Finite Group Databasebeta

← #29582286

Subgroups — #29582286

Order 15616.

462 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 103 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#21456970C211yesmaximal, characteristic
2#21456767C211yesmaximal, characteristic
3#21456343C211yesmaximal, characteristic
4#21444382C211yesmaximal, characteristic
5#21444382C211yesmaximal, characteristic
6#21443664C211yesmaximal, characteristic
7C61×#1993C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840319C2211yescharacteristic
10#16840171—21no—
11#16840171—21no—
12#16840171—21no—
13#16840171—21no—
14#16840165—21no—
15#16840141—21no—
16#16840114—21no—
17#16840112—21no—
18#16840097—22no—
19#16840054—22no—
20#16839969—22no—
21#16839918—22no—
22#16839815C2211yescharacteristic
23#16839815C2211yescharacteristic
24#16839800—22no—
25#16839498—22no—
26#16839498—22no—
27C2×C4⋊Q8×C61C2211yescharacteristic
28C2×#97×C61C2211yescharacteristic
29C61×#382C2211yescharacteristic
30C61×#340—22no—
31C2×#11758202—24no—
32#14519777D411yescharacteristic
33#14519777D411yescharacteristic
34#14519777D411yescharacteristic
35#14519777D411yescharacteristic
36#14519769—21no—
37#14519769—21no—
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41#14519756—21no—
42C4×Dic122—21no—
43C4×Dic122—21no—
44C4×Dic122—21no—
45C4×Dic122—21no—
46#14519736—21no—
47#14519736—21no—
48#14519711—22no—
49#14519711—22no—
50#14519710—24no—
51#14519709—22no—
52#14519709—22no—
53C4⋊#92418—21no—
54C4⋊#92418—21no—
55#14519701—24no—
56#14519700—24no—
57C22⋊#92418D412yes—
58C22⋊#92418D412yes—
59C4×#92418—21no—
60C4×#92418—21no—
61C22×Q8×C61—21no—
62C22×C8×C61D412yes—
63C4⋊Q8×C61—21no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61—21no—
66C4⋊Q8×C61D412yes—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C42×C61C2311yescharacteristic
71#97×C61—21no—
72#97×C61—21no—
73#95×C61—21no—
74#95×C61—21no—
75C2×Q8×C61—22no—
76C2×Q8×C61—41no—
77C2×Q8×C61—82no—
78C22×C4×C61—41no—
79C22×C4×C61—41no—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C2×Dic122—21no—
84C2×Dic122—21no—
85C2×Dic122—21no—
86C2×Dic122—21no—
87C2×C8×C61—22no—
88C2×C8×C61—22no—
89C2×C8×C61Q1614yescomplemented
90C4⋊C4×C61—22no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—42no—
95C42×C61C2×D411yescharacteristic
96C42×C61C2×D411yescharacteristic
97C42×C61C2×D411yescharacteristic
98C42×C61#4111yescharacteristic
99C22⋊Dic61—41no—
100C22⋊Dic61—41no—
101#11758202—84no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110#11758196—41no—
111#11758196—41no—
112#11758196—41no—
113#11758196—41no—
114C4×Dic61—41no—
115C4×Dic61—41no—
116C2×#92418—22no—
117C2×#92418—22no—
118C23×C61#12711yescharacteristic
119Q8×C61—44no—
120Q8×C61—82no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—42no—
124C2×C4×C61—82no—
125C2×C4×C61—81no—
126C2×C4×C61—81no—
127C2×C4×C61—81no—
128C2×C4×C61#12011yescharacteristic
129C2×C4×C61#12011yescharacteristic
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61#12111yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12111yescharacteristic
134C2×C4×C61C2×Q1612yes—
135C2×C4×C61#13712yes—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140Dic122—42no—
141Dic122—42no—
142Dic122—81no—
143Dic122—81no—
144C8×C61—24no—
145#92418—42no—
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—161no—
159C4×C61#45112yes—
160C4×C61#46212yes—
161Dic61—81no—
162Dic61—81no—
163Dic61—81no—
164Dic61—81no—
165#3230—611no—
166#3209—611no—
167#3162—611no—
168#1993D6111yescharacteristic
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#533—611no—
179C2×C4⋊Q8C2×D6111yescharacteristic
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#431—611no—
193#431—611no—
194#427—1222no—
195C2×#97C2×D6111yescharacteristic
196#382C2×D6111yescharacteristic
197#340—22no—
198#328—1222no—
199#328—1222no—
200C61#1022411yescharacteristic, complemented, Sylow
201C22×Q8—21no—
202C2×Q16—1224no—
203C22×C8C22⋊D6112yes—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—611no—
208C4⋊Q8—611no—
209C4⋊Q8—611no—
210C4⋊Q8—611no—
211C4⋊Q8D24412yes—
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—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C42C22×D6111yescharacteristic
228#107—1224no—
229#107—1222no—
230#107—1222no—
231#106—1222no—
232#106—1222no—
233C4⋊C8—1221no—
234C4⋊C8—1221no—
235#103—1224no—
236#103—1224no—
237C22⋊C8—612no—
238C22⋊C8—612no—
239#97—21no—
240#97—21no—
241C4×C8—1221no—
242C4×C8—1221no—
243#95—21no—
244#95—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—82no—
248C2×Q8—1221no—
249C2×Q8—1221no—
250C2×Q8—1221no—
251C2×Q8—1221no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4D4×D6111yescharacteristic
255C22×C4D4×D6111yescharacteristic
256C22×C4C2×C22⋊D6111yescharacteristic
257Q16—4884no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—1222no—
261C2×C8—1222no—
262C2×C8#1175820214yescomplemented
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—1221no—
269C4⋊C4—1221no—
270C4⋊C4—1221no—
271C4⋊C4—1221no—
272C4⋊C4—1221no—
273C4⋊C4—1221no—
274C4⋊C4—1221no—
275C4⋊C4—1221no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C22⋊C4—2441no—
281C22⋊C4—2441no—
282C42—2441no—
283C42—2441no—
284C42C2×D24411yescharacteristic
285C42#1175822211yescharacteristic
286C42D4×D6111yescharacteristic
287C42D4×D6111yescharacteristic
288C23#1451977311yescharacteristic
289Q8—44no—
290Q8—82no—
291Q8—4881no—
292Q8—4881no—
293Q8—2442no—
294Q8—2442no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4#1451974511yescharacteristic
307C2×C4#1451974711yescharacteristic
308C2×C4#1451974711yescharacteristic
309C2×C4#1451975011yescharacteristic
310C2×C4#1451975011yescharacteristic
311C2×C4#1451975111yescharacteristic
312C2×C4C2×#1175820212yes—
313C2×C4#1451980712yes—
314C8—24no—
315C8—2442no—
316C8—2442no—
317C22—21no—
318C22—21no—
319C22#1683967111yescharacteristic
320C22#1684014111yescharacteristic
321C22#1684015711yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—161no—
328C4—4881no—
329C4—4881no—
330C4—4881no—
331C4—4881no—
332C4#1683991712yes—
333C4#1683997212yes—
334C2—21no—
335C2—21no—
336C2#2144687411yescharacteristic
337C2#2144688111yescharacteristic
338C2#2144691811yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1993C2
socleC22×C61#392
prime radicalC23×C61#127
hypercenter#1993D61
hypocenterC61#10224
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
89. C2×C8×C61Q16semidirect
200. C61#10224semidirect
262. C2×C8#11758202semidirect

Maximal subgroups

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

subgroupindex
#214569702
#214567672
#214563432
#214443822
#214436642
C61×#19932
#1022461