Finite Group Databasebeta

← #29571739

Subgroups — #29571739

Order 15616.

966 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×Q8×D244C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic, complemented
3#21444490C212yesmaximal, complemented
4#21444245C212yesmaximal
5C61×#1461C211yesmaximal, characteristic, complemented
6C22×Q8×D61—21no—
7Q8×D244—24no—
8Q8×D244—24no—
9C2×#14519777C412yes—
10C2×#14519777C2211yescharacteristic
11C2×#14519757—22no—
12C2×#14519757—21no—
13C2×C4×D244C412yes—
14C2×C4×D244C2211yescharacteristic
15#16839786C2212yes—
16C2×#14519696—21no—
17#16839531—21no—
18#16839507—24no—
19#16839491—24no—
20C2×C4×Q8×C61C2211yescharacteristic
21C61×#431C2212yes—
22C61×#324—22no—
23C22×Dic122—22no—
24C22×Dic122—21no—
25C2×Q8×D61—28no—
26C2×Q8×D61—44no—
27C2×Q8×D61—42no—
28C2×Q8×D61—42no—
29C22×D244C2×C411yescharacteristic
30C22×C4×D61—22no—
31C22×C4×D61—21no—
32C2×#11758194—22no—
33#14519777—22no—
34#14519777—24no—
35#14519777—24no—
36#14519777D414yes—
37#14519757—28no—
38#14519757—216no—
39C4×D244—24no—
40C4×D244—22no—
41C4×D244—24no—
42C4×D244D414yes—
43C2×C4⋊Dic61—22no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61C2×C412yes—
46C2×C4⋊Dic61C2×C411yescharacteristic
47C2×#11758198—22no—
48C2×#11758198—21no—
49C4⋊#92418—22no—
50C4⋊#92418—22no—
51#14519696—42no—
52C22×Q8×C61D411yescharacteristic
53C4×Q8×C61—22no—
54C4×Q8×C61—22no—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C4⋊C4×C61D412yes—
57C2×C42×C61C2×C412yes—
58C2×C42×C61C2311yescharacteristic
59C4⋊C8×C61—22no—
60C22⋊C8×C61—22no—
61C4×C8×C61—22no—
62C23×D61—21no—
63C2×Q8×C61—22no—
64C2×Q8×C61—22no—
65C2×Q8×C61D812yes—
66C2×Q8×C61SD1612yes—
67C22×C4×C61C22⋊C412yes—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×Dic61—22no—
73C22×Dic61—21no—
74Q8×D61—416no—
75Q8×D61—416no—
76C2×D244—21no—
77C2×D244—24no—
78C2×D244C22⋊C412yes—
79C2×C4×D61—28no—
80C2×C4×D61—24no—
81C2×C4×D61—44no—
82C2×C4×D61—44no—
83C2×C4×D61—41no—
84C2×C4×D61—41no—
85C2×C4×D61—42no—
86C2×Dic122—28no—
87C2×Dic122—24no—
88C2×Dic122—44no—
89C2×Dic122—44no—
90C2×Dic122—41no—
91C2×Dic122—41no—
92C2×Dic122—42no—
93C2×C8×C61—42no—
94C4⋊C4×C61—24no—
95C4⋊C4×C61—22no—
96C42×C61—22no—
97C42×C61—22no—
98C42×C61C2×D412yes—
99C42×C61C2×D412yes—
100#11758198—28no—
101#11758198—24no—
102C4⋊Dic61—28no—
103C4⋊Dic61—21no—
104C4⋊Dic61—22no—
105C4⋊Dic61—24no—
106C4⋊Dic61C22⋊C414yes—
107C4⋊Dic61C22⋊C412yes—
108#11758194—42no—
109#11758194—42no—
110C2×#92418—42no—
111C23×C61C2×C22⋊C411yescharacteristic
112C22×D61—24no—
113C22×D61—22no—
114C22×D61—42no—
115C22×D61—41no—
116C22×D61—41no—
117Q8×C61—24no—
118Q8×C61—24no—
119C2×C4×C61—22no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—41no—
124C2×C4×C61—42no—
125C2×C4×C61#10214yes—
126C2×C4×C61C2×C22⋊C411yescharacteristic
127C2×C4×C61C2×C22⋊C411yescharacteristic
128C2×C4×C61#12012yes—
129C2×C4×C61#12012yes—
130C2×C4×C61C2×D811yescharacteristic
131C2×C4×C61C2×SD1611yescharacteristic
132C2×C4×C61AGL(1,ℤ/8)12yes—
133C2×Dic61—22no—
134C2×Dic61—24no—
135C2×Dic61—41no—
136C2×Dic61—42no—
137C2×Dic61—41no—
138C2×Dic61—42no—
139D244—24no—
140D244—22no—
141D244#10114yescomplemented
142C4×D61—416no—
143C4×D61—44no—
144C4×D61—44no—
145Dic122—416no—
146Dic122—44no—
147Dic122—44no—
148C8×C61—82no—
149#92418—82no—
150#8388—611nomaximal, Sylow
151C22×C61—21no—
152C22×C61—21no—
153C22×C61#37911yescharacteristic
154C22×C61C2×#10211yescharacteristic
155C22×C61#41811yescharacteristic
156C2×D61—24no—
157C2×D61—44no—
158C2×D61—44no—
159C2×D61—42no—
160C2×D61—42no—
161C4×C61—22no—
162C4×C61—21no—
163C4×C61—24no—
164C4×C61—42no—
165C4×C61C2×#10112yes—
166C4×C61#44712yes—
167C4×C61#45012yes—
168Dic61—42no—
169Dic61—44no—
170C2×D4×Q8—611no—
171#1801—611no—
172#1546—612no—
173#1486—612no—
174#1461D6111yescharacteristic, complemented
175C2×C61—21no—
176C2×C61—21no—
177C2×C61#178411yescharacteristic
178C2×C61#178611yescharacteristic
179C2×C61#179611yescharacteristic
180D61—44no—
181C23×Q8—1221no—
182D4×Q8—1224no—
183D4×Q8—1224no—
184C2×C4⋊Q8—612no—
185C2×C4⋊Q8—611no—
186C2×C22⋊Q8—1221no—
187C2×C22⋊Q8—1222no—
188C2×C4×Q8C2×D6111yescharacteristic
189C2×C4×D4—612no—
190C2×C4×D4—611no—
191#431C2×D6112yes—
192#423—612no—
193C2×#101—1221no—
194#343—1221no—
195#332—1224no—
196D4⋊C8—1224no—
197#324—22no—
198C61#838811yescharacteristic, complemented, Sylow
199C22×Q8—1228no—
200C22×Q8—1222no—
201C22×Q8—1221no—
202C22×Q8—2444no—
203C22×Q8—2442no—
204C22×Q8—2442no—
205C22×Q8D24411yescharacteristic
206C22×D4—611no—
207C23×C4—1222no—
208C23×C4—1221no—
209C2×M16—1222no—
210C4⋊Q8—614no—
211C4⋊Q8—1224no—
212C4⋊Q8—1224no—
213C4⋊Q8—1222no—
214C22⋊Q8—1228no—
215C22⋊Q8—12216no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×D4—614no—
219C4×D4—1224no—
220C4×D4—1224no—
221C4×D4—1222no—
222C2×C4⋊C4—611no—
223C2×C4⋊C4—612no—
224C2×C4⋊C4—1222no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4D24411yescharacteristic
227C2×C4⋊C4C22⋊D6112yes—
228C2×C22⋊C4—1222no—
229C2×C22⋊C4—1221no—
230C2×C42C4×D6112yes—
231C2×C42C22×D6111yescharacteristic
232C4⋊C8—22no—
233C4⋊C8—1222no—
234C4⋊C8—1222no—
235#101—2442no—
236C22⋊C8—22no—
237C4×C8—22no—
238C24—1221no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—1228no—
242C2×Q8—1224no—
243C2×Q8—2444no—
244C2×Q8—24416no—
245C2×Q8—24416no—
246C2×Q8—2444no—
247C2×Q8—2441no—
248C2×Q8—2441no—
249C2×Q8—2442no—
250C2×Q8#1175819012yes—
251C2×Q8D48812yes—
252C2×D4—612no—
253C2×D4—1224no—
254C2×D4—1221no—
255C22×C4—1228no—
256C22×C4—1222no—
257C22×C4—1221no—
258C22×C4—1224no—
259C22×C4—2444no—
260C22×C4—2444no—
261C22×C4—2441no—
262C22×C4—2441no—
263C22×C4—2442no—
264C22×C4#1175819812yes—
265C22×C4#1175819812yes—
266C22×C4C2×C4×D6111yescharacteristic
267C22×C4C2×D24411yescharacteristic
268C22×C4C2×C22⋊D6111yescharacteristic
269M16—2442no—
270M16—2442no—
271C2×C8—42no—
272C2×C8—2442no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—612no—
276C4⋊C4—614no—
277C4⋊C4—1228no—
278C4⋊C4—1221no—
279C4⋊C4—1222no—
280C4⋊C4—1224no—
281C22⋊C4—1228no—
282C22⋊C4—1224no—
283C42—22no—
284C42—22no—
285C42D4×D6112yes—
286C42D4×D6112yes—
287C23—1224no—
288C23—1222no—
289C23—2442no—
290C23—2441no—
291C23—2441no—
292C23C2×#1175819811yescharacteristic
293Q8—24no—
294Q8—24no—
295Q8—24416no—
296Q8—2444no—
297Q8—2444no—
298D4—614no—
299D4—1224no—
300D4—1222no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—1224no—
308C2×C4—1222no—
309C2×C4—24416no—
310C2×C4—2442no—
311C2×C4—2441no—
312C2×C4—2442no—
313C2×C4—2441no—
314C2×C4—2444no—
315C2×C4—2444no—
316C2×C4#1451969914yes—
317C2×C4C22⋊C4×D6111yescharacteristic
318C2×C4C22⋊C4×D6111yescharacteristic
319C2×C4#1451974512yes—
320C2×C4#1451974712yes—
321C2×C4C2×D48811yescharacteristic
322C2×C4C2×#1175819011yescharacteristic
323C2×C4#1451980512yes—
324C8—82no—
325C8—4882no—
326C22—21no—
327C22—21no—
328C22—1224no—
329C22—2444no—
330C22—2444no—
331C22—2442no—
332C22—2442no—
333C22#1683961711yescharacteristic
334C22C2×#1451969911yescharacteristic
335C22#1683976811yescharacteristic
336C4—21no—
337C4—24no—
338C4—22no—
339C4—42no—
340C4—2444no—
341C4—2442no—
342C4#101×D6112yes—
343C4#1683989812yes—
344C4#1683991512yes—
345C2—21no—
346C2—21no—
347C2—2444no—
348C2#2144574111yescharacteristic
349C2#2144575111yescharacteristic
350C2#2144579211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1461C2
socleC22×C61#379
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1461D61
hypocenterC61#8388
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #21445815C2semidirect
3. #21444490C2semidirect
5. C61×#1461C2semidirect
141. D244#101semidirect
174. #1461D61semidirect
198. C61#8388semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D2442
#214458152
#214444902
#214442452
C61×#14612
#838861