Finite Group Databasebeta

← #30597558

Subgroups — #30597558

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461605C211yesmaximal, characteristic
2C4×#14519832C211yesmaximal, characteristic
3C2×Q8⋊#92420C212yesmaximal
4#21460655C212yesmaximal
5C4×Q8×Dic61C211yesmaximal, characteristic
6C2×C4⋊#92420C2212yes—
7C2×#14519832—21no—
8C2×#14519832—21no—
9C2×#14519832—21no—
10C2×C4×#92420C2212yes—
11#16840628—21no—
12Q8⋊#92420C418yescomplemented
13C2×Q8×Dic61C411yescharacteristic
14C2×Q8×Dic61C2211yescharacteristic
15C4⋊C4×Dic61—21no—
16C4⋊C4×Dic61—21no—
17C4⋊C4×Dic61C411yescharacteristic
18C4⋊C4×Dic61C2211yescharacteristic
19C4×C4⋊Dic61—21no—
20C4×C4⋊Dic61C411yescharacteristic
21C42×Dic61—21no—
22C42×Dic61C2211yescharacteristic
23C2×C4×Q8×C61C411yescharacteristic
24C22×#92420—22no—
25C22×#92420—22no—
26C4⋊#92420—22no—
27C4⋊#92420C2×C414yes—
28#14519832—24no—
29#14519832—24no—
30#14519832—42no—
31C4×#92420—22no—
32C4×#92420C2×C414yes—
33Q8×Dic61—22no—
34Q8×Dic61—22no—
35Q8×Dic61—22no—
36Q8×Dic61C2×C414yes—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×C4×Dic61—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61C2×C411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61C2311yescharacteristic
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55C22×Q8×C61C2×C411yescharacteristic
56C4×Q8×C61—22no—
57C4×Q8×C61—22no—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C42×C61—21no—
61C2×C42×C61C2×C411yescharacteristic
62C2×Q8×C61—21no—
63C2×Q8×C61—21no—
64C2×Q8×C61—21no—
65C2×Q8×C61C4212yes—
66C2×Q8×C61M1612yes—
67C2×Q8×C61M1612yes—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C2×#92420—24no—
80C2×#92420—24no—
81C2×#92420—24no—
82C2×#92420—24no—
83C2×#92420—42no—
84C2×#92420—42no—
85C4⋊C4×C61—24no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61M1612yes—
88C42×C61—24no—
89C42×C61—21no—
90C42×C61—21no—
91C4⋊Dic61—24no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94C4⋊Dic61—24no—
95C4⋊Dic61C4212yes—
96C4⋊Dic61M1612yes—
97C4×Dic61—24no—
98C4×Dic61—22no—
99C4×Dic61—21no—
100C4×Dic61—21no—
101C4×Dic61—22no—
102C4×Dic61—24no—
103C4×Dic61—21no—
104C4×Dic61—22no—
105C4×Dic61—21no—
106C4×Dic61—42no—
107C4×Dic61—42no—
108C4×Dic61—41no—
109C4×Dic61—41no—
110C4×Dic61C22×C412yes—
111C4×Dic61#4112yes—
112C23×C61C2×C22⋊C411yescharacteristic
113Q8×C61—22no—
114Q8×C61—22no—
115Q8×C61—22no—
116Q8×C61#9714yescomplemented
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—21no—
120C2×C4×C61—22no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—22no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61#10412yes—
129C2×C4×C61C2×C4211yescharacteristic
130C2×C4×C61#11711yescharacteristic
131C2×C4×C61C2×M1611yescharacteristic
132C2×C4×C61C2×M1611yescharacteristic
133C2×C4×C61C2×M1611yescharacteristic
134C2×C4×C61C2×M1611yescharacteristic
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—22no—
140C2×Dic61—21no—
141C2×Dic61—22no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61#10412yes—
149C2×Dic61C4×D412yes—
150C2×Dic61C4×D412yes—
151C2×Dic61AGL(1,ℤ/8)11yescharacteristic
152C2×Dic61#13712yes—
153C2×Dic61#13711yescharacteristic
154#92420—28no—
155#92420—44no—
156#8173—611nomaximal, Sylow
157C22×C61C4×C22⋊C411yescharacteristic
158C22×C61#40711yescharacteristic
159C22×C61#40711yescharacteristic
160C22×C61#41711yescharacteristic
161C22×C61#41911yescharacteristic
162C22×C61C2×#10411yescharacteristic
163C22×C61C2×#10411yescharacteristic
164C4×C61—22no—
165C4×C61—24no—
166C4×C61—41no—
167C4×C61—42no—
168C4×C61C2×#9711yescharacteristic
169C4×C61C4×M1611yescharacteristic
170C4×C61#43211yescharacteristic
171C4×C61#43211yescharacteristic
172Dic61—22no—
173Dic61—22no—
174Dic61—41no—
175Dic61—42no—
176Dic61—42no—
177Dic61#44012yes—
178Dic61#44112yes—
179C42×Q8—611no—
180#1767—611no—
181C4×#97—611no—
182C2×Q8⋊C8—612no—
183#1279—612no—
184C2×C61#148911yescharacteristic
185C2×C61#148911yescharacteristic
186C2×C61#149111yescharacteristic
187C2×C61#149111yescharacteristic
188C2×C61#174911yescharacteristic
189C2×C61C4×#10411yescharacteristic
190C2×C61#176311yescharacteristic
191C2×C4×Q8—611no—
192C2×C4×Q8—611no—
193C2×C4×Q8#692311yescharacteristic
194C2×C4⋊C8—612no—
195C2×#97—1221no—
196C2×#97—1221no—
197C2×#97—1221no—
198C2×C4×C8—612no—
199C4×C4⋊C4—611no—
200C4×C4⋊C4—611no—
201C4×C4⋊C4—611no—
202C4×C4⋊C4—1221no—
203C4×C4⋊C4—1221no—
204C4×C4⋊C4—1221no—
205C43—611no—
206C43—1221no—
207#336—1221no—
208Q8⋊C8—618no—
209C61#817311yescharacteristic, complemented, Sylow
210C22×Q8C2×#692311yescharacteristic
211C22×C8—1222no—
212C22×C8—1222no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—614no—
216C4×Q8—1222no—
217C4×Q8—1222no—
218C4×Q8—1222no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—611no—
221C2×C4⋊C4—611no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4C2×#692311yescharacteristic
225C2×C42—21no—
226C2×C42—611no—
227C2×C42—611no—
228C2×C42—611no—
229C2×C42—611no—
230C2×C42—611no—
231C2×C42—611no—
232C2×C42—1221no—
233C2×C42—1221no—
234C2×C42—1221no—
235C2×C42—1221no—
236C2×C42—1221no—
237C2×C42C2×#692311yescharacteristic
238C4⋊C8—614no—
239C4⋊C8—1222no—
240#97—1224no—
241#97—1224no—
242#97—2442no—
243C4×C8—614no—
244C4×C8—1222no—
245#95—1221no—
246#95—1221no—
247#95—1221no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8#1175821312yes—
252C2×Q8C4×#692312yes—
253C2×Q8#1175821712yes—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4C22⋊#692311yescharacteristic
263C22×C4C22⋊#692311yescharacteristic
264C22×C4C22×#692311yescharacteristic
265C2×C8—1224no—
266C2×C8—1224no—
267C2×C8—1224no—
268C2×C8—1224no—
269C2×C8—2442no—
270C2×C8—2442no—
271C4⋊C4—24no—
272C4⋊C4—21no—
273C4⋊C4—612no—
274C4⋊C4—612no—
275C4⋊C4—1224no—
276C4⋊C4—1224no—
277C4⋊C4—1221no—
278C4⋊C4—1221no—
279C4⋊C4#1175821712yes—
280C42—24no—
281C42—21no—
282C42—21no—
283C42—612no—
284C42—612no—
285C42—1222no—
286C42—1221no—
287C42—1222no—
288C42—1221no—
289C42—1224no—
290C42—1224no—
291C42—1222no—
292C42—1221no—
293C42—1221no—
294C42—2441no—
295C42—2441no—
296C42—2442no—
297C42—2442no—
298C23C2×C22⋊#692311yescharacteristic
299Q8—22no—
300Q8—22no—
301Q8—22no—
302Q8#1451983214yescomplemented
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—611no—
315C2×C4—611no—
316C2×C4—612no—
317C2×C4—612no—
318C2×C4—612no—
319C2×C4—612no—
320C2×C4—1221no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—1222no—
325C2×C4—1221no—
326C2×C4—1221no—
327C2×C4—1221no—
328C2×C4—1222no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4#1451984212yes—
334C2×C4C2×C4×#692311yescharacteristic
335C2×C4#1451985611yescharacteristic
336C2×C4C2×#1175821711yescharacteristic
337C2×C4C2×#1175821711yescharacteristic
338C2×C4D61⋊M1611yescharacteristic
339C2×C4C2×#1175821311yescharacteristic
340C8—1228no—
341C8—2444no—
342C22C4×C22⋊#692311yescharacteristic
343C22#1684073211yescharacteristic
344C22#1684073311yescharacteristic
345C22#1684075311yescharacteristic
346C22#1684075511yescharacteristic
347C22D61⋊#10411yescharacteristic
348C22C2×#1451984211yescharacteristic
349C4—24no—
350C4—22no—
351C4—41no—
352C4—42no—
353C4—612no—
354C4—612no—
355C4—1222no—
356C4—1222no—
357C4—2442no—
358C4—2441no—
359C4—2442no—
360C4C2×#1451983211yescharacteristic
361C4C4×#1175821711yescharacteristic
362C4#1684078211yescharacteristic
363C4#1684078311yescharacteristic
364C2#2146115411yescharacteristic
365C2#2146115711yescharacteristic
366C2#2146116111yescharacteristic
367C2#2146116211yescharacteristic
368C2#2146154511yescharacteristic
369C2C4×#1451984211yescharacteristic
370C2#2146159111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socle, prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8173
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
12. Q8⋊#92420C4semidirect
116. Q8×C61#97semidirect
209. C61#8173semidirect
302. Q8#14519832semidirect

Maximal subgroups

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

subgroupindex
#214616052
C4×#145198322
C2×Q8⋊#924202
#214606552
C4×Q8×Dic612
#817361