Finite Group Databasebeta

← #29582285

Subgroups — #29582285

Order 15616.

462 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 101 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#21456767C211yesmaximal, characteristic
2#21456341C211yesmaximal, characteristic
3#21446793C211yesmaximal, characteristic
4#21444384C211yesmaximal, characteristic
5#21444383C211yesmaximal, characteristic
6#21443663C211yesmaximal, characteristic
7C61×#3230C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840172—21no—
10#16840172—21no—
11#16840168—21no—
12#16840168—21no—
13#16840141—21no—
14#16840112—21no—
15#16839969—22no—
16#16839918—22no—
17#16839815C2211yescharacteristic
18#16839798—22no—
19C2×#14519688C2211yescharacteristic
20#16839628C2211yescharacteristic
21#16839524—22no—
22#16839500—22no—
23#16839499—22no—
24C61×#533C2211yescharacteristic
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#500—21no—
27C61×#487—21no—
28C61×#484—22no—
29C61×#477—22no—
30C61×#431C2211yescharacteristic
31C2×#11758202—24no—
32C22×#92418D412yes—
33#14519777D411yescharacteristic
34#14519777D411yescharacteristic
35#14519769—21no—
36#14519756—21no—
37#14519756—21no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40#14519736—21no—
41C2×C4⋊Dic61—21no—
42#14519710—22no—
43#14519710—22no—
44#14519708—22no—
45#14519708—22no—
46C4⋊#92418—21no—
47#14519701—24no—
48C22⋊#92418D412yes—
49#14519688—21no—
50#14519688—21no—
51C4×#92418—21no—
52#14519684—21no—
53#14519684—21no—
54C22×Q8×C61—21no—
55C4⋊Q8×C61—21no—
56C4⋊Q8×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61D412yes—
59C4⋊Q8×C61D411yescharacteristic
60C4⋊Q8×C61D411yescharacteristic
61#125×C61—21no—
62C22⋊Q8×C61—21no—
63C22⋊Q8×C61—21no—
64C4×Q8×C61—21no—
65C4×Q8×C61—21no—
66#117×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C42×C61C2311yescharacteristic
70#107×C61—24no—
71C4⋊C8×C61—21no—
72#103×C61—24no—
73C22⋊C8×C61D412yes—
74C4×C8×C61—21no—
75C2×Q8×C61—21no—
76C2×Q8×C61—22no—
77C2×Q8×C61—21no—
78C2×Q8×C61—41no—
79C2×Q8×C61—82no—
80C22×C4×C61—41no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×Dic61—41no—
85C2×Dic122—21no—
86C2×Dic122—21no—
87C2×C8×C61—22no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C42×C61—41no—
100C42×C61C2×D411yescharacteristic
101C42×C61C2×D411yescharacteristic
102C42×C61C2×D411yescharacteristic
103C42×C61#4111yescharacteristic
104C22⋊Dic61—41no—
105#11758202—84no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110C4⋊Dic61—42no—
111#11758196—41no—
112#11758196—41no—
113C4×Dic61—41no—
114C2×#92418—22no—
115C2×#92418—22no—
116C2×#92418—22no—
117C2×#92418Q1614yescomplemented
118C23×C61#12711yescharacteristic
119Q8×C61—44no—
120Q8×C61—42no—
121Q8×C61—82no—
122Q8×C61—81no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×C4×C61—82no—
129C2×C4×C61#12011yescharacteristic
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61#12011yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12111yescharacteristic
134C2×C4×C61#12111yescharacteristic
135C2×C4×C61C2×Q1612yes—
136C2×C4×C61#13712yes—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—81no—
140C2×Dic61—81no—
141C2×Dic61—81no—
142Dic122—42no—
143Dic122—81no—
144C8×C61—42no—
145#92418—24no—
146#92418—42no—
147#10224—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61#39211yescharacteristic
151C22×C61#49411yescharacteristic
152C22×C61#49711yescharacteristic
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—82no—
158C4×C61—81no—
159C4×C61—81no—
160C4×C61#45112yes—
161C4×C61#46212yes—
162Dic61—81no—
163Dic61—81no—
164Dic61—161no—
165#3230D6111yescharacteristic
166#3209—611no—
167#3162—611no—
168#1993—611no—
169#1518—611no—
170#1518—611no—
171#1281—611no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61#200811yescharacteristic
175C2×C61#200911yescharacteristic
176C2×C61#201611yescharacteristic
177#533—611no—
178#533C2×D6111yescharacteristic
179C2×C4⋊Q8C2×D6111yescharacteristic
180#501—1221no—
181#501—1221no—
182#501—1221no—
183#501—1221no—
184#500—21no—
185#494—1221no—
186#487—21no—
187#487—1221no—
188#484—22no—
189#477—22no—
190#462—1222no—
191#451—1222no—
192#431—611no—
193#431C2×D6111yescharacteristic
194#427—1222no—
195C2×#97—611no—
196#382—611no—
197#340—1222no—
198#328—1222no—
199#328—1222no—
200C61#1022411yescharacteristic, complemented, Sylow
201C22×Q8—21no—
202C2×Q16—1224no—
203C22×C8—612no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—611no—
208C4⋊Q8—611no—
209C4⋊Q8C22⋊D6112yes—
210C4⋊Q8C22⋊D6111yescharacteristic
211C4⋊Q8C22⋊D6111yescharacteristic
212#125—21no—
213#125—1221no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—1221no—
217C22⋊Q8—1221no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—1221no—
221C4×Q8—1221no—
222#117—21no—
223#117—1221no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1221no—
227C2×C42C22×D6111yescharacteristic
228#107—24no—
229#107—1222no—
230#107—1222no—
231#106—1222no—
232#106—1222no—
233C4⋊C8—21no—
234C4⋊C8—1221no—
235#103—24no—
236#103—1224no—
237C22⋊C8—612no—
238C22⋊C8C22⋊D6112yes—
239#97—1221no—
240#97—1221no—
241C4×C8—21no—
242C4×C8—1221no—
243#95—1221no—
244#95—1221no—
245C2×Q8—22no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—82no—
250C2×Q8—1221no—
251C2×Q8—1221no—
252C22×C4—41no—
253C22×C4—2441no—
254C22×C4D4×D6111yescharacteristic
255C22×C4D4×D6111yescharacteristic
256C22×C4C2×C22⋊D6111yescharacteristic
257Q16—4884no—
258C2×C8—22no—
259C2×C8—614no—
260C2×C8—1222no—
261C2×C8—1222no—
262C2×C8—1222no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—1221no—
274C4⋊C4—1221no—
275C4⋊C4—1221no—
276C4⋊C4—1221no—
277C4⋊C4—2442no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C22⋊C4—41no—
281C22⋊C4—2441no—
282C42—41no—
283C42—2441no—
284C42D4×D6111yescharacteristic
285C42#1175822411yescharacteristic
286C42C2×C22⋊D6111yescharacteristic
287C42C2×C22⋊D6111yescharacteristic
288C23#1451977311yescharacteristic
289Q8—44no—
290Q8—42no—
291Q8—81no—
292Q8—82no—
293Q8—4881no—
294Q8—2442no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—4881no—
302C2×C4—4881no—
303C2×C4—4881no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4#1451974511yescharacteristic
307C2×C4#1451974511yescharacteristic
308C2×C4#1451974611yescharacteristic
309C2×C4#1451974811yescharacteristic
310C2×C4#1451974911yescharacteristic
311C2×C4#1451974911yescharacteristic
312C2×C4Q16×D6112yes—
313C2×C4#1451980912yes—
314C8—42no—
315C8—1224no—
316C8—2442no—
317C22—21no—
318C22—21no—
319C22#1683967011yescharacteristic
320C22#1684014011yescharacteristic
321C22#1684015511yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—9761no—
330C4—4881no—
331C4—4881no—
332C4#1683991612yes—
333C4#1683997012yes—
334C2—21no—
335C2—21no—
336C2#2144687011yescharacteristic
337C2#2144687911yescharacteristic
338C2#2144691711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3230C2
socleC22×C61#392
prime radicalC23×C61#127
hypercenter#3230D61
hypocenterC61#10224
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
117. C2×#92418Q16semidirect
200. C61#10224semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214563412
#214467932
#214443842
#214443832
#214436632
C61×#32302
#1022461