Finite Group Databasebeta

← #29582290

Subgroups — #29582290

Order 15616.

462 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 105 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#21456972C211yesmaximal, characteristic
2#21456769C211yesmaximal, characteristic
3#21456343C211yesmaximal, characteristic
4#21446792C211yesmaximal, characteristic
5#21444384C211yesmaximal, characteristic
6#21444384C211yesmaximal, characteristic
7C61×#1281C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840319C2211yescharacteristic
10C2×#14519777C2211yescharacteristic
11#16840171—21no—
12#16840171—21no—
13#16840171—21no—
14#16840171—21no—
15#16840167—21no—
16#16840142—21no—
17#16840113—21no—
18#16840113—21no—
19#16840098—22no—
20#16840058—22no—
21#16839973—22no—
22#16839920—22no—
23#16839800—22no—
24#16839628C2211yescharacteristic
25#16839525—22no—
26#16839500—22no—
27#16839500—22no—
28C61×#431C2211yescharacteristic
29C61×#431C2211yescharacteristic
30C2×#97×C61C2211yescharacteristic
31C22×Dic122—21no—
32C2×Dic244—24no—
33#14519777—21no—
34#14519777—21no—
35#14519777—21no—
36#14519777D412yes—
37#14519777D411yescharacteristic
38#14519777D411yescharacteristic
39#14519777D411yescharacteristic
40#14519777D411yescharacteristic
41#14519769—21no—
42#14519769—21no—
43#14519756—21no—
44#14519756—21no—
45#14519756—21no—
46#14519756—21no—
47C4×Dic122—21no—
48C4×Dic122—21no—
49C4×Dic122—21no—
50C4×Dic122—21no—
51#14519736—21no—
52#14519736—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56#14519711—22no—
57#14519711—22no—
58#14519711—24no—
59#14519709—22no—
60#14519709—22no—
61#14519702—24no—
62#14519702—24no—
63#14519684—21no—
64#14519684—21no—
65C22×C8×C61D412yes—
66C2×C42×C61C2311yescharacteristic
67C4⋊C8×C61—21no—
68C4⋊C8×C61—21no—
69C22⋊C8×C61D412yes—
70C22⋊C8×C61D412yes—
71#97×C61—21no—
72#97×C61—21no—
73C4×C8×C61—21no—
74C4×C8×C61—21no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×Dic61—41no—
79C22×Dic61—41no—
80C2×Dic122—22no—
81C2×Dic122—21no—
82C2×Dic122—21no—
83C2×Dic122—21no—
84C2×Dic122—21no—
85C2×Dic122—41no—
86C2×Dic122—82no—
87C2×C8×C61—22no—
88C2×C8×C61—22no—
89C2×C8×C61—22no—
90C2×C8×C61—22no—
91C2×C8×C61Q1614yescomplemented
92C42×C61C2×D411yescharacteristic
93C42×C61C2×D411yescharacteristic
94C42×C61C2×D411yescharacteristic
95C42×C61#4111yescharacteristic
96C22⋊Dic61—41no—
97C22⋊Dic61—41no—
98C4⋊Dic61—21no—
99C4⋊Dic61—22no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—22no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110C4⋊Dic61—42no—
111#11758196—41no—
112#11758196—41no—
113#11758196—41no—
114#11758196—41no—
115C4×Dic61—41no—
116C4×Dic61—41no—
117Dic244—84no—
118C23×C61#12711yescharacteristic
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61#12011yescharacteristic
122C2×C4×C61#12011yescharacteristic
123C2×C4×C61#12011yescharacteristic
124C2×C4×C61#12111yescharacteristic
125C2×C4×C61#12111yescharacteristic
126C2×C4×C61#12111yescharacteristic
127C2×C4×C61C2×Q1612yes—
128C2×C4×C61#13712yes—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—42no—
134C2×Dic61—82no—
135C2×Dic61—81no—
136C2×Dic61—81no—
137C2×Dic61—81no—
138Dic122—42no—
139Dic122—44no—
140Dic122—42no—
141Dic122—82no—
142Dic122—81no—
143Dic122—81no—
144C8×C61—24no—
145C8×C61—42no—
146C8×C61—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#45112yes—
158C4×C61#46212yes—
159Dic61—82no—
160Dic61—81no—
161Dic61—81no—
162Dic61—81no—
163Dic61—81no—
164Dic61—161no—
165#3230—611no—
166#3209—611no—
167#3162—611no—
168#1993—611no—
169#1518—611no—
170#1518—611no—
171#1281D6111yescharacteristic
172C2×C61—21no—
173C2×C61—21no—
174C2×C61#200811yescharacteristic
175C2×C61#200911yescharacteristic
176C2×C61#201611yescharacteristic
177#533—611no—
178#533—611no—
179C2×C4⋊Q8—611no—
180#501—1221no—
181#501—1221no—
182#501—1221no—
183#501—1221no—
184#500—1221no—
185#494—1221no—
186#487—1221no—
187#487—1221no—
188#484—1222no—
189#477—1222no—
190#462—1222no—
191#451—1222no—
192#431C2×D6111yescharacteristic
193#431C2×D6111yescharacteristic
194#427—1222no—
195C2×#97C2×D6111yescharacteristic
196#382—611no—
197#340—1222no—
198#328—1222no—
199#328—1222no—
200C61#1022411yescharacteristic, complemented, Sylow
201C22×Q8—1221no—
202C2×Q16—1224no—
203C22×C8D24412yes—
204C4⋊Q8—612no—
205C4⋊Q8—611no—
206C4⋊Q8—611no—
207C4⋊Q8—611no—
208C4⋊Q8—611no—
209C4⋊Q8—1221no—
210C4⋊Q8—1221no—
211C4⋊Q8—1221no—
212#125—1221no—
213#125—1221no—
214C22⋊Q8—1221no—
215C22⋊Q8—1221no—
216C22⋊Q8—1221no—
217C22⋊Q8—1221no—
218C4×Q8—1221no—
219C4×Q8—1221no—
220C4×Q8—1221no—
221C4×Q8—1221no—
222#117—1221no—
223#117—1221no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C42C22×D6111yescharacteristic
228#107—1222no—
229#107—1222no—
230#107—1224no—
231#106—1222no—
232#106—1222no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235#103—1224no—
236#103—1224no—
237C22⋊C8D24412yes—
238C22⋊C8D24412yes—
239#97—21no—
240#97—21no—
241C4×C8—21no—
242C4×C8—21no—
243#95—1221no—
244#95—1221no—
245C2×Q8—1221no—
246C2×Q8—1222no—
247C2×Q8—1221no—
248C2×Q8—1221no—
249C2×Q8—1221no—
250C2×Q8—4882no—
251C2×Q8—2441no—
252C22×C4—2441no—
253C22×C4—2441no—
254C22×C4C2×D24411yescharacteristic
255C22×C4C2×D24411yescharacteristic
256C22×C4C2×D24411yescharacteristic
257Q16—4884no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8Dic24414yescomplemented
263C4⋊C4—1221no—
264C4⋊C4—1222no—
265C4⋊C4—1221no—
266C4⋊C4—1221no—
267C4⋊C4—1221no—
268C4⋊C4—1221no—
269C4⋊C4—1221no—
270C4⋊C4—1222no—
271C4⋊C4—1221no—
272C4⋊C4—1221no—
273C4⋊C4—2442no—
274C4⋊C4—2441no—
275C4⋊C4—2441no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C22⋊C4—2441no—
281C22⋊C4—2441no—
282C42—2441no—
283C42—2441no—
284C42D4×D6111yescharacteristic
285C42D4×D6111yescharacteristic
286C42D4×D6111yescharacteristic
287C42#1175822611yescharacteristic
288C23#1451977411yescharacteristic
289Q8—4881no—
290Q8—4881no—
291Q8—4882no—
292Q8—2442no—
293Q8—2444no—
294Q8—2442no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—4881no—
298C2×C4—4881no—
299C2×C4—4881no—
300C2×C4—4882no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2442no—
306C2×C4#1451974711yescharacteristic
307C2×C4#1451974711yescharacteristic
308C2×C4#1451974711yescharacteristic
309C2×C4#1451975211yescharacteristic
310C2×C4#1451975211yescharacteristic
311C2×C4#1451975211yescharacteristic
312C2×C4C2×Dic24412yes—
313C2×C4#1451981012yes—
314C8—24no—
315C8—42no—
316C8—42no—
317C22—21no—
318C22—21no—
319C22#1683967211yescharacteristic
320C22#1684014211yescharacteristic
321C22#1684015611yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—9761no—
327C4—4882no—
328C4—4881no—
329C4—4881no—
330C4—4881no—
331C4—4881no—
332C4#1683992012yes—
333C4#1683997312yes—
334C2—21no—
335C2—21no—
336C2#2144687511yescharacteristic
337C2#2144688211yescharacteristic
338C2#2144692111yescharacteristic
subgroupG/H
centerC22#16839672
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1281C2
socleC22×C61#392
prime radicalC23×C61#127
hypercenter#1281D61
hypocenterC61#10224
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
91. C2×C8×C61Q16semidirect
200. C61#10224semidirect
262. C2×C8Dic244semidirect

Maximal subgroups

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

subgroupindex
#214569722
#214567692
#214563432
#214467922
#214443842
C61×#12812
#1022461