Finite Group Databasebeta

← #29579443

Subgroups — #29579443

Order 15616.

396 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21445570C211yesmaximal, characteristic, complemented
2#21444582C211yesmaximal, characteristic, complemented
3#21444434C211yesmaximal, characteristic, complemented
4#21444430C211yesmaximal, characteristic
5#21444331C211yesmaximal, characteristic, complemented
6#21444327C211yesmaximal, characteristic
7C61×#3167C211yesmaximal, characteristic
8#16839819C2211yescharacteristic
9#16839819C2211yescharacteristic
10C2×#14519688C2211yescharacteristic
11#16839608C2211yescharacteristic
12#16839516—21no—
13#16839516—21no—
14#16839515—21no—
15#16839513—21no—
16#16839504—22no—
17#16839502—22no—
18#16839499—22no—
19#16839496—22no—
20C61×#533C2211yescharacteristic
21C61×#532C2211yescharacteristic
22C61×#498C411yescharacteristic, complemented
23C61×#492C411yescharacteristic, complemented
24C61×#487C411yescharacteristic, complemented
25C61×#486C411yescharacteristic, complemented
26C61×#465—21no—
27C61×#465—21no—
28C4×SD16×C61—21no—
29C4×SD16×C61—21no—
30C2×C4×C8×C61C2211yescharacteristic
31C22×#92418—21no—
32C2×C4×Dic61—21no—
33C4⋊#92418—21no—
34C4⋊#92418—21no—
35C22⋊#92418—21no—
36C22⋊#92418—21no—
37#14519688—21no—
38#14519688—21no—
39#14519688—21no—
40#14519688D412yes—
41#14519684—21no—
42#14519684D412yes—
43C2×#41×C61—21no—
44C2×SD16×C61—21no—
45C2×SD16×C61—21no—
46C22×C8×C61D411yescharacteristic
47C22×C8×C61D411yescharacteristic
48C4⋊Q8×C61C2×C411yescharacteristic
49C4⋊Q8×C61C2×C411yescharacteristic
50C4⋊Q8×C61C2×C411yescharacteristic
51#127×C61C2×C411yescharacteristic
52#125×C61—21no—
53#124×C61—21no—
54C22⋊Q8×C61—21no—
55C22⋊Q8×C61—21no—
56#121×C61—21no—
57#121×C61—21no—
58C4×Q8×C61—21no—
59C4×Q8×C61—21no—
60C4×D4×C61—21no—
61C4×D4×C61—21no—
62#117×C61—21no—
63C2×C42×C61C2311yescharacteristic
64#106×C61—21no—
65#106×C61—21no—
66#103×C61—21no—
67#103×C61—21no—
68#102×C61—21no—
69#102×C61—21no—
70C4×C8×C61C2×C411yescharacteristic
71C4×C8×C61C2×C411yescharacteristic
72C4×C8×C61D411yescharacteristic
73C4×C8×C61D411yescharacteristic
74#41×C61—82no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61—21no—
78C2×D4×C61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—21no—
81C2×D4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61#4111yescharacteristic
86C22×Dic61—22no—
87SD16×C61—42no—
88SD16×C61—42no—
89C2×C8×C61—41no—
90C2×C8×C61—41no—
91C2×C8×C61C22⋊C411yescharacteristic
92C2×C8×C61C22⋊C411yescharacteristic
93C2×C8×C61C22⋊C411yescharacteristic
94C2×C8×C61C22⋊C411yescharacteristic
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C42×C61—41no—
107C42×C61C22×C411yescharacteristic
108C42×C61C2×D411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61#4111yescharacteristic
111C4×Dic61—41no—
112C4×Dic61—41no—
113C2×#92418—22no—
114C2×#92418—41no—
115C2×#92418—41no—
116C2×#92418—41no—
117C2×#92418—41no—
118C23×C61—41no—
119C23×C61#12111yescharacteristic
120Q8×C61—41no—
121Q8×C61—41no—
122Q8×C61—81no—
123Q8×C61—81no—
124D4×C61—41no—
125D4×C61—41no—
126D4×C61—81no—
127D4×C61—81no—
128D4×C61—82no—
129C2×C4×C61—22no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—82no—
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12411yescharacteristic
142C2×C4×C61#12711yescharacteristic
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C8×C61—21no—
147C8×C61—21no—
148C8×C61—21no—
149C8×C61—21no—
150#92418—42no—
151#92418—81no—
152#92418—81no—
153#9766—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—81no—
157C22×C61—81no—
158C22×C61—81no—
159C22×C61#39011yescharacteristic
160C22×C61#47112yes—
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—81no—
167C4×C61—81no—
168C4×C61—81no—
169C4×C61#42111yescharacteristic
170C4×C61#42111yescharacteristic
171Dic61—82no—
172#3167D6111yescharacteristic
173#1746—611no—
174#1568—611no—
175#1531—611no—
176#1530—611no—
177#1506—611no—
178#1505—611no—
179C2×C61—22no—
180C2×C61—161no—
181C2×C61#196311yescharacteristic
182C2×C61#197611yescharacteristic
183C2×C61#197711yescharacteristic
184#533C2×D6111yescharacteristic
185#532C2×D6111yescharacteristic
186#498Dic6111yescharacteristic, complemented
187#492Dic6111yescharacteristic, complemented
188#487Dic6111yescharacteristic, complemented
189#486Dic6111yescharacteristic, complemented
190#465—21no—
191#465—21no—
192C4×SD16—21no—
193C4×SD16—21no—
194#432—611no—
195#432—611no—
196C2×#97—611no—
197C2×C4×C8C2×D6111yescharacteristic
198#376—611no—
199#336—1221no—
200#336—1221no—
201#335—1221no—
202#334—1221no—
203#330—1222no—
204#329—1222no—
205#328—1222no—
206#327—1222no—
207C61#976611yescharacteristic, complemented, Sylow
208C2×#41—21no—
209C2×SD16—21no—
210C2×SD16—21no—
211C22×C8—1221no—
212C22×C8C22⋊D6111yescharacteristic
213C22×C8C22⋊D6111yescharacteristic
214C4⋊Q8C2×Dic6111yescharacteristic
215C4⋊Q8C2×Dic6111yescharacteristic
216C4⋊Q8C2×Dic6111yescharacteristic
217#127C2×Dic6111yescharacteristic
218#125—21no—
219#124—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222#121—21no—
223#121—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×D4—21no—
227C4×D4—21no—
228#117—21no—
229C2×C42—1221no—
230C2×C42C22×D6111yescharacteristic
231#106—21no—
232#106—21no—
233C4⋊C8—1221no—
234C4⋊C8—1221no—
235#103—21no—
236#103—21no—
237#102—21no—
238#102—21no—
239C22⋊C8—1221no—
240C22⋊C8—1221no—
241#97—612no—
242#97—1221no—
243#97—1221no—
244#97—1221no—
245C4×C8C2×Dic6111yescharacteristic
246C4×C8C2×Dic6111yescharacteristic
247C4×C8C22⋊D6111yescharacteristic
248C4×C8C22⋊D6111yescharacteristic
249#95—612no—
250#95—1221no—
251#41—82no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—41no—
260C22×C4—1222no—
261C22×C4D4×D6111yescharacteristic
262C22×C4#1175822411yescharacteristic
263C22×C4C2×C22⋊D6111yescharacteristic
264SD16—42no—
265SD16—42no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—1222no—
269C2×C8—2441no—
270C2×C8—2441no—
271C2×C8—2441no—
272C2×C8—2441no—
273C2×C8C22⋊Dic6111yescharacteristic
274C2×C8C22⋊Dic6111yescharacteristic
275C2×C8C22⋊Dic6111yescharacteristic
276C2×C8C22⋊Dic6111yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42—41no—
289C42—2441no—
290C42—2441no—
291C42D4×D6111yescharacteristic
292C42#1175822411yescharacteristic
293C42C22×Dic6111yescharacteristic
294C42C2×C22⋊D6111yescharacteristic
295C23—41no—
296C23#1451974811yescharacteristic
297Q8—41no—
298Q8—41no—
299Q8—81no—
300Q8—81no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—82no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—82no—
314C2×C4—2442no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4C2×C22⋊Dic6111yescharacteristic
318C2×C4D4×Dic6111yescharacteristic
319C2×C4D4×Dic6111yescharacteristic
320C2×C4#1451974811yescharacteristic
321C2×C4#1451976411yescharacteristic
322C2×C4#1451977311yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—21no—
327C8—4881no—
328C8—4881no—
329C8—2442no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22#1683966511yescharacteristic
336C22#1684002212yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—4882no—
346C4#1683977911yescharacteristic
347C4#1683977911yescharacteristic
348C2—22no—
349C2—161no—
350C2#2144665211yescharacteristic
351C2#2144671811yescharacteristic
352C2#2144672411yescharacteristic
subgroupG/H
centerC22#16839665
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3167C2
socleC22×C61#390
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#3167D61
hypocenterC61#9766
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21445570C2semidirect
2. #21444582C2semidirect
3. #21444434C2semidirect
5. #21444331C2semidirect
22. C61×#498C4semidirect
23. C61×#492C4semidirect
24. C61×#487C4semidirect
25. C61×#486C4semidirect
186. #498Dic61semidirect
187. #492Dic61semidirect
188. #487Dic61semidirect
189. #486Dic61semidirect
207. C61#9766semidirect

Maximal subgroups

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

subgroupindex
#214455702
#214445822
#214444342
#214444302
#214443312
#214443272
C61×#31672
#976661