Finite Group Databasebeta

← #30607807

Subgroups — #30607807

Order 15616.

394 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840683C211yesmaximal, characteristic, complemented
2C2×#16840657C211yesmaximal, characteristic
3C2×#16840657C211yesmaximal, characteristic
4C2×C4×#11758187C211yesmaximal, characteristic, complemented
5#21460858C211yesmaximal, characteristic, complemented
6#21460857C211yesmaximal, characteristic, complemented
7#21445589C211yesmaximal, characteristic
8C22×#11758187C2211yescharacteristic
9C22×#11758187C2211yescharacteristic
10C22×#11758187C2211yescharacteristic
11C22×#11758187C2211yescharacteristic
12#16840683—22no—
13#16840657—22no—
14#16840657—22no—
15C4×#11758187—22no—
16C22×C8×D61C2211yescharacteristic
17C2×#14519689C411yescharacteristic
18C2×C4×#92418C2211yescharacteristic
19C2×C8×Dic61C2211yescharacteristic
20C4×#11758198C411yescharacteristic
21#16839516C411yescharacteristic
22C61×#336C411yescharacteristic
23C2×#11758187—21no—
24C2×#11758187—21no—
25C2×#11758187—21no—
26C2×#11758187—21no—
27C2×#11758187—21no—
28C2×#11758187—21no—
29C2×#11758187—21no—
30C2×#11758187—21no—
31C2×#11758187—21no—
32C2×#11758187—21no—
33C2×#11758187D412yes—
34C2×#11758187D412yes—
35C22×C4×D61C2×C411yescharacteristic
36C22×#92418C2×C411yescharacteristic
37C22×#92418C2311yescharacteristic
38C2×C8×D61—21no—
39C2×C8×D61—21no—
40C2×C8×D61—21no—
41C2×C8×D61—21no—
42C2×C8×D61—21no—
43C2×C8×D61—21no—
44C2×#11758198C811yescharacteristic
45C2×#11758198C811yescharacteristic
46C2×C4×Dic61C2×C411yescharacteristic
47#14519689—22no—
48C4×#92418—21no—
49C4×#92418—21no—
50C8×Dic61—21no—
51C8×Dic61—21no—
52#14519684C811yescharacteristic
53#14519684C811yescharacteristic
54C22×C8×C61C2×C411yescharacteristic
55C22×C8×C61C2×C411yescharacteristic
56C2×C42×C61C2×C411yescharacteristic
57C23×D61C2×C811yescharacteristic
58C22×C4×C61C2×C811yescharacteristic
59C22×C4×C61C2×C811yescharacteristic
60C22×C4×C61C22×C411yescharacteristic
61C22×Dic61C2×C811yescharacteristic
62C22×Dic61C2×C811yescharacteristic
63C22×Dic61C2×C811yescharacteristic
64C2×C4×D61—21no—
65C2×C4×D61—21no—
66C2×C4×D61—21no—
67C2×C4×D61—21no—
68C2×C4×D61—21no—
69C2×C4×D61—21no—
70C2×C8×C61—21no—
71C2×C8×C61—21no—
72C2×C8×C61—21no—
73C2×C8×C61—21no—
74C2×C8×C61—21no—
75C2×C8×C61—21no—
76C42×C61—21no—
77C42×C61—21no—
78#11758198—22no—
79#11758198C1614yescomplemented
80C4×Dic61—21no—
81C4×Dic61—21no—
82C2×#92418—21no—
83C2×#92418—21no—
84C2×#92418M1612yes—
85C2×#92418C2×D411yescharacteristic
86C2×#92418C2×D411yescharacteristic
87C2×#92418#4111yescharacteristic
88C2×#92418#4111yescharacteristic
89C2×#92418#4111yescharacteristic
90C2×#92418#4111yescharacteristic
91C8×D61—41no—
92C8×D61—41no—
93C8×D61—41no—
94C8×D61—41no—
95#11758187—22no—
96#11758187—22no—
97#11758187—41no—
98#11758187—41no—
99C23×C61C22×C811yescharacteristic
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—21no—
106C22×D61C2×C1612yes—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61C2×C1612yes—
112C2×C4×C61M3212yes—
113C2×C4×C61#11711yescharacteristic
114C2×C4×C61C4×D411yescharacteristic
115C2×C4×C61C4×D411yescharacteristic
116C2×C4×C61C2×M1611yescharacteristic
117C2×C4×C61#13111yescharacteristic
118C2×C4×C61#13111yescharacteristic
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61C2×C1612yes—
128C4×D61—41no—
129C4×D61—41no—
130C4×D61—41no—
131C4×D61—41no—
132C8×C61—41no—
133C8×C61—41no—
134#92418—22no—
135#92418#12111yescharacteristic
136#92418#12311yescharacteristic
137#92418#12411yescharacteristic
138#92418#12611yescharacteristic
139#13120—611nomaximal, Sylow
140C22×C61#43111yescharacteristic
141C22×C61C8×D411yescharacteristic
142C22×C61C8×D411yescharacteristic
143C22×C61C22×C1611yescharacteristic
144C22×C61C2×M3211yescharacteristic
145C22×C61#50411yescharacteristic
146C22×C61#50411yescharacteristic
147C2×D61—22no—
148C2×D61—22no—
149C2×D61—22no—
150C2×D61—41no—
151C2×D61—41no—
152C2×D61—41no—
153C2×D61—41no—
154C4×C61—22no—
155C4×C61—22no—
156C4×C61#38811yescharacteristic
157C4×C61#43311yescharacteristic
158C4×C61#43511yescharacteristic
159C4×C61#43611yescharacteristic
160Dic61—22no—
161Dic61—41no—
162Dic61—41no—
163C2×C4⋊C16—611no—
164C2×C22⋊C16—611no—
165C2×C22⋊C16—611no—
166C2×C4×C16—611no—
167C8×C22⋊C4—611no—
168#1374—611no—
169#1374—611no—
170C2×C61#192311yescharacteristic
171C2×C61#216211yescharacteristic
172C2×C61#216411yescharacteristic
173C2×C61D4×C1611yescharacteristic
174C2×C61D4×C1611yescharacteristic
175C2×C61#216811yescharacteristic
176C2×C61#216911yescharacteristic
177D61—42no—
178C23×C8—611no—
179C22×C16—611no—
180C22×C16—611no—
181C22×C16—611no—
182C22×C16—611no—
183C2×C22⋊C8—611no—
184C2×C4×C8—611no—
185C2×C4×C8—611no—
186C4×C22⋊C4—611no—
187C4⋊C16—1222no—
188C22⋊C16—1222no—
189C22⋊C16—1222no—
190C4×C16—1222no—
191#336—611no—
192#336#692311yescharacteristic
193C61#1312011yescharacteristic, complemented, Sylow
194C23×C4—611no—
195C22×C8—611no—
196C22×C8—611no—
197C22×C8—1221no—
198C22×C8—1221no—
199C22×C8—1221no—
200C22×C8—1221no—
201C22×C8—1221no—
202C22×C8—1221no—
203C22×C8C2×#692311yescharacteristic
204C22×C8C2×#692311yescharacteristic
205C2×C22⋊C4—611no—
206C2×C22⋊C4—611no—
207C2×C42—611no—
208C2×C42C2×#692311yescharacteristic
209C2×C16—612no—
210C2×C16—612no—
211C2×C16—1221no—
212C2×C16—1221no—
213C2×C16—1221no—
214C2×C16—1221no—
215C2×C16—1221no—
216C2×C16—1221no—
217C2×C16—1221no—
218C2×C16—1221no—
219C2×C16—1221no—
220C2×C16—1221no—
221C22⋊C8—1222no—
222C4×C8—1221no—
223C4×C8—1221no—
224C4×C8—1221no—
225C4×C8—1221no—
226#95—611no—
227#95—611no—
228C24—611no—
229C22×C4—611no—
230C22×C4—611no—
231C22×C4—611no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1221no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—1221no—
238C22×C4D61⋊C811yescharacteristic
239C22×C4D61⋊C811yescharacteristic
240C22×C4C22×#692311yescharacteristic
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—611no—
248C2×C8—611no—
249C2×C8—611no—
250C2×C8—611no—
251C2×C8—611no—
252C2×C8—611no—
253C2×C8—612no—
254C2×C8—1221no—
255C2×C8—1221no—
256C2×C8—2441no—
257C2×C8—2441no—
258C2×C8—2441no—
259C2×C8—2441no—
260C22⋊C4—614no—
261C22⋊C4—1222no—
262C42—21no—
263C42—21no—
264C42—1221no—
265C42—1221no—
266C16—1222no—
267C16—1222no—
268C16—2441no—
269C16—2441no—
270C23—612no—
271C23—1221no—
272C23—1221no—
273C23—1221no—
274C23—1221no—
275C23—1221no—
276C23—1221no—
277C23C2×D61⋊C811yescharacteristic
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—612no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4D61⋊C1612yes—
296C2×C4#1451985212yes—
297C2×C4#1451985611yescharacteristic
298C2×C4D4×#692311yescharacteristic
299C2×C4D4×#692311yescharacteristic
300C2×C4D61⋊M1611yescharacteristic
301C2×C4#1451986411yescharacteristic
302C2×C4#1451986511yescharacteristic
303C8—41no—
304C8—41no—
305C8—611no—
306C8—611no—
307C8—611no—
308C8—611no—
309C8—1222no—
310C22—1222no—
311C22—1222no—
312C22—1222no—
313C22—2441no—
314C22—2441no—
315C22—2441no—
316C22—2441no—
317C22#1684078011yescharacteristic
318C22#1684078911yescharacteristic
319C22#1684079011yescharacteristic
320C22C2×D61⋊C1611yescharacteristic
321C22C2×#1451985211yescharacteristic
322C22#1684082111yescharacteristic
323C22#1684082111yescharacteristic
324C4—22no—
325C4—22no—
326C4—1222no—
327C4—2441no—
328C4—2441no—
329C4#1684071211yescharacteristic
330C4#1684078711yescharacteristic
331C4#1684079211yescharacteristic
332C4#1684079511yescharacteristic
333C2—2442no—
334C2#2146189011yescharacteristic
335C2#2146217811yescharacteristic
336C2#2146218611yescharacteristic
337C2#2146219711yescharacteristic
338C2#2146219811yescharacteristic
339C2#2146220111yescharacteristic
340C2#2146220511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C22×#6923
commutatorC22×C61C22×C16
FittingC61×#336C4
socleC23×C61C22×C8
prime radicalC23×D61C2×C8
cosocleC22×#92418C23
hypercenter#336#6923
hypocenterC61#13120
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #336

Complements

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

NH ≅ G/Ntype
1. C2×#16840683C2semidirect
4. C2×C4×#11758187C2semidirect
5. #21460858C2semidirect
6. #21460857C2semidirect
79. #11758198C16semidirect
193. C61#13120semidirect

Maximal subgroups

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

subgroupindex
C2×#168406832
C2×#168406572
C2×C4×#117581872
#214608582
#214608572
#214455892
#1312061