Finite Group Databasebeta

← #29575257

Subgroups — #29575257

Order 15616.

432 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454768C211yesmaximal, characteristic
2#21446764C211yesmaximal, characteristic
3#21444318C211yesmaximal, characteristic
4#21444222C211yesmaximal, characteristic
5#21443659C211yesmaximal, characteristic
6#21443656C211yesmaximal, characteristic
7C61×#3170C211yesmaximal, characteristic
8#16839893—21no—
9#16839893—22no—
10Q8×#92418—21no—
11#16839815C2211yescharacteristic
12#16839789—21no—
13C2×C4⋊#92418C2211yescharacteristic
14#16839718—21no—
15C4×#11758194C2211yescharacteristic
16C4×C4⋊Dic61C2211yescharacteristic
17#16839521—21no—
18#16839521—21no—
19#16839499—22no—
20Q8⋊#92418—21no—
21Q8⋊#92418—21no—
22C61×#533C2211yescharacteristic
23C61×#533C2211yescharacteristic
24C61×#500C411yescharacteristic, complemented
25C61×#494C411yescharacteristic, complemented
26C61×#487C411yescharacteristic, complemented
27C61×#487C411yescharacteristic, complemented
28C61×#467—22no—
29C4×Q16×C61—22no—
30C2×C4×C8×C61C2211yescharacteristic
31C2×#11758194D411yescharacteristic
32C2×#11758194D411yescharacteristic
33C22×#92418—21no—
34C2×C4⋊Dic61—21no—
35C2×C4×Dic61—21no—
36C4⋊#92418—22no—
37C4⋊#92418—22no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418D411yescharacteristic
41C4⋊#92418D411yescharacteristic
42C22⋊#92418—21no—
43#14519688—22no—
44#14519688—21no—
45C4×#92418—22no—
46C4×#92418—21no—
47C4×#92418—21no—
48#14519684—21no—
49C2×Q16×C61—22no—
50C22×C8×C61—21no—
51C4⋊Q8×C61C2×C411yescharacteristic
52C4⋊Q8×C61C2×C411yescharacteristic
53C4⋊Q8×C61C2×C411yescharacteristic
54C4⋊Q8×C61C2×C411yescharacteristic
55#125×C61—21no—
56#125×C61—21no—
57C22⋊Q8×C61—21no—
58C22⋊Q8×C61—21no—
59C22⋊Q8×C61—21no—
60C22⋊Q8×C61D412yes—
61C4×Q8×C61—21no—
62C4×Q8×C61—21no—
63C4×Q8×C61—21no—
64C4×Q8×C61D412yes—
65#117×C61—21no—
66#117×C61—21no—
67C2×C42×C61C2311yescharacteristic
68#107×C61—22no—
69#103×C61—22no—
70#103×C61—22no—
71C4×C8×C61—21no—
72C4×C8×C61C2×C411yescharacteristic
73C4×C8×C61C2×C411yescharacteristic
74C2×Q8×C61—21no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61C22⋊C412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61#4111yescharacteristic
81C22×Dic61—41no—
82Q16×C61—44no—
83C2×C8×C61—21no—
84C2×C8×C61—21no—
85C2×C8×C61—42no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61—21no—
88C4⋊C4×C61—21no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—22no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61C22⋊C412yes—
98C22⋊C4×C61—22no—
99C22⋊C4×C61—41no—
100C42×C61—22no—
101C42×C61—41no—
102C42×C61C22×C411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61#4111yescharacteristic
106C4⋊Dic61—22no—
107C4⋊Dic61—41no—
108C4×Dic61—41no—
109C4×Dic61—41no—
110#11758194—21no—
111#11758194—21no—
112#11758194—21no—
113#11758194Q1612yescomplemented
114C2×#92418—21no—
115C2×#92418—21no—
116C2×#92418—21no—
117C2×#92418—21no—
118C2×#92418—41no—
119C2×#92418—41no—
120C2×#92418—41no—
121C23×C61#12111yescharacteristic
122Q8×C61—42no—
123Q8×C61—42no—
124Q8×C61—41no—
125Q8×C61—41no—
126Q8×C61—41no—
127Q8×C61—41no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61C2×C22⋊C411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61C4×D411yescharacteristic
138C2×C4×C61#12011yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12311yescharacteristic
141C2×C4×C61C2×Q1611yescharacteristic
142C2×C4×C61#13511yescharacteristic
143C2×C4×C61#13711yescharacteristic
144C2×C4×C61#13711yescharacteristic
145C2×Dic61—42no—
146C2×Dic61—81no—
147C2×Dic61—81no—
148C8×C61—22no—
149C8×C61—41no—
150#92418—22no—
151#92418—41no—
152#92418—41no—
153#92418—41no—
154#92418—81no—
155#9056—611nomaximal, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61#38611yescharacteristic
159C22×C61#46211yescharacteristic
160C22×C61#46411yescharacteristic
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—42no—
165C4×C61—42no—
166C4×C61—81no—
167C4×C61—81no—
168C4×C61C4×Q1611yescharacteristic
169C4×C61#44111yescharacteristic
170C4×C61#44311yescharacteristic
171C4×C61#44411yescharacteristic
172C4×C61#45111yescharacteristic
173C4×C61#45211yescharacteristic
174Dic61—82no—
175#3170D6111yescharacteristic
176#2962—611no—
177#1986—611no—
178#1503—611no—
179#1481—611no—
180#1279—611no—
181#1278—611no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61#187611yescharacteristic
185C2×C61#187811yescharacteristic
186C2×C61#188411yescharacteristic
187#533C2×D6111yescharacteristic
188#533C2×D6111yescharacteristic
189#500Dic6111yescharacteristic, complemented
190#494Dic6111yescharacteristic, complemented
191#487Dic6111yescharacteristic, complemented
192#487Dic6111yescharacteristic, complemented
193#467—22no—
194#446—1221no—
195#446—1222no—
196C8×Q8—1221no—
197C4×Q16—22no—
198#431—611no—
199#424—1221no—
200C2×C4⋊C8—611no—
201#405—1221no—
202C4×M16—611no—
203C2×C4×C8C2×D6111yescharacteristic
204C4×C4⋊C4—611no—
205#339—1221no—
206#339—1221no—
207#328—1222no—
208Q8⋊C8—1221no—
209Q8⋊C8—1221no—
210C61#905611yescharacteristic, complemented, Sylow
211C2×Q16—22no—
212C2×M16—611no—
213C2×M16—611no—
214C22×C8—21no—
215C22×C8—1221no—
216C4⋊Q8C2×Dic6111yescharacteristic
217C4⋊Q8C2×Dic6111yescharacteristic
218C4⋊Q8C2×Dic6111yescharacteristic
219C4⋊Q8C2×Dic6111yescharacteristic
220#125—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8C22⋊D6112yes—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C22⋊D6112yes—
230#117—21no—
231#117—21no—
232C2×C4⋊C4—1221no—
233C2×C42—1221no—
234C2×C42C22×D6111yescharacteristic
235#107—22no—
236C4⋊C8—611no—
237C4⋊C8—611no—
238C4⋊C8—1222no—
239C4⋊C8—1222no—
240C4⋊C8—1221no—
241C4⋊C8—1221no—
242#103—22no—
243#103—22no—
244C22⋊C8—1221no—
245#97—1222no—
246#97—1221no—
247C4×C8—21no—
248C4×C8—1222no—
249C4×C8—1221no—
250C4×C8—1221no—
251C4×C8C2×Dic6111yescharacteristic
252C4×C8C2×Dic6111yescharacteristic
253#95—1221no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8C22⋊Dic6112yes—
258C22×C4—2441no—
259C22×C4D4×D6111yescharacteristic
260C22×C4#1175822411yescharacteristic
261C22×C4C2×C22⋊D6111yescharacteristic
262Q16—44no—
263M16—612no—
264M16—1221no—
265M16—1221no—
266M16—1221no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—1221no—
271C2×C8—1221no—
272C2×C8—1221no—
273C2×C8—1221no—
274C2×C8—2441no—
275C2×C8—2441no—
276C2×C8—2441no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—1222no—
289C4⋊C4—2441no—
290C4⋊C4C22⋊Dic6112yes—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C42—22no—
294C42—41no—
295C42—2441no—
296C42—2441no—
297C42D4×D6111yescharacteristic
298C42#1175822411yescharacteristic
299C42C22×Dic6111yescharacteristic
300C42C2×C22⋊D6111yescharacteristic
301C23#1451974911yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—4881no—
316C2×C4—4881no—
317C2×C4—2442no—
318C2×C4C2×C22⋊Dic6111yescharacteristic
319C2×C4D4×Dic6111yescharacteristic
320C2×C4D4×Dic6111yescharacteristic
321C2×C4#1451974511yescharacteristic
322C2×C4#1451974911yescharacteristic
323C2×C4#1451975811yescharacteristic
324C2×C4Q16×D6111yescharacteristic
325C2×C4#1451980011yescharacteristic
326C2×C4#1451980911yescharacteristic
327C2×C4#1451980911yescharacteristic
328C8—22no—
329C8—41no—
330C8—1222no—
331C8—4881no—
332C8—2441no—
333C8—2441no—
334C8—2441no—
335C22—21no—
336C22—21no—
337C22#1683964211yescharacteristic
338C22#1683997011yescharacteristic
339C22#1683998611yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—4882no—
348C4Q16×Dic6111yescharacteristic
349C4#1683986911yescharacteristic
350C4#1683987911yescharacteristic
351C4#1683988411yescharacteristic
352C4#1683991611yescharacteristic
353C4#1683992311yescharacteristic
354C2—21no—
355C2—21no—
356C2#2144614811yescharacteristic
357C2#2144616211yescharacteristic
358C2#2144620311yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3170C2
socleC22×C61#386
prime radicalC23×C61#121
cosocleC2×C42×C61C23
hypercenter#3170D61
hypocenterC61#9056
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
24. C61×#500C4semidirect
25. C61×#494C4semidirect
26. C61×#487C4semidirect
27. C61×#487C4semidirect
113. #11758194Q16semidirect
189. #500Dic61semidirect
190. #494Dic61semidirect
191. #487Dic61semidirect
192. #487Dic61semidirect
210. C61#9056semidirect

Maximal subgroups

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

subgroupindex
#214547682
#214467642
#214443182
#214442222
#214436592
#214436562
C61×#31702
#905661