Finite Group Databasebeta

← #30606548

Subgroups — #30606548

Order 15616.

628 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462311C211yesmaximal, characteristic
2D61⋊#353C211yesmaximal, characteristic
3#99×#6923C212yesmaximal, complemented
4#21460912C212yesmaximal, complemented
5#353×D61C211yesmaximal, characteristic
6C2×D4×#6923C411yescharacteristic, complemented
7C2×D4×#6923C2211yescharacteristic
8D61⋊#99C2212yes—
9#16840714C411yescharacteristic, complemented
10#16840714C2211yescharacteristic
11#16840710—22no—
12#16840710—22no—
13#16840706C411yescharacteristic, complemented
14C22⋊C4×#6923—22no—
15#16840698C411yescharacteristic, complemented
16#16840667C412yescomplemented
17#16840666C412yescomplemented
18#16840639—22no—
19#127×D61C2211yescharacteristic
20#99×D61C2212yes—
21#16839555C411yescharacteristic, complemented
22#16839554C412yescomplemented
23C61×#353C411yescharacteristic, complemented
24C23×#6923—21no—
25C23×#6923D412yes—
26D4×#6923—42no—
27D4×#6923—42no—
28C2×C4⋊#6923C2×C411yescharacteristic
29C2×C4⋊#6923C2×C411yescharacteristic
30C2×C22⋊#6923—22no—
31C2×C22⋊#6923—21no—
32C2×C22⋊#6923—22no—
33C2×C22⋊#6923—22no—
34C2×C22⋊#6923D412yes—
35C2×C4×#6923—42no—
36C2×C4×#6923C2×C411yescharacteristic
37C2×C4×#6923C2×C411yescharacteristic
38#14519836C2×C414yescomplemented
39#14519835—22no—
40(C2×#6923)⋊C4—22no—
41(C2×#6923)⋊C4—21no—
42(C2×#6923)⋊C4—21no—
43(C2×#6923)⋊C4—42no—
44C2×D4×D61—22no—
45C2×D4×D61C2×C411yescharacteristic
46C2×D4×D61C2311yescharacteristic
47#14519774C2×C411yescharacteristic
48#14519773—22no—
49#14519773C2×C411yescharacteristic
50#14519773C2×C411yescharacteristic
51C22⋊C4×D61—22no—
52C42×D61C2×C411yescharacteristic
53#14519692C2×C412yescomplemented
54#14519691—22no—
55#127×C61C2×C411yescharacteristic
56#99×C61C2×C412yescomplemented
57C23×D61—21no—
58C23×D61C2×D412yes—
59C22×#6923—22no—
60C22×#6923—22no—
61C22×#6923—21no—
62C22×#6923—21no—
63C22×#6923—42no—
64C22×#6923—42no—
65C22×#6923—82no—
66C22×#6923—82no—
67C22×#6923C22⋊C411yescharacteristic
68C22×#6923C22⋊C412yes—
69C22×#6923C22⋊C411yescharacteristic
70C2×D4×C61—22no—
71C2×D4×C61C4211yescharacteristic, complemented
72C2×D4×C61C22×C411yescharacteristic
73C2×C22⋊D61—22no—
74C2×C22⋊D61—22no—
75C2×C22⋊D61—21no—
76C2×C22⋊D61#4112yes—
77D4×D61—41no—
78D4×D61—41no—
79D4×D61—41no—
80D4×D61—41no—
81D4×D61—82no—
82D4×D61—82no—
83C2×D244—22no—
84C2×D244C4211yescharacteristic, complemented
85C2×D244C22×C411yescharacteristic
86C2×C4×D61—21no—
87C2×C4×D61—42no—
88C2×C4×D61C22×C411yescharacteristic
89C22⋊#6923—24no—
90C22⋊#6923—42no—
91C22⋊#6923—42no—
92C22⋊#6923—42no—
93C22⋊#6923—82no—
94C22⋊#6923—82no—
95C4⋊#6923—41no—
96C4⋊#6923—41no—
97C4×#6923—22no—
98C4×#6923—41no—
99C4×#6923—84no—
100C22⋊C4×C61—22no—
101C42×C61C4211yescharacteristic, complemented
102C22⋊Dic61—22no—
103#11758198—42no—
104C4×Dic61—21no—
105C4×Dic61C4211yescharacteristic, complemented
106C23×C61—21no—
107C23×C61C4×D412yes—
108C22×D61—22no—
109C22×D61—22no—
110C22×D61—21no—
111C22×D61—42no—
112C22×D61—42no—
113C22×D61—42no—
114C22×D61—42no—
115C22×D61—41no—
116C22×D61—41no—
117C22×D61C2×C22⋊C411yescharacteristic
118C22×D61C4×D412yes—
119C2×#6923—21no—
120C2×#6923—22no—
121C2×#6923—41no—
122C2×#6923—44no—
123C2×#6923—42no—
124C2×#6923—41no—
125C2×#6923—44no—
126C2×#6923—41no—
127C2×#6923—41no—
128C2×#6923—42no—
129C2×#6923—41no—
130C2×#6923—41no—
131C2×#6923—82no—
132C2×#6923—84no—
133C2×#6923—82no—
134C2×#6923—82no—
135C2×#6923#9912yes—
136D4×C61—41no—
137D4×C61—41no—
138D4×C61—82no—
139C2×C4×C61—21no—
140C2×C4×C61—42no—
141C2×C4×C61C2×C4211yescharacteristic
142C22⋊D61—42no—
143C22⋊D61—42no—
144C22⋊D61—82no—
145C22⋊D61—82no—
146C2×Dic61—21no—
147C2×Dic61—42no—
148C2×Dic61#11711yescharacteristic
149D244—41no—
150D244—41no—
151D244—82no—
152C4×D61—21no—
153C4×D61—21no—
154C4×D61—41no—
155C4×D61—41no—
156C4×D61—82no—
157C4×D61—82no—
158#12628—611nomaximal, Sylow
159C22×C61—22no—
160C22×C61—42no—
161C22×C61—42no—
162C22×C61—41no—
163C22×C61C4×C22⋊C411yescharacteristic
164C2×D61—21no—
165C2×D61—21no—
166C2×D61—22no—
167C2×D61—42no—
168C2×D61—42no—
169C2×D61—42no—
170C2×D61—42no—
171C2×D61—41no—
172C2×D61—82no—
173C2×D61—82no—
174C2×D61—81no—
175C2×D61—81no—
176C2×D61C2×#9911yescharacteristic
177C2×D61#41011yescharacteristic
178#6923—22no—
179#6923—42no—
180#6923—41no—
181#6923—81no—
182#6923—84no—
183#6923—81no—
184C4×C61—21no—
185C4×C61—41no—
186C4×C61—82no—
187Dic61—21no—
188Dic61—41no—
189Dic61—82no—
190#2328—611no—
191C2×#353—611no—
192C2×#353—611no—
193C4×#99—612no—
194#1393—612no—
195C2×C61—21no—
196C2×C61—42no—
197C2×C61—81no—
198C2×C61C4×#9911yescharacteristic
199D61—21no—
200D61—42no—
201D61—81no—
202D61#212711yescharacteristic
203D61#212811yescharacteristic
204C2×#127—611no—
205C2×C4×D4—611no—
206C2×C4×D4—611no—
207C2×#99—612no—
208C2×#99—612no—
209#390—611no—
210#390—611no—
211#386—1222no—
212#386—1222no—
213#382—611no—
214C4×C22⋊C4—1222no—
215#376—611no—
216#353—612no—
217#353—612no—
218#353—611no—
219#353—612no—
220#353#692311yescharacteristic, complemented
221#342—1222no—
222C61#1262811yescharacteristic, complemented, Sylow
223C22×D4—611no—
224C22×D4—611no—
225C22×D4—1222no—
226C23×C4—612no—
227C23×C4—1221no—
228#127—611no—
229#127—611no—
230#127—611no—
231#127—1222no—
232#127C2×#692311yescharacteristic
233C4×D4—2442no—
234C4×D4—2442no—
235C2×C4⋊C4—611no—
236C2×C4⋊C4—611no—
237C2×C22⋊C4—612no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4—1222no—
242C2×C22⋊C4—1221no—
243C2×C42—611no—
244C2×C42—611no—
245C2×C42—611no—
246C2×C42—2442no—
247#99—614no—
248#99—612no—
249#99—1222no—
250#99—1222no—
251#99C2×#692312yescomplemented
252#95—1221no—
253#95—1221no—
254#95—1222no—
255#95—2442no—
256C24—612no—
257C24—1221no—
258C2×D4—22no—
259C2×D4—612no—
260C2×D4—611no—
261C2×D4—611no—
262C2×D4—1222no—
263C2×D4—1221no—
264C2×D4—1222no—
265C2×D4—1222no—
266C2×D4—4882no—
267C2×D4—4882no—
268C2×D4—2441no—
269C2×D4—2441no—
270C2×D4—2441no—
271C2×D4—2441no—
272C2×D4C4×#692311yescharacteristic, complemented
273C2×D4C22×#692311yescharacteristic
274C22×C4—611no—
275C22×C4—611no—
276C22×C4—611no—
277C22×C4—612no—
278C22×C4—1221no—
279C22×C4—1222no—
280C22×C4—1222no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—4882no—
284C22×C4—4882no—
285C22×C4—2442no—
286C22×C4—2442no—
287C22×C4—2442no—
288C4⋊C4—2441no—
289C4⋊C4—2441no—
290C22⋊C4—22no—
291C22⋊C4—1224no—
292C22⋊C4—1222no—
293C22⋊C4—4882no—
294C22⋊C4—4882no—
295C22⋊C4—2442no—
296C22⋊C4—2442no—
297C22⋊C4—2442no—
298C22⋊C4—2442no—
299C42—611no—
300C42—1221no—
301C42—1222no—
302C42—4884no—
303C42—2441no—
304C42C4×#692311yescharacteristic, complemented
305C23—21no—
306C23—611no—
307C23—612no—
308C23—1222no—
309C23—1222no—
310C23—1221no—
311C23—2441no—
312C23—2441no—
313C23—2442no—
314C23—2442no—
315C23—2442no—
316C23—2442no—
317C23D4×#692312yes—
318D4—41no—
319D4—41no—
320D4—82no—
321D4—4882no—
322D4—4882no—
323D4—4882no—
324D4—2441no—
325D4—2441no—
326D4—2442no—
327D4—2442no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—612no—
331C2×C4—611no—
332C2×C4—1221no—
333C2×C4—1221no—
334C2×C4—1221no—
335C2×C4—1222no—
336C2×C4—1221no—
337C2×C4—4882no—
338C2×C4—4882no—
339C2×C4—4882no—
340C2×C4—4882no—
341C2×C4—4882no—
342C2×C4—4884no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2444no—
347C2×C4—2442no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2444no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2441no—
356C2×C4C2×C4×#692311yescharacteristic
357C22—22no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—611no—
362C22—611no—
363C22—1221no—
364C22—1221no—
365C22—1222no—
366C22—4882no—
367C22—4881no—
368C22—4881no—
369C22—4882no—
370C22—2442no—
371C22—2442no—
372C22—2442no—
373C22—2442no—
374C22—2441no—
375C22C22⋊C4×#692311yescharacteristic
376C4—21no—
377C4—41no—
378C4—82no—
379C4—1222no—
380C4—1221no—
381C4—4882no—
382C4—4881no—
383C4—4881no—
384C4—4884no—
385C4—2441no—
386C4—2441no—
387C4—2442no—
388C2—21no—
389C2—42no—
390C2—81no—
391C2—611no—
392C2—611no—
393C2—1221no—
394C2—4881no—
395C2—2442no—
396C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×D4C22×#6923
FittingC61×#353C4
socleC2×C61C4×#99
prime radical#127×D61C22
cosocleC2×D4×D61C23
hypercenter#353#6923
hypocenterC61#12628
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
3. #99×#6923C2semidirect
4. #21460912C2semidirect
6. C2×D4×#6923C4semidirect
9. #16840714C4semidirect
13. #16840706C4semidirect
15. #16840698C4semidirect
16. #16840667C4semidirect
17. #16840666C4semidirect
21. #16839555C4semidirect
22. #16839554C4semidirect
23. C61×#353C4semidirect
38. #14519836C2×C4semidirect
53. #14519692C2×C4semidirect
56. #99×C61C2×C4semidirect
71. C2×D4×C61C42semidirect
84. C2×D244C42semidirect
101. C42×C61C42semidirect
105. C4×Dic61C42semidirect
220. #353#6923semidirect
222. C61#12628semidirect
251. #99C2×#6923semidirect
272. C2×D4C4×#6923semidirect
304. C42C4×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623112
D61⋊#3532
#99×#69232
#214609122
#353×D612
#1262861