Finite Group Databasebeta

← #30601744

Subgroups — #30601744

Order 15616.

720 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 134 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#21461605C211yesmaximal, characteristic, complemented
2C4×#14519844C211yesmaximal, characteristic, complemented
3C2×#16840610C214yesmaximal, complemented
4#21448381C211yesmaximal, characteristic
5C2×#14519844—21no—
6C2×C4⋊#92420C2212yes—
7C2×#14519832—21no—
8C2×C4×#92420C2212yes—
9#16840628—21no—
10#16840628—21no—
11#16840610—28no—
12C2×#14519774C411yescharacteristic
13C2×#14519739—21no—
14C2×#14519739—21no—
15C2×#14519739C2212yes—
16#16839663—21no—
17#16839663—21no—
18#16839663C412yes—
19C42×Dic61C2211yescharacteristic
20C61×#384C411yescharacteristic
21C22×#92420—22no—
22C22×#92420—22no—
23#14519844—24no—
24C4⋊#92420—22no—
25C4⋊#92420D414yescomplemented
26#14519832—42no—
27C4×#92420—24no—
28C22×D244—21no—
29C22×D244—21no—
30C22×D244C2×C412yes—
31C22×C4×D61—22no—
32#14519774—24no—
33#14519739—24no—
34#14519739—24no—
35#14519739—44no—
36#14519739—42no—
37#14519739—42no—
38C2×#11758198—22no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×C4×Dic61C2×C411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61C2311yescharacteristic
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61C2×C412yes—
50C2×C42×C61C2×C411yescharacteristic
51C23×D61—22no—
52C22×C4×C61—22no—
53C22×C4×C61C22⋊C411yescharacteristic
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22×C411yescharacteristic
56C22×Dic61C22⋊C411yescharacteristic
57C22×Dic61C22⋊C411yescharacteristic
58C22×Dic61C2×D411yescharacteristic
59C22×Dic61C2×D411yescharacteristic
60C2×D244—24no—
61C2×D244—24no—
62C2×D244—24no—
63C2×D244—24no—
64C2×D244—22no—
65C2×D244—42no—
66C2×D244—42no—
67C2×D244—42no—
68C2×D244—42no—
69C2×D244M1614yes—
70C2×C4×D61—44no—
71C2×C4×D61—42no—
72C2×C4×D61—42no—
73C2×C4×D61—44no—
74C2×#92420—24no—
75C2×#92420—24no—
76C2×#92420—42no—
77C2×#92420—42no—
78C2×#92420—42no—
79C2×#92420—42no—
80C4⋊C4×C61—24no—
81C4⋊C4×C61—42no—
82C4⋊C4×C61—42no—
83C42×C61—22no—
84#11758198—44no—
85#11758198—44no—
86#11758198—42no—
87#11758198—42no—
88C4×Dic61—22no—
89C4×Dic61—22no—
90C4×Dic61—22no—
91C4×Dic61—41no—
92C4×Dic61D812yes—
93C4×Dic61SD1612yes—
94C4×Dic61C2×D412yes—
95C4×Dic61#4112yes—
96C23×C61C2×C22⋊C411yescharacteristic
97C22×D61—24no—
98C22×D61—42no—
99C22×D61—42no—
100C22×D61—42no—
101C22×D61—42no—
102C22×D61—42no—
103C22×D61—42no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61#10212yes—
112C2×C4×C61#10412yes—
113C2×C4×C61C4×D412yes—
114C2×C4×C61C2×M1612yes—
115C2×C4×C61#13112yes—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61#10212yes—
124C2×Dic61#12112yes—
125C2×Dic61#12411yescharacteristic
126C2×Dic61#12711yescharacteristic
127C2×Dic61C2×D811yescharacteristic
128C2×Dic61C2×SD1611yescharacteristic
129C2×Dic61AGL(1,ℤ/8)12yes—
130D244—28no—
131D244—48no—
132D244—48no—
133D244—44no—
134C4×D61—84no—
135C4×D61—84no—
136#92420—44no—
137#92420—44no—
138#9678—611nomaximal, Sylow
139C22×C61#39011yescharacteristic
140C22×C61#40711yescharacteristic
141C22×C61#40811yescharacteristic
142C22×C61C2×#10211yescharacteristic
143C22×C61#41811yescharacteristic
144C22×C61C2×#10411yescharacteristic
145C22×C61#42111yescharacteristic
146C2×D61—44no—
147C2×D61—44no—
148C2×D61—44no—
149C2×D61—82no—
150C2×D61—82no—
151C2×D61—82no—
152C2×D61—82no—
153C4×C61—22no—
154C4×C61—22no—
155C4×C61—82no—
156C4×C61#43514yes—
157Dic61—21no—
158Dic61—21no—
159Dic61—22no—
160Dic61—41no—
161Dic61#45912yes—
162Dic61#46012yes—
163C4×#127—611no—
164#1767—611no—
165C4×C4⋊C8—611no—
166C2×D4⋊C8—614no—
167C2×C61#148611yescharacteristic
168C2×C61#149011yescharacteristic
169C2×C61#149211yescharacteristic
170C2×C61#149611yescharacteristic
171C2×C61#195511yescharacteristic
172C2×C61#198011yescharacteristic
173C2×C61#198211yescharacteristic
174D61—84no—
175C2×#127—611no—
176C2×C4×D4—612no—
177C2×C4×D4—1221no—
178C2×C4×D4—1221no—
179C2×C4⋊C8—612no—
180C2×C4⋊C8—1221no—
181C2×#97—1221no—
182C2×C4×C8—612no—
183#390—612no—
184#390—1221no—
185#390—1221no—
186#384#692311yescharacteristic
187C43—611no—
188#336—1221no—
189#336—1221no—
190D4⋊C8—1228no—
191C61#967811yescharacteristic, complemented, Sylow
192C22×D4—612no—
193C22×D4—1221no—
194C22×D4—1221no—
195C23×C4—1222no—
196C22×C8—1222no—
197C22×C8—1222no—
198#127—1224no—
199C4×D4—1224no—
200C4×D4—1224no—
201C4×D4—2442no—
202C4×D4—2442no—
203C4×D4—2444no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×#692312yes—
207C2×C22⋊C4—1222no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1222no—
210C2×C42—611no—
211C2×C42—611no—
212C2×C42—611no—
213C2×C42—611no—
214C2×C42—611no—
215C2×C42—611no—
216C2×C42C2×#692311yescharacteristic
217C4⋊C8—614no—
218C4⋊C8—1222no—
219C4⋊C8—1224no—
220#97—2442no—
221C4×C8—1224no—
222C24—1222no—
223C2×D4—614no—
224C2×D4—1224no—
225C2×D4—1224no—
226C2×D4—1224no—
227C2×D4—1222no—
228C2×D4—1224no—
229C2×D4—2442no—
230C2×D4—2442no—
231C2×D4—2442no—
232C2×D4—2442no—
233C22×C4—22no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—611no—
238C22×C4—2444no—
239C22×C4—2442no—
240C22×C4—2442no—
241C22×C4—2444no—
242C22×C4C22⋊#692311yescharacteristic
243C22×C4C22⋊#692311yescharacteristic
244C22×C4C22×#692311yescharacteristic
245C2×C8—1224no—
246C2×C8—1224no—
247C2×C8—2442no—
248C2×C8—2442no—
249C2×C8—2442no—
250C2×C8—2442no—
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—2442no—
255C22⋊C4—2442no—
256C22⋊C4—2444no—
257C22⋊C4—2444no—
258C42—22no—
259C42—612no—
260C42—612no—
261C42—612no—
262C42—612no—
263C42—1222no—
264C42—1222no—
265C42—1222no—
266C42—2441no—
267C23—1224no—
268C23—2442no—
269C23—2442no—
270C23—2442no—
271C23—2442no—
272C23—2442no—
273C23—2442no—
274C23C2×C22⋊#692311yescharacteristic
275D4—1228no—
276D4—2448no—
277D4—2448no—
278D4—2444no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—612no—
287C2×C4—611no—
288C2×C4—611no—
289C2×C4—612no—
290C2×C4—611no—
291C2×C4—611no—
292C2×C4—612no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—4884no—
301C2×C4—4884no—
302C2×C4#1451983912yes—
303C2×C4#1451984212yes—
304C2×C4D4×#692312yes—
305C2×C4D61⋊M1612yes—
306C2×C4#1451986512yes—
307C8—2444no—
308C8—2444no—
309C22—4882no—
310C22—4882no—
311C22—4882no—
312C22—4882no—
313C22—2444no—
314C22—2444no—
315C22—2444no—
316C22#1684071411yescharacteristic
317C22#1684073411yescharacteristic
318C22#1684073711yescharacteristic
319C22C2×#1451983911yescharacteristic
320C22#1684075411yescharacteristic
321C22C2×#1451984211yescharacteristic
322C22#1684076111yescharacteristic
323C4—22no—
324C4—22no—
325C4—82no—
326C4—612no—
327C4—612no—
328C4—1222no—
329C4—1221no—
330C4—1221no—
331C4—2441no—
332C4#1684079214yes—
333C2—4884no—
334C2#2146114611yescharacteristic
335C2#2146116011yescharacteristic
336C2#2146116611yescharacteristic
337C2#2146118011yescharacteristic
338C2#2146194511yescharacteristic
339C2#2146198211yescharacteristic
340C2#2146198611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519774C4
cosocleC2×C4×Dic61C23
hypercenter#384#6923
hypocenterC61#9678
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
1. #21461605C2semidirect
2. C4×#14519844C2semidirect
3. C2×#16840610C2semidirect
25. C4⋊#92420D4semidirect
191. C61#9678semidirect

Maximal subgroups

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

subgroupindex
#214616052
C4×#145198442
C2×#168406102
#214483812
#967861