Finite Group Databasebeta

← #29699708

Subgroups — #29699708

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519799C211yesmaximal, characteristic
2#21454484C212yesmaximal, complemented
3#21446707C212yesmaximal
4#21446468C214yesmaximal, complemented
5#21445991C211yesmaximal, characteristic, complemented
6#21445840C212yesmaximal, complemented
7#21445698C212yesmaximal, complemented
8C8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
9C2×#14519799C2211yescharacteristic
10C4×#11758222C2212yes—
11C4×Dic244C411yescharacteristic, complemented
12C4×Dic244C411yescharacteristic, complemented
13C4×#11758190C412yescomplemented
14C4×#11758190C412yescomplemented
15C4×D488C411yescharacteristic, complemented
16C4×D488C411yescharacteristic, complemented
17#16839822—24no—
18#16839818—22no—
19#16839814—22no—
20C2×#14519713C2212yes—
21C2×#14519713C2212yes—
22#16839800C2211yescharacteristic
23#16839786C2212yes—
24C2×#14519705C2214yes—
25#16839759C2212yes—
26#16839728C2214yes—
27#16839717C2214yes—
28#16839522C2214yes—
29C2×C4⋊C8×C61C2211yescharacteristic
30C2×C4×C8×C61C2212yes—
31C2×C4×C8×C61C2211yescharacteristic
32C4×C4⋊C4×C61C2211yescharacteristic
33C61×#336C2212yes—
34C2×#11758222C2312yes—
35#14519799—22no—
36#14519799—22no—
37C2×Dic244C2×C411yescharacteristic
38C2×#11758190C2×C412yes—
39C2×D488C2×C411yescharacteristic
40C2×#11758194C2314yes—
41C4×Dic122C2×C412yes—
42C4×Dic122C2×C412yes—
43C4×D244C2×C412yes—
44C4×D244C2×C412yes—
45C4×C22⋊D61—22no—
46C4×C22⋊D61—22no—
47#14519736C2312yes—
48#14519734—22no—
49C42×D61—22no—
50#14519713—24no—
51#14519713—42no—
52#14519711C2×C411yescharacteristic
53#14519711C2×C411yescharacteristic
54#14519709C2×C412yes—
55C4⋊#92418—22no—
56C4⋊#92418—22no—
57#14519705—44no—
58#14519702C2×C412yes—
59#14519702C2×C412yes—
60#14519699C2×C412yes—
61#14519699C2×C412yes—
62#14519689—24no—
63#14519689—24no—
64#14519687—24no—
65C8×Dic61—24no—
66C22×C8×C61C2311yescharacteristic
67C22×C8×C61C2312yes—
68C22×C8×C61C2311yescharacteristic
69C2×C4⋊C4×C61—21no—
70C2×C42×C61C2312yes—
71C2×C42×C61C2311yescharacteristic
72C4⋊C8×C61D411yescharacteristic
73C4⋊C8×C61D411yescharacteristic
74C4⋊C8×C61D412yes—
75C4×C8×C61—24no—
76C4×C8×C61C2×C411yescharacteristic
77C4×C8×C61C2×C411yescharacteristic
78C4×C8×C61C2×C411yescharacteristic
79C4×C8×C61C2×C411yescharacteristic
80#95×C61—21no—
81C22×C4×C61—22no—
82C22×C4×C61#4111yescharacteristic
83C22×C4×C61#4111yescharacteristic
84C22×C4×C61C2411yescharacteristic
85C2×C22⋊D61—22no—
86#11758222—22no—
87#11758222—22no—
88#11758222—22no—
89#11758222—22no—
90C2×D244C22×C412yes—
91C2×C4×D61—22no—
92C2×C4×D61—22no—
93C2×C4×D61—22no—
94C2×Dic122C22×C412yes—
95C2×C8×C61—22no—
96C2×C8×C61—21no—
97C2×C8×C61—21no—
98C2×C8×C61—22no—
99C2×C8×C61C22×C411yescharacteristic
100C2×C8×C61C22×C411yescharacteristic
101C2×C8×C61C22×C411yescharacteristic
102C2×C8×C61C22×C411yescharacteristic
103C2×C8×C61C2×D412yes—
104C2×C8×C61C2×D412yes—
105C2×C8×C61#4112yes—
106C2×C8×C61#4112yes—
107C4⋊C4×C61—24no—
108C42×C61—22no—
109C42×C61—22no—
110C42×C61C22×C411yescharacteristic
111C42×C61C22×C411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C22⋊Dic61—22no—
115#11758198—22no—
116#11758198—22no—
117C4⋊Dic61C22×C412yes—
118C4⋊Dic61C22×C412yes—
119#11758196—22no—
120#11758196—22no—
121C4×Dic61—22no—
122C4×Dic61—22no—
123C4×Dic61—22no—
124#11758194—24no—
125#11758194—24no—
126C2×#92418—24no—
127Dic244—22no—
128D488—22no—
129#11758190—24no—
130C23×C61C2×#4111yescharacteristic
131C22×D61—22no—
132C2×C4×C61—22no—
133C2×C4×C61—24no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61C4×D411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61C4×D411yescharacteristic
141C2×C4×C61C23×C411yescharacteristic
142C2×C4×C61C22×D411yescharacteristic
143C2×C4×C61C2×#4111yescharacteristic
144C2×C4×C61C2×#4111yescharacteristic
145C2×C4×C61H(5,𝔽2)11yescharacteristic
146C2×C4×C612−1+411yescharacteristic
147C22⋊D61—42no—
148C22⋊D61—42no—
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152D244—22no—
153D244—22no—
154C4×D61—42no—
155C4×D61—42no—
156C4×D61—44no—
157C4×D61—82no—
158Dic122—22no—
159Dic122—22no—
160C8×C61—21no—
161C8×C61—21no—
162C8×C61—22no—
163C8×C61—22no—
164C8×C61#11712yes—
165C8×C61#11712yes—
166#92418—44no—
167#22802—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61C2×C4×D411yescharacteristic
171C22×C61#55511yescharacteristic
172C22×C61#55611yescharacteristic
173C2×D61—42no—
174C2×D61—42no—
175C2×D61—42no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—44no—
180C4×C61#44312yes—
181C4×C61C2×#11711yescharacteristic
182C4×C61#52011yescharacteristic
183C4×C61#54711yescharacteristic
184C4×C61#55011yescharacteristic
185Dic61—42no—
186Dic61—42no—
187Dic61—42no—
188C4×#135—611no—
189#2925—612no—
190#1974—612no—
191#1929—614no—
192#1848—611no—
193#1806—612no—
194#1772—612no—
195C8×C4⋊C4D6111yescharacteristic, complemented
196C2×C61—21no—
197C2×C61—21no—
198C2×C61#235411yescharacteristic
199C2×C61C2×#44311yescharacteristic
200C2×C61#295711yescharacteristic
201D61—82no—
202C2×#135—611no—
203C4×#41—612no—
204C4×Q16—611no—
205C4×Q16—611no—
206C4×SD16—612no—
207C4×SD16—612no—
208C4×D8—611no—
209C4×D8—611no—
210#433—1224no—
211#432—1222no—
212#431—1222no—
213C2×#108—612no—
214C2×#108—612no—
215#427—611no—
216#423—612no—
217C2×C4⋊C8C2×D6111yescharacteristic
218C2×#104—614no—
219#416—612no—
220#408—614no—
221#405—614no—
222C2×C4×C8C2×D6112yes—
223C2×C4×C8C2×D6111yescharacteristic
224C4×C4⋊C4C2×D6111yescharacteristic
225#339—614no—
226#336C2×D6112yes—
227C61#2280211yescharacteristic, complemented, Sylow
228C2×#41—612no—
229#135—1222no—
230#135—1222no—
231C2×Q16—611no—
232C2×SD16—612no—
233C2×D8—611no—
234C2×M16—614no—
235C22×C8C22×D6111yescharacteristic
236C22×C8C22×D6112yes—
237C22×C8C22×D6111yescharacteristic
238C4×Q8—612no—
239C4×Q8—612no—
240C4×D4—612no—
241C4×D4—612no—
242C4×D4—1222no—
243C4×D4—1222no—
244#117—612no—
245#117—1222no—
246C2×C4⋊C4—21no—
247C2×C42—1222no—
248C2×C42C22×D6112yes—
249C2×C42C22×D6111yescharacteristic
250#108—1224no—
251#108—2442no—
252#107—611no—
253#107—611no—
254#106—612no—
255C4⋊C8—1222no—
256C4⋊C8—1222no—
257C4⋊C8D24411yescharacteristic
258C4⋊C8D24411yescharacteristic
259C4⋊C8D24412yes—
260#104—2444no—
261#103—612no—
262#103—612no—
263#102—612no—
264#102—612no—
265C22⋊C8—1224no—
266C22⋊C8—1224no—
267#97—1224no—
268C4×C8—24no—
269C4×C8—1224no—
270C4×C8C4×D6111yescharacteristic
271C4×C8C4×D6111yescharacteristic
272C4×C8C4×D6111yescharacteristic
273C4×C8C4×D6111yescharacteristic
274#95—21no—
275#41—1222no—
276#41—1222no—
277#41—1222no—
278#41—1222no—
279C2×Q8—612no—
280C2×D4—612no—
281C2×D4—1222no—
282C22×C4—22no—
283C22×C4—1222no—
284C22×C4—1222no—
285C22×C4—1222no—
286C22×C4#1175822211yescharacteristic
287C22×C4#1175822211yescharacteristic
288C22×C4C23×D6111yescharacteristic
289Q16—1222no—
290SD16—1224no—
291D8—1222no—
292M16—1224no—
293M16—1224no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—1224no—
299C2×C8C2×C4×D6111yescharacteristic
300C2×C8C2×C4×D6111yescharacteristic
301C2×C8C2×C4×D6111yescharacteristic
302C2×C8C2×C4×D6111yescharacteristic
303C2×C8C2×D24412yes—
304C2×C8C2×D24412yes—
305C2×C8#1175822412yes—
306C2×C8#1175822612yes—
307C4⋊C4—24no—
308C4⋊C4—612no—
309C4⋊C4—612no—
310C4⋊C4—1222no—
311C4⋊C4—1222no—
312C22⋊C4—1222no—
313C22⋊C4—1222no—
314C22⋊C4—1222no—
315C42—22no—
316C42—22no—
317C42—1222no—
318C42—1222no—
319C42—1222no—
320C42C2×C4×D6111yescharacteristic
321C42C2×C4×D6111yescharacteristic
322C42C2×D24411yescharacteristic
323C42C2×D24411yescharacteristic
324C23—1222no—
325C23C2×#1175822211yescharacteristic
326Q8—1222no—
327Q8—1222no—
328D4—1222no—
329D4—1222no—
330D4—2442no—
331D4—2442no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—1222no—
340C2×C4—4882no—
341C2×C4—2444no—
342C2×C4—2442no—
343C2×C4—2442no—
344C2×C4C4×D24411yescharacteristic
345C2×C4C4×D24411yescharacteristic
346C2×C4C4×D24411yescharacteristic
347C2×C4C4×D24411yescharacteristic
348C2×C4C22×C4×D6111yescharacteristic
349C2×C4C22×D24411yescharacteristic
350C2×C4C2×#1175822411yescharacteristic
351C2×C4C2×#1175822611yescharacteristic
352C2×C4#1451982311yescharacteristic
353C2×C4#1451982411yescharacteristic
354C8—21no—
355C8—21no—
356C8—22no—
357C8—22no—
358C8—2444no—
359C8#1451973512yes—
360C8#1451973512yes—
361C22—21no—
362C22—21no—
363C22—2442no—
364C22—2442no—
365C22—2442no—
366C22C2×C4×D24411yescharacteristic
367C22#1684047711yescharacteristic
368C22#1684048511yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—44no—
373C4—2442no—
374C4—2442no—
375C4—2442no—
376C4#1683988012yes—
377C4C2×#1451973511yescharacteristic
378C4#1684024211yescharacteristic
379C4#1684041611yescharacteristic
380C4#1684043411yescharacteristic
381C2—21no—
382C2—21no—
383C2—4882no—
384C2#2144876611yescharacteristic
385C2C2×#1683988011yescharacteristic
386C2#2145473711yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC8×C4⋊C4×C61C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519799C22
cosocleC22×C4×C61C24
hypercenterC8×C4⋊C4D61
hypocenterC61#22802
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21454484C2semidirect
4. #21446468C2semidirect
5. #21445991C2semidirect
6. #21445840C2semidirect
7. #21445698C2semidirect
8. C8×C4⋊C4×C61C2semidirect
11. C4×Dic244C4semidirect
12. C4×Dic244C4semidirect
13. C4×#11758190C4semidirect
14. C4×#11758190C4semidirect
15. C4×D488C4semidirect
16. C4×D488C4semidirect
195. C8×C4⋊C4D61semidirect
227. C61#22802semidirect

Maximal subgroups

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

subgroupindex
C4×#145197992
#214544842
#214467072
#214464682
#214459912
#214458402
#214456982
C8×C4⋊C4×C612
#2280261