Finite Group Databasebeta

← #29736766

Subgroups — #29736766

Order 15616.

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

normal subgroups only

This group has 954 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
1#21459629C211yesmaximal, characteristic
2#21456199C212yesmaximal
3#21446980C214yesmaximal
4#21446762C212yesmaximal
5#21446213C214yesmaximal
6C4×#14519696C211yesmaximal, characteristic
7C61×#2546C211yesmaximal, characteristic
8Q8×Dic122—22no—
9#16840491—22no—
10#16840488—24no—
11#16840393—22no—
12#16840393C2212yes—
13#16840319—22no—
14#16840319C2212yes—
15#16840240C2212yes—
16#16840130—24no—
17#16840112—22no—
18#16840112—22no—
19#16839771C2218yes—
20C2×#14519696C2212yes—
21C4×#11758194C2212yes—
22#16839531C2212yes—
23#16839522C2218yes—
24C2×C4⋊Q8×C61C2212yes—
25C2×C4⋊Q8×C61C2211yescharacteristic
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#398—22no—
28C61×#391C2212yes—
29C61×#389—22no—
30C4×C4⋊C4×C61C2211yescharacteristic
31C2×#11758194C2314yes—
32#14519777—22no—
33#14519777—22no—
34#14519777D412yes—
35#14519777D412yes—
36#14519776—28no—
37#14519776—24no—
38#14519769—24no—
39#14519769D414yes—
40#14519768—28no—
41#14519768—24no—
42#14519756—22no—
43#14519756—24no—
44#14519756—22no—
45Q8×Dic61—24no—
46Q8×Dic61—24no—
47C4×Dic122—24no—
48C4×Dic122—24no—
49C4×Dic122—22no—
50C4×Dic122—22no—
51#14519736—24no—
52#14519736C2314yes—
53#14519701—28no—
54#14519701—28no—
55#14519696—24no—
56#14519688—22no—
57C4×#92418—22no—
58C22×Q8×C61C2312yes—
59C22×Q8×C61C2311yescharacteristic
60C4⋊Q8×C61—28no—
61C4⋊Q8×C61—22no—
62C4⋊Q8×C61—22no—
63C4×Q8×C61—22no—
64C4×Q8×C61—22no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—24no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D412yes—
70C2×C4⋊C4×C61C2311yescharacteristic
71C2×C42×C61D412yes—
72C2×C42×C61C2312yes—
73C2×C42×C61C2311yescharacteristic
74#95×C61—21no—
75#95×C61—24no—
76C2×Q8×C61—22no—
77C2×Q8×C61—22no—
78C2×Q8×C61—24no—
79C2×Q8×C61—24no—
80C2×Q8×C61—24no—
81C2×Q8×C61#4114yes—
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D412yes—
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2411yescharacteristic
89C2×Dic122—22no—
90C2×Dic122—24no—
91C2×Dic122—22no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—24no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—42no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—48no—
99C42×C61—22no—
100C42×C61—22no—
101C42×C61—24no—
102C42×C61C2×D412yes—
103C42×C61C2×D412yes—
104C22⋊Dic61—24no—
105C4⋊Dic61—24no—
106C4⋊Dic61—24no—
107C4⋊Dic61—24no—
108C4⋊Dic61—22no—
109C4⋊Dic61—22no—
110C4⋊Dic61C2×D414yes—
111C4⋊Dic61C2×D414yes—
112#11758196—28no—
113#11758196—24no—
114#11758196—24no—
115C4×Dic61—28no—
116C4×Dic61—24no—
117#11758194—24no—
118#11758194—24no—
119C2×#92418—24no—
120C23×C61C22×D411yescharacteristic
121Q8×C61—28no—
122Q8×C61—48no—
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—22no—
126C2×C4×C61—24no—
127C2×C4×C61—22no—
128C2×C4×C61—41no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61#12011yescharacteristic
136C2×C4×C61#12012yes—
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12414yes—
139C2×C4×C61C22×D412yes—
140C2×C4×C61C2×#4112yes—
141C2×C4×C612−1+412yes—
142C2×Dic61—24no—
143C2×Dic61—24no—
144Dic122—44no—
145Dic122—82no—
146Dic122—84no—
147#92418—44no—
148#25919—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61C2×#12011yescharacteristic
152C22×C61C2×#12411yescharacteristic
153C22×C61#53611yescharacteristic
154C4×C61—24no—
155C4×C61—21no—
156C4×C61—22no—
157C4×C61—44no—
158C4×C61—42no—
159C4×C61—82no—
160C4×C61#45612yes—
161C4×C61#54814yes—
162Dic61—44no—
163Dic61—44no—
164#3502—611no—
165#3139—612no—
166#2546D6111yescharacteristic
167#2028—614no—
168#1986—612no—
169#1886—614no—
170C4×#101—611no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61#243611yescharacteristic
174C2×C61C2×#45611yescharacteristic
175C2×C61#302211yescharacteristic
176Q82—1222no—
177#557—1224no—
178#557—1222no—
179#544—612no—
180#544—1222no—
181#533—612no—
182#533—1222no—
183C2×C4⋊Q8C2×D6112yes—
184C2×C4⋊Q8C2×D6111yescharacteristic
185#519—612no—
186C2×C4×Q8C2×D6111yescharacteristic
187#491—1224no—
188#487—1222no—
189#487—1222no—
190#419—618no—
191C2×#101—612no—
192C4×M16—612no—
193#398—22no—
194#391C2×D6112yes—
195#389—22no—
196C4×C4⋊C4C2×D6111yescharacteristic
197#343—612no—
198#339—618no—
199C61#2591911yescharacteristic, complemented, Sylow
200C22×Q8C22×D6112yes—
201C22×Q8C22×D6111yescharacteristic
202C2×M16—614no—
203C4⋊Q8—28no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—612no—
207C4⋊Q8—612no—
208C4⋊Q8—1222no—
209C4⋊Q8—1222no—
210C4⋊Q8—1224no—
211C4⋊Q8—1228no—
212#125—614no—
213#125—1224no—
214#125—1224no—
215#125—1228no—
216C22⋊Q8—1222no—
217C22⋊Q8—1222no—
218C22⋊Q8—1224no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—1224no—
222C4×Q8—1224no—
223C4×Q8—1224no—
224C4×Q8—1224no—
225C4×Q8—1222no—
226C4×Q8—1222no—
227#117—614no—
228#117—1224no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C22⋊D6112yes—
234C2×C4⋊C4C22×D6111yescharacteristic
235C2×C42C22⋊D6112yes—
236C2×C42C22×D6112yes—
237C2×C42C22×D6111yescharacteristic
238#103—1228no—
239#103—1228no—
240#101—1224no—
241#97—1222no—
242C4×C8—1222no—
243#95—21no—
244#95—24no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—1224no—
251C2×Q8—1222no—
252C2×Q8—1222no—
253C2×Q8#1175822414yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4C2×C22⋊D6111yescharacteristic
257C22×C4C2×C22⋊D6112yes—
258C22×C4C2×C22⋊D6112yes—
259C22×C4C2×C22⋊D6111yescharacteristic
260C22×C4C23×D6111yescharacteristic
261M16—1224no—
262M16—1224no—
263C2×C8—1224no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—48no—
271C4⋊C4—614no—
272C4⋊C4—614no—
273C4⋊C4—1228no—
274C4⋊C4—1224no—
275C4⋊C4—1224no—
276C4⋊C4—1224no—
277C4⋊C4—1224no—
278C4⋊C4—1224no—
279C4⋊C4—1222no—
280C4⋊C4—1222no—
281C22⋊C4—1224no—
282C42—22no—
283C42—22no—
284C42—24no—
285C42—1228no—
286C42—1224no—
287C42D4×D6112yes—
288C42D4×D6112yes—
289C23C22×C22⋊D6111yescharacteristic
290Q8—28no—
291Q8—48no—
292Q8—4882no—
293Q8—4884no—
294Q8—2444no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—1224no—
308C2×C4—1224no—
309C2×C4#1451974511yescharacteristic
310C2×C4#1451974512yes—
311C2×C4#1451974511yescharacteristic
312C2×C4#1451976414yes—
313C2×C4C2×D4×D6112yes—
314C2×C4C2×#1175822412yes—
315C2×C4#1451982412yes—
316C8—2444no—
317C22—21no—
318C22—21no—
319C22C2×#1451974511yescharacteristic
320C22C2×#1451976411yescharacteristic
321C22#1684033311yescharacteristic
322C4—24no—
323C4—21no—
324C4—22no—
325C4—44no—
326C4—42no—
327C4—82no—
328C4—2444no—
329C4—2444no—
330C4#1683994412yes—
331C4#1684041814yes—
332C2—21no—
333C2—21no—
334C2#2144947811yescharacteristic
335C2C2×#1683994411yescharacteristic
336C2#2145519711yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2546C2
socleC22×C61C2×#120
prime radicalC23×C61C22×D4
hypercenter#2546D61
hypocenterC61#25919
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

Complements

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

NH ≅ G/Ntype
199. C61#25919semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214561992
#214469802
#214467622
#214462132
C4×#145196962
C61×#25462
#2591961