Finite Group Databasebeta

← #30619291

Subgroups — #30619291

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840797C211yesmaximal, characteristic, complemented
2C2×D4×#92420C211yesmaximal, characteristic, complemented
3#21461940C211yesmaximal, characteristic, complemented
4#21461940C211yesmaximal, characteristic, complemented
5#21461796C214yesmaximal, complemented
6#21461792C214yesmaximal, complemented
7#21461736C211yesmaximal, characteristic, complemented
8#21461534C211yesmaximal, characteristic, complemented
9#127×Dic61C211yesmaximal, characteristic
10C22×#11758217C2212yes—
11C23×#92420C2212yes—
12#16840797—24no—
13D4×#92420—24no—
14C2×C4⋊#92420C2211yescharacteristic
15C2×C4⋊#92420C2211yescharacteristic
16C2×C22⋊#92420C2212yes—
17C2×C22⋊#92420C2212yes—
18C2×C22⋊#92420C2214yes—
19C2×C22⋊#92420C2214yes—
20C2×C4×#92420C2211yescharacteristic
21C2×C4×#92420C2211yescharacteristic
22#16840628C2212yes—
23#16840628C2212yes—
24#16840628C2214yes—
25C2×D4×Dic61C2211yescharacteristic
26C2×D4×Dic61C2211yescharacteristic
27C2×D4×Dic61C2214yes—
28#16839665C411yescharacteristic
29#16839665C414yes—
30#16839665C411yescharacteristic
31#16839632C411yescharacteristic
32C42×Dic61C2211yescharacteristic
33C2×#127×C61C411yescharacteristic
34C2×#11758217—24no—
35C2×#11758217—44no—
36C22×#92420—24no—
37C22×#92420—44no—
38C22×#92420C2312yes—
39C22×#92420C2312yes—
40C22×#92420C2312yes—
41C22×#92420C2311yescharacteristic
42C22×#92420C2311yescharacteristic
43C4⋊#92420—22no—
44C4⋊#92420—22no—
45C22⋊#92420—24no—
46C22⋊#92420—24no—
47C22⋊#92420—44no—
48C22⋊#92420—44no—
49C4×#92420—22no—
50C4×#92420D414yescomplemented
51C23×Dic61C2312yes—
52C23×Dic61C2312yes—
53D4×Dic61—22no—
54D4×Dic61—22no—
55D4×Dic61—22no—
56D4×Dic61—22no—
57D4×Dic61—44no—
58D4×Dic61—44no—
59C2×C4⋊Dic61C2×C411yescharacteristic
60C2×C4⋊Dic61C2×C411yescharacteristic
61C2×C4⋊Dic61C2×C414yes—
62C2×C22⋊Dic61C2×C412yes—
63C2×C22⋊Dic61C2×C412yes—
64C2×C22⋊Dic61C2×C414yes—
65C2×C22⋊Dic61C2×C414yes—
66C2×C4×Dic61C2×C411yescharacteristic
67C2×C4×Dic61C2×C412yes—
68C2×C4×Dic61C2312yes—
69C2×C4×Dic61C2311yescharacteristic
70C22×D4×C61C2×C411yescharacteristic
71C22×D4×C61C2×C414yes—
72C22×D4×C61C2×C411yescharacteristic
73#127×C61—24no—
74C2×C42×C61C2×C411yescharacteristic
75C24×C61C22×C412yes—
76C24×C61C22×C412yes—
77C2×D4×C61—28no—
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61—28no—
83C2×D4×C61—22no—
84C2×D4×C61—44no—
85C2×D4×C61M1614yescomplemented
86C22×C4×C61C22×C412yes—
87C22×C4×C61C22×C411yescharacteristic
88C22×Dic61—22no—
89C22×Dic61—22no—
90C22×Dic61—22no—
91C22×Dic61—22no—
92C22×Dic61—24no—
93C22×Dic61—24no—
94C22×Dic61—24no—
95C22×Dic61—24no—
96C22×Dic61C22×C412yes—
97C22×Dic61C22×C412yes—
98C22×Dic61C22×C412yes—
99C22×Dic61C22×C411yescharacteristic
100C22×Dic61C2411yescharacteristic
101#11758217—48no—
102C2×#92420—22no—
103C2×#92420—22no—
104C2×#92420—22no—
105C2×#92420—22no—
106C2×#92420—22no—
107C2×#92420—22no—
108C2×#92420—22no—
109C2×#92420—24no—
110C2×#92420—24no—
111C2×#92420—48no—
112C2×#92420C2×D412yes—
113C2×#92420C2×D412yes—
114C42×C61—22no—
115C22⋊Dic61—24no—
116C22⋊Dic61—24no—
117C22⋊Dic61—44no—
118C22⋊Dic61—44no—
119C4⋊Dic61—22no—
120C4⋊Dic61—22no—
121C4⋊Dic61—44no—
122C4×Dic61—24no—
123C4×Dic61—22no—
124C4×Dic61—22no—
125C4×Dic61—22no—
126C4×Dic61C2×D412yes—
127C4×Dic61#4112yes—
128C23×C61—22no—
129C23×C61—24no—
130C23×C61—22no—
131C23×C61—24no—
132C23×C61—22no—
133C23×C61—24no—
134C23×C61—22no—
135C23×C61—22no—
136C23×C61—24no—
137C23×C61C2×M1614yes—
138C23×C61C23×C411yescharacteristic
139D4×C61—28no—
140D4×C61—416no—
141D4×C61—44no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61C4×D412yes—
145C2×C4×C61C2×M1612yes—
146C2×C4×C61#13114yes—
147C2×C4×C61#13112yes—
148C2×Dic61—22no—
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—24no—
153C2×Dic61—24no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—44no—
161C2×Dic61—44no—
162C2×Dic61C4×D412yes—
163C2×Dic61C22×D411yescharacteristic
164C2×Dic61C2×#4111yescharacteristic
165C2×Dic61H(5,𝔽2)12yes—
166C2×Dic61H(5,𝔽2)12yes—
167#92420—22no—
168#92420—22no—
169#92420—42no—
170#92420—42no—
171#92420—42no—
172#22262—611nomaximal, Sylow
173C22×C61—24no—
174C22×C61—28no—
175C22×C61—42no—
176C22×C61—42no—
177C22×C61—44no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—44no—
182C22×C61—44no—
183C22×C61C2×C4×D411yescharacteristic
184C22×C61#51812yes—
185C22×C61C22×M1611yescharacteristic
186C22×C61C2×#13111yescharacteristic
187C22×C61C2×#13112yes—
188C4×C61—24no—
189C4×C61#43514yes—
190Dic61—22no—
191Dic61—42no—
192Dic61—42no—
193Dic61—42no—
194Dic61#53511yescharacteristic
195Dic61#53511yescharacteristic
196Dic61#54012yes—
197C2×#436—611no—
198C2×C8×D4—611no—
199C4×#127—611no—
200#1954—611no—
201#1954—611no—
202#1870—614no—
203#1869—614no—
204#1831—611no—
205#1744—611no—
206C2×C61—44no—
207C2×C61—82no—
208C2×C61#232811yescharacteristic
209C2×C61C2×#43511yescharacteristic
210C2×C61#293111yescharacteristic
211C2×C61#297412yes—
212C2×C61#297912yes—
213C22×M16—612no—
214C23×C8—612no—
215C2×#127#692311yescharacteristic
216C2×C4×D4—611no—
217C2×C4×D4—611no—
218C2×C4×D4—614no—
219#436—1224no—
220C8×D4—1224no—
221C2×C4⋊C8—611no—
222C2×C4⋊C8—611no—
223C2×C22⋊C8—612no—
224C2×C22⋊C8—612no—
225C2×C22⋊C8—614no—
226C2×C22⋊C8—614no—
227C2×C4×C8—611no—
228C2×C4×C8—611no—
229#390—614no—
230#390—611no—
231#390—611no—
232#384—611no—
233C43—611no—
234#336—614no—
235#336—612no—
236#336—612no—
237C61#2226211yescharacteristic, complemented, Sylow
238C22×D4C2×#692311yescharacteristic
239C22×D4C2×#692314yes—
240C22×D4C2×#692311yescharacteristic
241C23×C4—612no—
242C23×C4—612no—
243C2×M16—1224no—
244C2×M16—2444no—
245C22×C8—611no—
246C22×C8—611no—
247C22×C8—612no—
248C22×C8—612no—
249C22×C8—612no—
250C22×C8—1224no—
251C22×C8—2444no—
252#127—24no—
253C4×D4—1222no—
254C4×D4—1222no—
255C4×D4—1222no—
256C4×D4—1222no—
257C4×D4—2444no—
258C4×D4—2444no—
259C2×C4⋊C4—611no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—614no—
262C2×C22⋊C4—612no—
263C2×C22⋊C4—612no—
264C2×C22⋊C4—614no—
265C2×C22⋊C4—614no—
266C2×C42—611no—
267C2×C42—612no—
268C2×C42—612no—
269C2×C42—611no—
270C2×C42C2×#692311yescharacteristic
271C4⋊C8—1222no—
272C4⋊C8—1222no—
273C22⋊C8—1224no—
274C22⋊C8—1224no—
275C22⋊C8—2444no—
276C22⋊C8—2444no—
277C4×C8—614no—
278C4×C8—1222no—
279C24C22×#692312yes—
280C24C22×#692312yes—
281C2×D4—28no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—28no—
287C2×D4—22no—
288C2×D4—44no—
289C2×D4#1175821714yescomplemented
290C22×C4—611no—
291C22×C4—611no—
292C22×C4—612no—
293C22×C4—612no—
294C22×C4—612no—
295C22×C4—1222no—
296C22×C4—1222no—
297C22×C4—1224no—
298C22×C4—1224no—
299C22×C4—1222no—
300C22×C4—1222no—
301C22×C4—1224no—
302C22×C4—1224no—
303C22×C4C22×#692312yes—
304C22×C4C22×#692311yescharacteristic
305M16—2448no—
306C2×C8—612no—
307C2×C8—612no—
308C2×C8—1222no—
309C2×C8—1222no—
310C2×C8—1222no—
311C2×C8—1222no—
312C2×C8—1222no—
313C2×C8—1222no—
314C2×C8—1222no—
315C2×C8—1224no—
316C2×C8—1224no—
317C2×C8—2448no—
318C4⋊C4—1222no—
319C4⋊C4—1222no—
320C4⋊C4—2444no—
321C22⋊C4—1224no—
322C22⋊C4—1224no—
323C22⋊C4—2444no—
324C22⋊C4—2444no—
325C42—22no—
326C42—612no—
327C42—612no—
328C42—1222no—
329C42—1222no—
330C42—1224no—
331C42—1222no—
332C23—22no—
333C23—24no—
334C23—24no—
335C23—22no—
336C23—22no—
337C23—24no—
338C23—22no—
339C23—24no—
340C23—22no—
341C23C2×#1175821714yes—
342C23C23×#692311yescharacteristic
343D4—28no—
344D4—416no—
345D4—44no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—611no—
349C2×C4—611no—
350C2×C4—612no—
351C2×C4—612no—
352C2×C4—612no—
353C2×C4—1222no—
354C2×C4—1222no—
355C2×C4—1222no—
356C2×C4—1222no—
357C2×C4—1224no—
358C2×C4—1224no—
359C2×C4—1222no—
360C2×C4—1222no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4—2444no—
366C2×C4—2444no—
367C2×C4D4×#692312yes—
368C2×C4C2×#1175821712yes—
369C2×C4#1451986414yes—
370C2×C4#1451986412yes—
371C8—1222no—
372C8—1222no—
373C8—2442no—
374C8—2442no—
375C8—2442no—
376C22—28no—
377C22—24no—
378C22—42no—
379C22—42no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22—44no—
384C22—44no—
385C22—42no—
386C22C2×D4×#692311yescharacteristic
387C22#1684082912yes—
388C22C22×#1175821711yescharacteristic
389C22C2×#1451986412yes—
390C22C2×#1451986411yescharacteristic
391C4—24no—
392C4—611no—
393C4—611no—
394C4—612no—
395C4—1222no—
396C4—2442no—
397C4—2442no—
398C4—2442no—
399C4#1684079314yes—
400C2—44no—
401C2—82no—
402C2#2146231111yescharacteristic
403C2C2×#1684079311yescharacteristic
404C2#2146254111yescharacteristic
405C2#2146267512yes—
406C2#2146270012yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
Fitting, prime radicalC2×#127×C61C4
cosocleC22×Dic61C24
hypercenterC2×#127#6923
hypocenterC61#22262
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#16840797C2semidirect
2. C2×D4×#92420C2semidirect
3. #21461940C2semidirect
4. #21461940C2semidirect
5. #21461796C2semidirect
6. #21461792C2semidirect
7. #21461736C2semidirect
8. #21461534C2semidirect
50. C4×#92420D4semidirect
85. C2×D4×C61M16semidirect
237. C61#22262semidirect
289. C2×D4#11758217semidirect

Maximal subgroups

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

subgroupindex
C2×#168407972
C2×D4×#924202
#214619402
#214617962
#214617922
#214617362
#214615342
#127×Dic612
#2226261