Finite Group Databasebeta

← #30613104

Subgroups — #30613104

Order 15616.

544 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462723C212yesmaximal
2#21461444C212yesmaximal, complemented
3#21461443C212yesmaximal, complemented
4#21461433C212yesmaximal, complemented
5#21461430C212yesmaximal, complemented
6#21461381C211yesmaximal, characteristic
7#21461181C212yesmaximal
8#21461032C211yesmaximal, characteristic
9#21456400C211yesmaximal, characteristic
10#16840814—22no—
11#16840813C2212yes—
12#16840812—22no—
13#16840812C2212yes—
14#16840811C2212yes—
15Q8×#92420C2212yes—
16#16840786—22no—
17#16840786—22no—
18#16840783C2212yes—
19#16840778C2211yescharacteristic
20#16840778C2211yescharacteristic
21#16840625C2211yescharacteristic
22#16840625C2211yescharacteristic
23#16840624C2212yes—
24#16840612C2212yes—
25#16840612C2212yes—
26Q8⋊#92420C2212yes—
27Q8⋊#92420C2212yes—
28#92420⋊C8C2212yes—
29#92420⋊C8C2212yes—
30#16840314C2212yes—
31#16840166C411yescharacteristic
32#16840140C411yescharacteristic
33#16840111C412yes—
34#16840003C411yescharacteristic
35#16840001C412yes—
36Q16×Dic61C2211yescharacteristic
37#16839856C2211yescharacteristic
38#16839854C2212yes—
39C2×C8×Dic61C2211yescharacteristic
40C61×#467C411yescharacteristic
41#14519844—22no—
42#14519844—22no—
43C4⋊#92420—22no—
44C4⋊#92420—22no—
45C4⋊#92420—21no—
46C4⋊#92420D411yescharacteristic
47C4⋊#92420D411yescharacteristic
48C4⋊#92420C2312yes—
49C4⋊#92420C2312yes—
50C22⋊#92420—22no—
51C22⋊#92420—21no—
52C22⋊#92420D412yes—
53#14519832—22no—
54#14519832—22no—
55#14519832—22no—
56C4×#92420—22no—
57C4×#92420—22no—
58C4×#92420C2312yes—
59C4×#92420C2312yes—
60C2×Dic244C2×C411yescharacteristic
61C2×#11758202C2×C412yes—
62C22×#92418C2×C411yescharacteristic
63#14519776C2×C412yes—
64#14519775C2×C412yes—
65#14519767—22no—
66#14519756C2×C412yes—
67#14519754—22no—
68Q8×Dic61C2312yes—
69#14519742—22no—
70#14519742C2312yes—
71#14519733—22no—
72C2×C4×Dic61C2311yescharacteristic
73#14519711C2×C411yescharacteristic
74#14519710C2×C412yes—
75#14519702C2×C412yes—
76#14519701C2×C412yes—
77#14519700C2×C412yes—
78C8×Dic61C2×C411yescharacteristic
79C8×Dic61C2×C411yescharacteristic
80C8×Dic61C2311yescharacteristic
81C8×Dic61C2311yescharacteristic
82C2×Q16×C61C2×C411yescharacteristic
83C22×C8×C61C2×C411yescharacteristic
84C22⋊Q8×C61C2×C412yes—
85#107×C61C2×C411yescharacteristic
86#103×C61C2×C412yes—
87C2×Q8×C61C22×C412yes—
88C22×C4×C61C22×C411yescharacteristic
89C22×Dic61C2×D411yescharacteristic
90C22×Dic61#4111yescharacteristic
91C2×Dic122—22no—
92C2×Dic122C22×C412yes—
93C2×#92420—22no—
94C2×#92420—22no—
95C2×#92420—22no—
96C2×#92420C2×D412yes—
97C2×#92420C2×D412yes—
98Q16×C61—41no—
99C2×C8×C61—41no—
100C2×C8×C61C22×C411yescharacteristic
101C2×C8×C61C22×C411yescharacteristic
102C4⋊C4×C61—22no—
103C4⋊C4×C61C22×C412yes—
104C22⋊C4×C61—22no—
105C22⋊Dic61—22no—
106#11758202—42no—
107C4⋊Dic61—22no—
108C4⋊Dic61C22×C412yes—
109#11758196—22no—
110#11758196—22no—
111#11758196—22no—
112#11758196—22no—
113#11758196—22no—
114C4×Dic61—22no—
115C4×Dic61—22no—
116C4×Dic61C22×C411yescharacteristic
117C4×Dic61C2×D411yescharacteristic
118C4×Dic61#4111yescharacteristic
119C4×Dic61C2411yescharacteristic
120C2×#92418—41no—
121C2×#92418C22×C411yescharacteristic
122C2×#92418C22×C411yescharacteristic
123Dic244—41no—
124C23×C61C4×D411yescharacteristic
125Q8×C61—22no—
126Q8×C61—22no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61C4×D411yescharacteristic
132C2×C4×C61C23×C411yescharacteristic
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—22no—
136C2×Dic61—22no—
137C2×Dic61—41no—
138C2×Dic61C4×D411yescharacteristic
139C2×Dic61C4×D411yescharacteristic
140C2×Dic61C22×D411yescharacteristic
141C2×Dic61C2×#4111yescharacteristic
142Dic122—22no—
143Dic122—22no—
144Dic122—82no—
145#92420—22no—
146#92420—22no—
147#92420—42no—
148#92420—42no—
149#92420—42no—
150C8×C61—21no—
151C8×C61—21no—
152#92418—21no—
153#92418—21no—
154#17991—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61C2×C4×D411yescharacteristic
159C4×C61—21no—
160C4×C61—42no—
161C4×C61—42no—
162C4×C61#56811yescharacteristic
163C4×C61#56811yescharacteristic
164Dic61—21no—
165Dic61—21no—
166Dic61—21no—
167Dic61—42no—
168Dic61—42no—
169Dic61#57811yescharacteristic
170Dic61#57811yescharacteristic
171#3170—611no—
172#2985—612no—
173#1587—612no—
174#1587—612no—
175#1584—612no—
176#1584—612no—
177#1568—611no—
178#1497—612no—
179#1452—611no—
180C2×C61—41no—
181C2×C61#293211yescharacteristic
182C2×C61#295011yescharacteristic
183C2×C61#295711yescharacteristic
184#533—612no—
185#500—611no—
186#494—611no—
187#487—612no—
188#467—611no—
189#467—612no—
190#467#692311yescharacteristic
191#446—612no—
192#446—612no—
193#446—1222no—
194#446—1222no—
195C8×Q8—612no—
196C8×Q8—612no—
197C4×Q16—611no—
198C4×Q16—611no—
199C4×Q16—612no—
200#433—1222no—
201#433—1222no—
202#432—612no—
203#431—611no—
204#431—611no—
205C2×C4×C8—611no—
206#335—611no—
207#335—611no—
208#334—612no—
209Q8⋊C8—612no—
210Q8⋊C8—612no—
211Q8⋊C8—612no—
212Q8⋊C8—612no—
213#322—612no—
214#322—612no—
215C61#1799111yescharacteristic, complemented, Sylow
216C2×Q16—612no—
217C2×Q16—611no—
218C2×Q16C2×#692311yescharacteristic
219C22×C8—611no—
220C22×C8C2×#692311yescharacteristic
221C4⋊Q8—612no—
222C4⋊Q8—612no—
223#125—1222no—
224C22⋊Q8—612no—
225C22⋊Q8—1222no—
226C22⋊Q8C2×#692312yes—
227C4×Q8—612no—
228C4×Q8—612no—
229C4×Q8—1222no—
230#117—1222no—
231C2×C42—611no—
232#107—611no—
233#107—612no—
234#107C2×#692311yescharacteristic
235C4⋊C8—611no—
236C4⋊C8—611no—
237C4⋊C8—612no—
238C4⋊C8—612no—
239C4⋊C8—1221no—
240C4⋊C8—1222no—
241C4⋊C8—1222no—
242C4⋊C8—1222no—
243C4⋊C8—1222no—
244#103—612no—
245#103—612no—
246#103—612no—
247#103C2×#692312yes—
248C22⋊C8—612no—
249C22⋊C8—1221no—
250C22⋊C8—1222no—
251#97—1222no—
252#97—1222no—
253#97—1222no—
254C4×C8—612no—
255C4×C8—611no—
256C4×C8—611no—
257C4×C8—611no—
258C4×C8—611no—
259C4×C8—612no—
260C4×C8—1222no—
261C4×C8—1222no—
262C2×Q8—612no—
263C2×Q8—1222no—
264C2×Q8C22×#692312yes—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4C22×#692311yescharacteristic
268Q16—41no—
269Q16—2441no—
270Q16—2442no—
271C2×C8—41no—
272C2×C8—612no—
273C2×C8—612no—
274C2×C8—611no—
275C2×C8—611no—
276C2×C8—1222no—
277C2×C8—1222no—
278C2×C8—1222no—
279C2×C8—2441no—
280C2×C8C22×#692311yescharacteristic
281C2×C8C22×#692311yescharacteristic
282C4⋊C4—22no—
283C4⋊C4—612no—
284C4⋊C4—1222no—
285C4⋊C4—1222no—
286C4⋊C4—1222no—
287C4⋊C4—1222no—
288C4⋊C4—1222no—
289C4⋊C4—1222no—
290C4⋊C4C22×#692312yes—
291C22⋊C4—22no—
292C22⋊C4—1222no—
293C42—611no—
294C42—611no—
295C42—611no—
296C42—611no—
297C42—1222no—
298C42—1222no—
299C23D4×#692311yescharacteristic
300Q8—22no—
301Q8—22no—
302Q8—1222no—
303Q8—1222no—
304Q8—4882no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—611no—
312C2×C4—611no—
313C2×C4—1221no—
314C2×C4—1221no—
315C2×C4—1222no—
316C2×C4—1222no—
317C2×C4—2441no—
318C2×C4D4×#692311yescharacteristic
319C2×C4C23×#692311yescharacteristic
320C8—21no—
321C8—21no—
322C8—1222no—
323C8—1222no—
324C8—1221no—
325C8—1221no—
326C8—2442no—
327C8—2442no—
328C8—2442no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22C2×D4×#692311yescharacteristic
333C4—21no—
334C4—42no—
335C4—42no—
336C4—611no—
337C4—611no—
338C4—1221no—
339C4—1221no—
340C4—1221no—
341C4—2442no—
342C4—2442no—
343C4#1684084011yescharacteristic
344C4#1684084111yescharacteristic
345C2—41no—
346C2#2146254711yescharacteristic
347C2#2146258011yescharacteristic
348C2#2146259611yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#467C4
socleC22×C61C2×C4×D4
prime radicalC23×C61C4×D4
cosocleC4×Dic61C24
hypercenter#467#6923
hypocenterC61#17991
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #21461444C2semidirect
3. #21461443C2semidirect
4. #21461433C2semidirect
5. #21461430C2semidirect
215. C61#17991semidirect

Maximal subgroups

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

subgroupindex
#214627232
#214614442
#214614432
#214614332
#214614302
#214613812
#214611812
#214610322
#214564002
#1799161