximal, characteristic, complemented
6#21446639C211yesmaximal, characteristic, complemented
7#21446157C214yesmaximal, complemented
8#21445588C211yesmaximal, characteristic
9C2×C61×#491C211yesmaximal, characteristic, complemented
10C22×#11758202C2214yes—
11#16840375—24no—
12#16840375—24no—
13C2×#14519777C2211yescharacteristic
14C2×#14519772—22no—
15C2×#14519768—22no—
16C2×#14519757C2212yes—
17C2×#14519753C2212yes—
18C2×#14519742C2212yes—
19C2×#14519734C2211yescharacteristic
20#16840141—24no—
21C2×#14519702C2214yes—
22C2×#14519689C2212yes—
23C2×C4×#92418C2211yescharacteristic
24#16839693—22no—
25#16839688—22no—
26#16839669C2212yes—
27#16839628C2211yescharacteristic
28#16839524C2214yes—
29#16839516C2212yes—
30C2×C4⋊Q8×C61C2211yescharacteristic
31C2×#125×C61C2211yescharacteristic
32C61×#491—24no—
33C2×#103×C61C2214yes—
34C2×#97×C61C2211yescharacteristic
35C22×Dic122C2312yes—
36C22×C4×D61C2311yescharacteristic
37C2×#11758202—28no—
38C2×#11758202—28no—
39C2×#11758202—44no—
40C22×#92418C2312yes—
41#14519777—22no—
42#14519777D414yes—
43#14519772—216no—
44#14519768—216no—
45#14519757—24no—
46#14519757—24no—
47#14519753—44no—
48#14519742—24no—
49#14519742—24no—
50#14519734—22no—
51#14519734—22no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61C2312yes—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×#11758196D412yes—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×#11758198D412yes—
61C2×C4×Dic61—22no—
62C2×C4×Dic61C2311yescharacteristic
63#14519702—44no—
64#14519689—24no—
65C4×#92418—22no—
66#14519684—22no—
67#14519684—24no—
68C22×Q8×C61C2312yes—
69C22×C8×C61C2312yes—
70C4⋊Q8×C61—22no—
71C4⋊Q8×C61—22no—
72#125×C61—22no—
73#125×C61D414yes—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61C2312yes—
77C2×C42×C61C2311yescharacteristic
78#103×C61—28no—
79#97×C61—22no—
80C23×D61C2×D411yescharacteristic
81C2×Q8×C61—22no—
82C2×Q8×C61—22no—
83C2×Q8×C61—24no—
84C2×Q8×C61—24no—
85C22×C4×C61—22no—
86C22×C4×C61—22no—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2411yescharacteristic
89C22×Dic61—22no—
90C22×Dic61—22no—
91C22×Dic61C2×D411yescharacteristic
92C22×Dic61C2×D412yes—
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—22no—
96C2×C4×D61—22no—
97C2×Dic122—24no—
98C2×Dic122—24no—
99C2×Dic122—22no—
100C2×Dic122C2×D414yes—
101C2×C8×C61—22no—
102C2×C8×C61—22no—
103C2×C8×C61—22no—
104C4⋊C4×C61—28no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—44no—
107C4⋊C4×C61C2×D414yes—
108C42×C61C2×D412yes—
109C42×C61C2×D412yes—
110#11758202—416no—
111#11758198—28no—
112#11758198—24no—
113#11758198—44no—
114C4⋊Dic61—28no—
115C4⋊Dic61—28no—
116C4⋊Dic61—24no—
117#11758196—28no—
118#11758196—24no—
119#11758196—44no—
120C4×Dic61—28no—
121C4×Dic61—21no—
122C4×Dic61—21no—
123C2×#92418—22no—
124C2×#92418—22no—
125C2×#92418#4114yes—
126C23×C61C22×D411yescharacteristic
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—21no—
130C22×D61—24no—
131Q8×C61—44no—
132Q8×C61—44no—
133C2×C4×C61—24no—
134C2×C4×C61—24no—
135C2×C4×C61—44no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61#12014yes—
140C2×C4×C61C22×D411yescharacteristic
141C2×C4×C61C22×D411yescharacteristic
142C2×C4×C61C2×#4112yes—
143C2×C4×C61H(5,𝔽2)12yes—
144C2×Dic61—22no—
145C2×Dic61—24no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—24no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—44no—
154C4×D61—42no—
155C4×D61—42no—
156Dic122—28no—
157Dic122—44no—
158C8×C61—42no—
159#92418—24no—
160#25609—611nomaximal, Sylow
161C22×C61C2×#12011yescharacteristic
162C22×C61#53211yescharacteristic
163C22×C61#53511yescharacteristic
164C22×C61#57814yes—
165C2×D61—42no—
166C2×D61—41no—
167C2×D61—42no—
168C2×D61—42no—
169C4×C61—24no—
170C4×C61—44no—
171C4×C61—82no—
172C4×C61#54612yes—
173C4×C61#55012yes—
174Dic61—44no—
175Dic61—41no—
176Dic61—42no—
177Dic61—82no—
178C2×#541—611no—
179C2×#494—611no—
180C2×#491D6111yescharacteristic, complemented
181#2621—611no—
182#2001—614no—
183#1991—611no—
184#1960—611no—
185#1878—614no—
186#1750—611no—
187C2×C61#243111yescharacteristic
188C2×C61#301112yes—
189C2×C61#317712yes—
190C2×C61#322312yes—
191D61—81no—
192C22×Q16—614no—
193#541—1224no—
194#541—1224no—
195C2×C4⋊Q8—611no—
196C2×C4⋊Q8C2×D6111yescharacteristic
197C2×#126—1222no—
198C2×#125—1222no—
199C2×#125C2×D6111yescharacteristic
200C2×C22⋊Q8—612no—
201C2×C22⋊Q8—612no—
202C2×C4×Q8—612no—
203C2×#117—611no—
204#494—1224no—
205#491—24no—
206C2×#103—614no—
207C2×#103C2×D6114yes—
208C2×C22⋊C8—612no—
209C2×#97C2×D6111yescharacteristic
210C2×C4×C8—611no—
211#398—1222no—
212#397—1222no—
213#391—612no—
214#382—611no—
215#340—614no—
216#336—612no—
217C61#2560911yescharacteristic, complemented, Sylow
218C22×Q8—612no—
219C22×Q8C22×D6112yes—
220C23×C4—611no—
221C2×Q16—1228no—
222C2×Q16—1228no—
223C2×Q16—2444no—
224C22×C8—612no—
225C22×C8C22×D6112yes—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—614no—
229C4⋊Q8—1222no—
230#126—12216no—
231#125—22no—
232#125—12216no—
233#125C22⋊D6114yes—
234C22⋊Q8—1224no—
235C22⋊Q8—1224no—
236C22⋊Q8—2444no—
237C4×Q8—1224no—
238C4×Q8—1224no—
239#117—1222no—
240#117—1222no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—612no—
244C2×C4⋊C4—612no—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4—1222no—
247C2×C4⋊C4—1222no—
248C2×C4⋊C4—1222no—
249C2×C4⋊C4C22×D6112yes—
250C2×C22⋊C4—612no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4—1222no—
253C2×C42—611no—
254C2×C42—1222no—
255C2×C42C22×D6111yescharacteristic
256#103—28no—
257#103—2444no—
258C22⋊C8—1224no—
259#97—22no—
260C4×C8—1222no—
261#95—1224no—
262#95—1222no—
263C24—611no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—614no—
269C2×Q8—1222no—
270C2×Q8—1224no—
271C2×Q8—1224no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—612no—
275C22×C4—611no—
276C22×C4—1222no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—1222no—
280C22×C4—1222no—
281C22×C4—1222no—
282C22×C4D4×D6112yes—
283C22×C4C23×D6111yescharacteristic
284Q16—24416no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—614no—
289C2×C8—1222no—
290C2×C8—1222no—
291C4⋊C4—22no—
292C4⋊C4—28no—
293C4⋊C4—44no—
294C4⋊C4—1228no—
295C4⋊C4—1228no—
296C4⋊C4—1224no—
297C4⋊C4—1228no—
298C4⋊C4—1224no—
299C4⋊C4—2444no—
300C4⋊C4C2×C22⋊D6114yes—
301C22⋊C4—1228no—
302C22⋊C4—1224no—
303C22⋊C4—2444no—
304C42—1228no—
305C42—1221no—
306C42—1221no—
307C42D4×D6112yes—
308C42C2×C22⋊D6112yes—
309C23—1221no—
310C23—1221no—
311C23—1224no—
312C23—1221no—
313C23C2×D4×D6111yescharacteristic
314Q8—44no—
315Q8—44no—
316Q8—1228no—
317Q8—2444no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—1222no—
325C2×C4—1224no—
326C2×C4—1221no—
327C2×C4—1221no—
328C2×C4—1224no—
329C2×C4—1221no—
330C2×C4—2444no—
331C2×C4—2442no—
332C2×C4—2442no—
333C2×C4—2442no—
334C2×C4—2442no—
335C2×C4—2442no—
336C2×C4#1451974514yes—
337C2×C4C2×D4×D6111yescharacteristic
338C2×C4C22×C22⋊D6111yescharacteristic
339C2×C4#41×D6112yes—
340C2×C4#1451982312yes—
341C8—42no—
342C8—1224no—
343C22—2442no—
344C22—2442no—
345C22—2442no—
346C22—2441no—
347C22C2×#1451974511yescharacteristic
348C22#1684031111yescharacteristic
349C22#1684033011yescharacteristic
350C22#1684057814yes—
351C4—24no—
352C4—44no—
353C4—82no—
354C4—4882no—
355C4—2442no—
356C4—2441no—
357C4—2444no—
358C4#1684040512yes—
359C4#1684043112yes—
360C2—4881no—
361C2#2144944111yescharacteristic
362C2#2145511312yes—
363C2#2145647712yes—
364C2#2145690312yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#491C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#491D61
hypocenterC61#25609
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
2. C2×#16840141C2semidirect
5. #21446785C2semidirect
6. #21446639C2semidirect
7. #21446157C2semidirect
9. C2×C61×#491C2semidirect
180. C2×#491D61semidirect
217. C61#25609semidirect

Maximal subgroups

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

subgroupindex
C2×#168403752
C2×#168401412
#214515862
#214468282
#214467852
#214466392
#214461572
#214455882
C2×C61×#4912
#2560961