Finite Group Databasebeta

← #30604804

Subgroups — #30604804

Order 15616.

440 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 84 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#21462047C211yesmaximal, characteristic
2C2×#16840638C211yesmaximal, characteristic
3#21460812C212yesmaximal, complemented
4#21460811C212yesmaximal, complemented
5#467×D61C211yesmaximal, characteristic
6C22×#11758213—21no—
7C2×#14519852—21no—
8#16840657C2212yes—
9C22⋊#11758187—21no—
10#16840638—21no—
11#16840638C2212yes—
12C2×Q16×D61C411yescharacteristic
13C22×C8×D61C2211yescharacteristic
14C22⋊Q8×D61—21no—
15#16840004C2211yescharacteristic
16#16840003C411yescharacteristic
17#16840002—21no—
18#16840001—21no—
19#16840000C2211yescharacteristic
20#107×D61C411yescharacteristic
21#103×D61—21no—
22C61×#467C411yescharacteristic
23C2×#11758213—24no—
24C2×#11758213—22no—
25C2×#11758213—42no—
26C2×#11758213—41no—
27C22×#92420—41no—
28#14519852—42no—
29C2×#11758187—22no—
30C2×Q8×D61—21no—
31C22×C4×D61D411yescharacteristic
32C2×Dic244C2×C411yescharacteristic
33C2×#11758202—21no—
34Q16×D61—42no—
35C22×#92418D411yescharacteristic
36C2×C8×D61—42no—
37C2×C8×D61C2×C411yescharacteristic
38C2×C8×D61D411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61C2311yescharacteristic
41#14519757—21no—
42#14519756—21no—
43#14519755—41no—
44#14519754—41no—
45#14519753—21no—
46C4⋊C4×D61—21no—
47C4⋊C4×D61—41no—
48C22⋊C4×D61—41no—
49#14519711C2×C411yescharacteristic
50#14519710—21no—
51#14519702—21no—
52#14519701—21no—
53#14519700—21no—
54C2×Q16×C61C2×C411yescharacteristic
55C22×C8×C61C2×C411yescharacteristic
56C22⋊Q8×C61—21no—
57#107×C61C2×C411yescharacteristic
58#103×C61—21no—
59C23×D61SD1611yescharacteristic
60C2×Q8×C61—21no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×Dic61D811yescharacteristic
63Q8×D61—42no—
64Q8×D61—42no—
65C2×C4×D61—22no—
66C2×C4×D61—22no—
67C2×C4×D61—41no—
68C2×C4×D61—41no—
69C2×C4×D61C22⋊C411yescharacteristic
70C2×C4×D61D811yescharacteristic
71C2×C4×D61SD1611yescharacteristic
72C2×C4×D61C2×D411yescharacteristic
73C2×Dic122—21no—
74C2×Dic122—41no—
75C2×#92420—44no—
76C2×#92420—42no—
77#11758213—24no—
78#11758213—44no—
79#11758213—42no—
80Q16×C61—41no—
81C2×C8×C61—41no—
82C2×C8×C61C22⋊C411yescharacteristic
83C2×C8×C61C22×C411yescharacteristic
84C4⋊C4×C61—21no—
85C4⋊C4×C61—41no—
86C22⋊C4×C61—41no—
87C22⋊Dic61—41no—
88#11758202—42no—
89#11758198—41no—
90#11758198—41no—
91C4⋊Dic61—21no—
92C4⋊Dic61—41no—
93#11758196—41no—
94#11758196—41no—
95#11758196—41no—
96C2×#92418—41no—
97C2×#92418C22⋊C411yescharacteristic
98C2×#92418C2×D411yescharacteristic
99Dic244—41no—
100C8×D61—22no—
101C8×D61—22no—
102C8×D61—41no—
103C8×D61—41no—
104#11758187—42no—
105C23×C61#10211yescharacteristic
106C22×D61—22no—
107C22×D61—22no—
108C22×D61—41no—
109C22×D61#10211yescharacteristic
110C22×D61C2×SD1611yescharacteristic
111Q8×C61—41no—
112Q8×C61—41no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61#10211yescharacteristic
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×Dic61—22no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61#10211yescharacteristic
123C2×Dic61C2×D811yescharacteristic
124C4×D61—22no—
125C4×D61—21no—
126C4×D61—21no—
127C4×D61—22no—
128C4×D61—82no—
129C4×D61—82no—
130C4×D61#9912yes—
131C4×D61#10412yes—
132Dic122—41no—
133Dic122—41no—
134Dic122—82no—
135#92420—44no—
136C8×C61—21no—
137C8×C61—21no—
138#92418—21no—
139#92418—21no—
140#11712—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61#36212yes—
144C22×C61C2×#10211yescharacteristic
145C2×D61—21no—
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—44no—
151C2×D61#32712yes—
152C4×C61—21no—
153C4×C61—81no—
154C4×C61—81no—
155C4×C61C2×#9911yescharacteristic
156C4×C61C2×#10411yescharacteristic
157Dic61—21no—
158Dic61—81no—
159Dic61—81no—
160Dic61#32712yes—
161C2×#467—611no—
162#2112—611no—
163C2×#341—611no—
164#1350—612no—
165#1350—612no—
166C2×C61—22no—
167C2×C61C2×#32711yescharacteristic
168C2×C61#214311yescharacteristic
169C2×C61C2×#36211yescharacteristic
170D61—22no—
171D61—41no—
172C22×Q16—611no—
173C22×M16—1221no—
174C23×C8—611no—
175C2×C22⋊Q8—1221no—
176C2×M32—1221no—
177#467—611no—
178#467—611no—
179#467—611no—
180#467—1221no—
181#467—1221no—
182#467#692311yescharacteristic
183C2×#107—611no—
184C2×#103—1221no—
185C22⋊C16—612no—
186C22⋊C16—1221no—
187#341—612no—
188#341—1221no—
189C61#1171211yescharacteristic, complemented, Sylow
190C22×Q8—1221no—
191C23×C4—611no—
192C2×Q16—611no—
193C2×Q16—1221no—
194C2×Q16—2442no—
195C2×Q16C2×#692311yescharacteristic
196C2×M16—1222no—
197C2×M16—1224no—
198C2×M16—2441no—
199C2×M16—2442no—
200C22×C8—611no—
201C22×C8—611no—
202C22×C8—611no—
203C22×C8—611no—
204C22×C8—611no—
205C22×C8—2442no—
206C22×C8—2441no—
207C22×C8C2×#692311yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8—1221no—
210C22⋊Q8—1221no—
211C22⋊Q8—1221no—
212C22⋊Q8—2441no—
213C22⋊Q8—2441no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4—2441no—
216C2×C22⋊C4—2441no—
217M32—2442no—
218C2×C16—1222no—
219#107—611no—
220#107—1221no—
221#107C2×#692311yescharacteristic
222#103—21no—
223#103—1221no—
224#103—1221no—
225#103—1221no—
226C24—611no—
227C2×Q8—21no—
228C2×Q8—1221no—
229C2×Q8—2441no—
230C2×Q8—2442no—
231C2×Q8—2442no—
232C22×C4—611no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—1222no—
238C22×C4—1222no—
239C22×C4—2441no—
240C22×C4—2441no—
241C22×C4C22⋊#692311yescharacteristic
242Q16—41no—
243Q16—2441no—
244Q16—2442no—
245M16—1224no—
246M16—2444no—
247M16—2442no—
248C2×C8—41no—
249C2×C8—611no—
250C2×C8—611no—
251C2×C8—1222no—
252C2×C8—1222no—
253C2×C8—2441no—
254C2×C8—2441no—
255C2×C8—2441no—
256C2×C8—2442no—
257C2×C8—2444no—
258C2×C8C22⋊#692311yescharacteristic
259C2×C8C22×#692311yescharacteristic
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—1221no—
263C4⋊C4—2441no—
264C4⋊C4—2441no—
265C4⋊C4—2441no—
266C4⋊C4—2441no—
267C22⋊C4—41no—
268C22⋊C4—2441no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C16—2442no—
272C23—611no—
273C23—611no—
274C23—1222no—
275C23—1222no—
276C23—2441no—
277C23#1451983911yescharacteristic
278Q8—41no—
279Q8—41no—
280Q8—4882no—
281Q8—2441no—
282Q8—2441no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—611no—
288C2×C4—612no—
289C2×C4—612no—
290C2×C4—611no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1222no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—4882no—
297C2×C4—4882no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4#1451983911yescharacteristic
301C2×C4C2×C22⋊#692311yescharacteristic
302C8—21no—
303C8—21no—
304C8—1221no—
305C8—1221no—
306C8—2444no—
307C22—21no—
308C22—21no—
309C22—612no—
310C22—1221no—
311C22—1221no—
312C22—1221no—
313C22—1221no—
314C22—1221no—
315C22—2444no—
316C22#1684068112yes—
317C22C2×#1451983911yescharacteristic
318C4—21no—
319C4—81no—
320C4—81no—
321C4—612no—
322C4—1221no—
323C4—4881no—
324C4—4881no—
325C4D61⋊#9911yescharacteristic
326C4D61⋊#10411yescharacteristic
327C2—22no—
328C2—1222no—
329C2—2441no—
330C2D61⋊#32711yescharacteristic
331C2#2146213311yescharacteristic
332C2C2×#1684068111yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#467C4
socleC22×C61C2×#102
prime radicalC23×D61SD16
cosocleC2×C8×D61C23
hypercenter#467#6923
hypocenterC61#11712
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #467

Complements

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

NH ≅ G/Ntype
3. #21460812C2semidirect
4. #21460811C2semidirect
189. C61#11712semidirect

Maximal subgroups

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

subgroupindex
#214620472
C2×#168406382
#214608122
#214608112
#467×D612
#1171261