Finite Group Databasebeta

← #29596255

Subgroups — #29596255

Order 15616.

656 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 107 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#21445747C211yesmaximal, characteristic
2#21445737C211yesmaximal, characteristic
3#21443937C214yesmaximal, complemented
4C61×#2297C211yesmaximal, characteristic
5C2×#14519696—21no—
6C2×#14519694C2212yes—
7C2×#14519692—21no—
8C2×#14519692C2212yes—
9#16839726—21no—
10#16839625—21no—
11C22×#41×C61C2211yescharacteristic
12C2×#117×C61C411yescharacteristic, complemented
13C22×C42×C61C411yescharacteristic, complemented
14C61×#391—21no—
15C61×#390—21no—
16C61×#386C412yescomplemented
17C61×#386C2212yes—
18C4×C22⋊C4×C61—21no—
19C4×C22⋊C4×C61—21no—
20C2×#11758194—22no—
21C2×C22⋊Dic61—22no—
22#14519696—42no—
23#14519694—24no—
24#14519692—24no—
25#14519692—42no—
26C22⋊#92418—42no—
27#14519684—42no—
28C2×#41×C61—24no—
29C2×#41×C61—42no—
30C2×#41×C61—42no—
31C22×Q8×C61D411yescharacteristic
32C22×D4×C61C2×C411yescharacteristic
33C22×D4×C61D411yescharacteristic
34C22×D4×C61C2311yescharacteristic
35C23×C4×C61C2×C411yescharacteristic
36C23×C4×C61C2×C412yes—
37C23×C4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39#117×C61—42no—
40C2×C4⋊C4×C61—21no—
41C2×C22⋊C4×C61—22no—
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61C2×C412yes—
44C2×C42×C61—21no—
45C2×C42×C61—21no—
46C2×C42×C61—21no—
47C2×C42×C61—42no—
48#95×C61—22no—
49#95×C61—22no—
50C24×C61C22⋊C411yescharacteristic
51C24×C61C22×C411yescharacteristic
52C24×C61C2×D411yescharacteristic
53#41×C61—416no—
54C2×Q8×C61—22no—
55C2×Q8×C61—22no—
56C2×Q8×C61—41no—
57C2×D4×C61—22no—
58C2×D4×C61—22no—
59C2×D4×C61—22no—
60C2×D4×C61—41no—
61C2×D4×C61—41no—
62C2×D4×C61—41no—
63C2×D4×C61—41no—
64C2×D4×C61—41no—
65C2×D4×C61#4114yes—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—22no—
69C22×C4×C61—22no—
70C22×C4×C61—22no—
71C22×C4×C61—22no—
72C22×C4×C61—22no—
73C22×C4×C61—22no—
74C22×C4×C61—22no—
75C22×C4×C61—41no—
76C22×C4×C61—41no—
77C22×C4×C61—44no—
78C22×C4×C61—42no—
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×Dic61—42no—
84C4⋊C4×C61—42no—
85C22⋊C4×C61—44no—
86C22⋊C4×C61—44no—
87C22⋊C4×C61—42no—
88C42×C61—42no—
89C42×C61—42no—
90C42×C61—42no—
91C22⋊Dic61—42no—
92C22⋊Dic61—42no—
93#11758194—42no—
94#11758194—42no—
95C2×#92418—42no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—42no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61C2×C22⋊C411yescharacteristic
107C23×C61#11712yes—
108C23×C61#12312yes—
109Q8×C61—44no—
110D4×C61—44no—
111D4×C61—44no—
112D4×C61—44no—
113C2×C4×C61—28no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—24no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—44no—
125C2×C4×C61—44no—
126C2×C4×C61—42no—
127C2×C4×C61—44no—
128C2×C4×C61—42no—
129C2×C4×C61—44no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61#9912yes—
134C2×C4×C61#9912yes—
135C2×C4×C61#12312yes—
136C2×Dic61—82no—
137C2×Dic61—84no—
138#92418—82no—
139#12853—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61—22no—
142C22×C61—22no—
143C22×C61—42no—
144C22×C61—42no—
145C22×C61—42no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61—41no—
150C22×C61—42no—
151C22×C61—81no—
152C22×C61—81no—
153C22×C61#38111yescharacteristic
154C22×C61C2×#9911yescharacteristic
155C22×C61C2×#9911yescharacteristic
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—41no—
160C4×C61—44no—
161C4×C61—82no—
162Dic61—162no—
163#2297D6111yescharacteristic
164#1785—611no—
165#1782—611no—
166#1390—614no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61—42no—
170C2×C61—42no—
171C2×C61—81no—
172C2×C61#181911yescharacteristic
173C2×C61#212012yes—
174C22×#41C2×D6111yescharacteristic
175C2×#117Dic6111yescharacteristic, complemented
176C22×C42Dic6111yescharacteristic, complemented
177C2×#101—1221no—
178C2×#100—612no—
179C2×#99—612no—
180C2×#99—1221no—
181#407—1221no—
182#391—21no—
183#390—21no—
184#386Dic6112yescomplemented
185#386C2×D6112yes—
186#381—1221no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C61#1285311yescharacteristic, complemented, Sylow
190C2×#41—24no—
191C2×#41—42no—
192C2×#41—42no—
193C22×Q8C22⋊D6111yescharacteristic
194C22×D4C2×Dic6111yescharacteristic
195C22×D4C22⋊D6111yescharacteristic
196C22×D4C22×D6111yescharacteristic
197C23×C4C2×Dic6111yescharacteristic
198C23×C4C2×Dic6112yes—
199C23×C4C22⋊D6111yescharacteristic
200C23×C4C22⋊D6111yescharacteristic
201C2×M16—1222no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—1222no—
207C2×C22⋊C4C2×Dic6112yes—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—42no—
212#101—2442no—
213#100—1224no—
214#99—1224no—
215#99—2442no—
216C22⋊C8—2442no—
217#95—22no—
218#95—22no—
219#95—2442no—
220C24C22⋊Dic6111yescharacteristic
221C24C22×Dic6111yescharacteristic
222C24C2×C22⋊D6111yescharacteristic
223#41—416no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4#1175822214yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—2442no—
250C22×C4C22⋊Dic6111yescharacteristic
251C22×C4C22⋊Dic6111yescharacteristic
252C22×C4C22⋊Dic6111yescharacteristic
253C22×C4C2×C22⋊D6111yescharacteristic
254M16—2442no—
255M16—2442no—
256C2×C8—2442no—
257C4⋊C4—42no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—2442no—
262C22⋊C4—2442no—
263C42—42no—
264C42—42no—
265C42—42no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23C2×C22⋊Dic6111yescharacteristic
277C23#1451973612yes—
278C23#1451976012yes—
279Q8—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283C2×C4—28no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—4882no—
304C2×C4—4884no—
305C2×C4#1451969212yes—
306C2×C4#1451969212yes—
307C2×C4#1451976012yes—
308C8—4882no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22#1683962511yescharacteristic
323C22C2×#1451969211yescharacteristic
324C22C2×#1451969211yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—44no—
330C4—82no—
331C4—9762no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—42no—
336C2—81no—
337C2#2144589311yescharacteristic
338C2#2144746012yes—
subgroupG/H
centerC22C2×#14519692
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#2297C2
socleC22×C61C2×#99
prime radicalC22×#41×C61C22
cosocleC22×D4×C61C23
hypercenter#2297D61
hypocenterC61#12853
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
3. #21443937C2semidirect
12. C2×#117×C61C4semidirect
13. C22×C42×C61C4semidirect
16. C61×#386C4semidirect
175. C2×#117Dic61semidirect
176. C22×C42Dic61semidirect
184. #386Dic61semidirect
189. C61#12853semidirect

Maximal subgroups

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

subgroupindex
#214457472
#214457372
#214439372
C61×#22972
#1285361