Finite Group Databasebeta

← #30602189

Subgroups — #30602189

Order 15616.

654 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C4×#14519834C211yesmaximal, characteristic
2#21460997C211yesmaximal, characteristic
3#21460692C214yesmaximal
4#21454166C211yesmaximal, characteristic
5C22×C4⋊#6923—21no—
6C2×#14519834C2212yes—
7C2×C4×#92420—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4C2212yes—
10#16840628—21no—
11C2×#14519753—21no—
12C2×C42×D61C2211yescharacteristic
13#16839669—21no—
14#16839658—21no—
15#16839658C412yescomplemented
16#16839632C411yescharacteristic, complemented
17#16839619—21no—
18#16839619C2212yes—
19C2×C4⋊Q8×C61C411yescharacteristic, complemented
20C23×#6923—22no—
21C22×#92420—22no—
22C2×C4⋊#6923—24no—
23C2×C4⋊#6923—42no—
24C2×C4⋊#6923—42no—
25#14519834—22no—
26#14519834Q814yescomplemented
27C4×#92420—42no—
28(C2×#6923)⋊C4—24no—
29(C2×#6923)⋊C4—42no—
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×C4×D61C2311yescharacteristic
33#14519753—42no—
34#14519753—42no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2×C412yes—
38C2×#11758196—22no—
39C2×#11758198—22no—
40C2×C4×Dic61C2×C411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43C42×D61—42no—
44#14519684—22no—
45#14519684—22no—
46C22×Q8×C61—21no—
47C4⋊Q8×C61—42no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61C2×C412yes—
50C2×C42×C61C2×C411yescharacteristic
51C23×D61C2×D411yescharacteristic
52C22×#6923—24no—
53C22×#6923—42no—
54C22×#6923—42no—
55C22×#6923—42no—
56C22×#6923—42no—
57C22×#6923—42no—
58C22×#6923—42no—
59C2×Q8×C61—42no—
60C2×Q8×C61—42no—
61C2×Q8×C61—42no—
62C22×C4×C61—22no—
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×Dic61—22no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61—41no—
73C2×C4×D61—41no—
74C2×C4×D61—41no—
75C2×C4×D61—41no—
76C2×C4×D61D812yes—
77C2×C4×D61SD1612yes—
78C2×C4×D61SD1612yes—
79C2×C4×D61Q1612yes—
80C2×C4×D61C2×Q812yes—
81C2×C4×D61#4112yes—
82C2×#92420—24no—
83C2×#92420—42no—
84C2×#92420—42no—
85C4⋊#6923—48no—
86C4⋊C4×C61—42no—
87C4⋊C4×C61—42no—
88C42×C61—22no—
89#11758198—42no—
90#11758198—42no—
91C4⋊Dic61—42no—
92C4⋊Dic61—42no—
93C4⋊Dic61—42no—
94#11758196—44no—
95C4×Dic61—22no—
96C4×Dic61—41no—
97C4×Dic61—41no—
98C23×C61C2×C22⋊C411yescharacteristic
99C22×D61—21no—
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—21no—
103C22×D61#12411yescharacteristic
104C22×D61C4⋊Q811yescharacteristic
105C22×D61C2×D811yescharacteristic
106C22×D61C2×SD1611yescharacteristic
107C22×D61C2×SD1611yescharacteristic
108C22×D61C2×Q1611yescharacteristic
109C2×#6923—44no—
110C2×#6923—44no—
111C2×#6923—44no—
112C2×#6923—82no—
113C2×#6923—82no—
114C2×#6923—82no—
115C2×#6923—82no—
116Q8×C61—84no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61#9912yes—
125C2×C4×C61#10212yes—
126C2×C4×C61#10312yes—
127C2×C4×C61#10412yes—
128C2×C4×C61C4×Q812yes—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61#10212yes—
139C2×Dic61#10312yes—
140C2×Dic61C22⋊Q812yes—
141C4×D61—24no—
142C4×D61—24no—
143C4×D61—42no—
144C4×D61—42no—
145C4×D61—42no—
146C4×D61—42no—
147#92420—44no—
148#9885—611nomaximal, Sylow
149C22×C61#39111yescharacteristic
150C22×C61C2×#9911yescharacteristic
151C22×C61#41011yescharacteristic
152C22×C61C2×#10211yescharacteristic
153C22×C61C2×#10311yescharacteristic
154C22×C61C2×#10411yescharacteristic
155C22×C61#42111yescharacteristic
156C2×D61—21no—
157C2×D61—21no—
158C2×D61—21no—
159C2×D61—21no—
160C2×D61—21no—
161C2×D61—21no—
162C2×D61—21no—
163C2×D61—21no—
164C2×D61—21no—
165C2×D61—21no—
166C2×D61—21no—
167C2×D61—21no—
168C2×D61#48611yescharacteristic
169C2×D61#48711yescharacteristic
170C2×D61#49811yescharacteristic
171C2×D61#50011yescharacteristic
172#6923—84no—
173C4×C61—22no—
174C4×C61—22no—
175C4×C61—82no—
176C4×C61#32714yes—
177Dic61—22no—
178Dic61—22no—
179Dic61—22no—
180Dic61—82no—
181#2857—611no—
182C4×C22⋊C8—611no—
183C4⋊#95—611no—
184#1294—614no—
185C2×C61C2×#32711yescharacteristic
186C2×C61#150311yescharacteristic
187C2×C61#151711yescharacteristic
188C2×C61#152011yescharacteristic
189C2×C61#198611yescharacteristic
190C2×C61#199011yescharacteristic
191C2×C61#199611yescharacteristic
192D61—21no—
193D61—21no—
194D61—21no—
195D61—21no—
196C2×C4⋊Q8#692311yescharacteristic, complemented
197C2×C22⋊Q8—1221no—
198C22×C4⋊C4—1221no—
199C22×C42—611no—
200C2×C22⋊C8—612no—
201C2×C4×C8—1221no—
202#391—1221no—
203#389—612no—
204#389—1221no—
205#384—611no—
206#380—612no—
207#380—1221no—
208C2×#95—612no—
209C2×#95—1221no—
210#336—1221no—
211C61#988511yescharacteristic, complemented, Sylow
212C22×Q8—21no—
213C23×C4—611no—
214C23×C4—611no—
215C23×C4—611no—
216C23×C4—1222no—
217C22×C8—1222no—
218C4⋊Q8—42no—
219C22⋊Q8—2442no—
220C22⋊Q8—2442no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—612no—
223C2×C4⋊C4—1222no—
224C2×C4⋊C4—1224no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4—2442no—
228C2×C4⋊C4—2442no—
229C2×C4⋊C4C2×#692312yes—
230C2×C22⋊C4—1222no—
231C2×C42—611no—
232C2×C42—611no—
233C2×C42—611no—
234C2×C42—2442no—
235C2×C42C2×#692311yescharacteristic
236C22⋊C8—614no—
237C22⋊C8—1222no—
238C4×C8—2442no—
239#95—1222no—
240#95—1222no—
241#95—1224no—
242#95—2442no—
243C24—611no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C22×C4—22no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—612no—
252C22×C4—612no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—612no—
256C22×C4—612no—
257C22×C4—1222no—
258C22×C4—1222no—
259C22×C4—1224no—
260C22×C4—1222no—
261C22×C4—2441no—
262C22×C4—2441no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4—2442no—
266C22×C4—2442no—
267C22×C4—2442no—
268C22×C4—2442no—
269C22×C4—2442no—
270C22×C4—2442no—
271C22×C4C22⋊#692311yescharacteristic
272C22×C4C22⋊#692311yescharacteristic
273C22×C4C22×#692311yescharacteristic
274C2×C8—1224no—
275C2×C8—2442no—
276C2×C8—2442no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—2442no—
280C4⋊C4—2442no—
281C4⋊C4—2444no—
282C4⋊C4—2448no—
283C4⋊C4—2442no—
284C22⋊C4—2442no—
285C22⋊C4—2442no—
286C42—22no—
287C42—1222no—
288C42—2441no—
289C42—2441no—
290C23—611no—
291C23—611no—
292C23—611no—
293C23—611no—
294C23—611no—
295C23—611no—
296C23—1221no—
297C23—1221no—
298C23—1221no—
299C23—1221no—
300C23C2×C22⋊#692311yescharacteristic
301Q8—84no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—612no—
310C2×C4—612no—
311C2×C4—612no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—1221no—
315C2×C4—1221no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—1224no—
319C2×C4—1224no—
320C2×C4—4882no—
321C2×C4—4882no—
322C2×C4—4882no—
323C2×C4—4882no—
324C2×C4—2442no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2444no—
329C2×C4—2444no—
330C2×C4—2444no—
331C2×C4—2442no—
332C2×C4—2442no—
333C2×C4—2442no—
334C2×C4#1451983512yes—
335C2×C4#1451983912yes—
336C2×C4#1451984012yes—
337C2×C4#1451984112yes—
338C2×C4Q8×#692312yes—
339C8—2444no—
340C22—611no—
341C22—611no—
342C22—611no—
343C22—611no—
344C22—1221no—
345C22—1221no—
346C22—1221no—
347C22—1221no—
348C22—1221no—
349C22—1221no—
350C22—1221no—
351C22—1221no—
352C22—1221no—
353C22—1221no—
354C22—1221no—
355C22—1221no—
356C22(C2×#6923)⋊Q811yescharacteristic
357C22C2×#1451983511yescharacteristic
358C22#1684074311yescharacteristic
359C22C2×#1451983911yescharacteristic
360C22C2×#1451984011yescharacteristic
361C22C2×#1451984111yescharacteristic
362C22#1684076111yescharacteristic
363C4—22no—
364C4—22no—
365C4—82no—
366C4—1222no—
367C4—1222no—
368C4—1222no—
369C4—4884no—
370C4—4882no—
371C4#1684061314yes—
372C2—1221no—
373C2—1221no—
374C2—1221no—
375C2—1221no—
376C2C2×#1684061311yescharacteristic
377C2#2146120211yescharacteristic
378C2#2146125011yescharacteristic
379C2#2146125911yescharacteristic
380C2#2146199511yescharacteristic
381C2#2146200011yescharacteristic
382C2#2146200811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#9885
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
15. #16839658C4semidirect
16. #16839632C4semidirect
19. C2×C4⋊Q8×C61C4semidirect
26. #14519834Q8semidirect
196. C2×C4⋊Q8#6923semidirect
211. C61#9885semidirect

Maximal subgroups

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

subgroupindex
C4×#145198342
#214609972
#214606922
#214541662
#988561