Finite Group Databasebeta

← #29582288

Subgroups — #29582288

Order 15616.

462 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 99 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#21456764C211yesmaximal, characteristic
3#21456341C211yesmaximal, characteristic
4#21446791C211yesmaximal, characteristic
5#21444382C211yesmaximal, characteristic
6#21443663C211yesmaximal, characteristic
7C61×#1518C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×#14519777C2211yescharacteristic
10#16840172—21no—
11#16840172—21no—
12#16840168—21no—
13#16840168—21no—
14#16840167—21no—
15#16840141—21no—
16#16840114—21no—
17#16840113—21no—
18#16840098—22no—
19#16840058—22no—
20#16839972—22no—
21#16839917—22no—
22#16839815C2211yescharacteristic
23#16839798—22no—
24C2×#14519688C2211yescharacteristic
25#16839524—22no—
26#16839498—22no—
27C61×#533C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C61×#382C2211yescharacteristic
30C61×#328—22no—
31C22×Dic122—21no—
32C2×#11758202—24no—
33C22×#92418D412yes—
34#14519777—21no—
35#14519777—21no—
36#14519777—21no—
37#14519777D412yes—
38#14519777D411yescharacteristic
39#14519777D411yescharacteristic
40#14519769—21no—
41#14519756—21no—
42#14519756—21no—
43C4×Dic122—21no—
44C4×Dic122—21no—
45#14519736—21no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48#14519711—24no—
49#14519710—22no—
50#14519710—22no—
51#14519708—22no—
52#14519708—22no—
53C4⋊#92418—21no—
54#14519702—24no—
55#14519700—24no—
56C22⋊#92418D412yes—
57#14519688—21no—
58#14519688—21no—
59C4×#92418—21no—
60C4⋊Q8×C61D411yescharacteristic
61C4⋊Q8×C61D411yescharacteristic
62#125×C61—21no—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67#117×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C42×C61C2311yescharacteristic
70C4⋊C8×C61—21no—
71C22⋊C8×C61D412yes—
72C4×C8×C61—21no—
73#95×C61—21no—
74#95×C61—21no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C22×C4×C61—41no—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×Dic61—41no—
82C2×Dic122—22no—
83C2×Dic122—21no—
84C2×Dic122—21no—
85C2×Dic122—41no—
86C2×Dic122—82no—
87C2×C8×C61—22no—
88C4⋊C4×C61—21no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—42no—
95C22⋊C4×C61—41no—
96C42×C61—41no—
97C42×C61C2×D411yescharacteristic
98C42×C61C2×D411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61#4111yescharacteristic
101C22⋊Dic61—41no—
102#11758202—84no—
103C4⋊Dic61—22no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—22no—
108C4⋊Dic61—21no—
109C4⋊Dic61—41no—
110C4⋊Dic61—41no—
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—42no—
120Q8×C61—81no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
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—42no—
139C2×Dic61—82no—
140Dic122—44no—
141Dic122—42no—
142Dic122—82no—
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—81no—
158C4×C61—81no—
159C4×C61—161no—
160C4×C61#45112yes—
161C4×C61#46212yes—
162Dic61—82no—
163Dic61—81no—
164Dic61—81no—
165#3230—611no—
166#3209—611no—
167#3162—611no—
168#1993—611no—
169#1518—611no—
170#1518D6111yescharacteristic
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⋊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#431—611no—
193#431C2×D6111yescharacteristic
194#427—1222no—
195C2×#97—611no—
196#382C2×D6111yescharacteristic
197#340—1222no—
198#328—22no—
199#328—1222no—
200C61#1022411yescharacteristic, complemented, Sylow
201C22×Q8—1221no—
202C2×Q16—1224no—
203C22×C8—612no—
204C4⋊Q8—612no—
205C4⋊Q8—611no—
206C4⋊Q8—611no—
207C4⋊Q8—1221no—
208C4⋊Q8—1221no—
209C4⋊Q8—1221no—
210C4⋊Q8D24411yescharacteristic
211C4⋊Q8D24411yescharacteristic
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—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—1221no—
235#103—1224no—
236#103—1224no—
237C22⋊C8—612no—
238C22⋊C8C22⋊D6112yes—
239#97—1221no—
240#97—1221no—
241C4×C8—21no—
242C4×C8—1221no—
243#95—21no—
244#95—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—1221no—
248C2×Q8—1222no—
249C2×Q8—1221no—
250C2×Q8—4882no—
251C2×Q8—2441no—
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—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—1222no—
271C4⋊C4—1221no—
272C4⋊C4—1221no—
273C4⋊C4—1221no—
274C4⋊C4—1222no—
275C4⋊C4—1221no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C22⋊C4—41no—
281C22⋊C4—2441no—
282C42—41no—
283C42—2441no—
284C42C2×D24411yescharacteristic
285C42#1175822211yescharacteristic
286C42D4×D6111yescharacteristic
287C42D4×D6111yescharacteristic
288C23#1451977311yescharacteristic
289Q8—42no—
290Q8—81no—
291Q8—4881no—
292Q8—4882no—
293Q8—2444no—
294Q8—2442no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—4882no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2442no—
306C2×C4#1451974511yescharacteristic
307C2×C4#1451974711yescharacteristic
308C2×C4#1451974711yescharacteristic
309C2×C4#1451975011yescharacteristic
310C2×C4#1451975011yescharacteristic
311C2×C4#1451975111yescharacteristic
312C2×C4Q16×D6112yes—
313C2×C4#1451980612yes—
314C8—42no—
315C8—1224no—
316C8—2442no—
317C22—21no—
318C22—21no—
319C22#1683967111yescharacteristic
320C22#1684014011yescharacteristic
321C22#1684015311yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—81no—
327C4—81no—
328C4—161no—
329C4—4882no—
330C4—4881no—
331C4—4881no—
332C4#1683991912yes—
333C4#1683997112yes—
334C2—21no—
335C2—21no—
336C2#2144687311yescharacteristic
337C2#2144687811yescharacteristic
338C2#2144692011yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1518C2
socleC22×C61#392
prime radicalC23×C61#127
hypercenter#1518D61
hypocenterC61#10224
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

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
#214569722
#214567642
#214563412
#214467912
#214443822
#214436632
C61×#15182
#1022461