Finite Group Databasebeta

← #29835361

Subgroups — #29835361

Order 15616.

570 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#21456313C211yesmaximal, characteristic
2#21448066C212yesmaximal, complemented
3#21448058C212yesmaximal, complemented
4#21448053C212yesmaximal, complemented
5#21448048C212yesmaximal, complemented
6#21447538C211yesmaximal, characteristic, complemented
7#21447529C211yesmaximal, characteristic
8#21447524C211yesmaximal, characteristic
9C2×#16839565C211yesmaximal, characteristic, complemented
10#21447416C211yesmaximal, characteristic, complemented
11C61×#3164C211yesmaximal, characteristic, complemented
12C2×#14519810—21no—
13C2×#14519799C2211yescharacteristic
14C22×Dic244C2211yescharacteristic
15#16840338—21no—
16C2×#14519717C2211yescharacteristic
17C22×#11758185C2211yescharacteristic
18#16840156—22no—
19#16840146C2212yes—
20#16840142C2212yes—
21#16839565C2212yes—
22#16839565C2212yes—
23#16839565C2211yescharacteristic
24#16839565C2211yescharacteristic
25#16839565C2212yes—
26#16839565C2212yes—
27#16839565C2212yes—
28#16839562C2211yescharacteristic
29#16839562C2211yescharacteristic
30#16839562C2212yes—
31#16839538C2212yes—
32C4×#11758185C2212yes—
33C61×#544—21no—
34C61×#501—22no—
35C61×#500C2212yes—
36C61×#499C2212yes—
37C61×#428—21no—
38C61×#427C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#405C2211yescharacteristic
41C2×#11758222—21no—
42C22×Dic122—21no—
43#14519810—44no—
44#14519799—22no—
45#14519799—22no—
46C2×Dic244—22no—
47C2×Dic244—22no—
48C2×Dic244—21no—
49C2×Dic244—22no—
50C2×Dic244C2312yes—
51C2×Dic244C2311yescharacteristic
52C2×#11758190—21no—
53C2×#11758190—22no—
54C2×D488C2311yescharacteristic
55#14519777—22no—
56#14519765—22no—
57#14519761—42no—
58#14519757—41no—
59#14519757—41no—
60#14519717—22no—
61C2×#11758185—22no—
62C2×#11758185C2311yescharacteristic
63C2×#11758185C2312yes—
64C2×#11758185C2311yescharacteristic
65C2×M16×C61D411yescharacteristic
66C22×C8×C61C2311yescharacteristic
67C4⋊Q8×C61—22no—
68C4⋊Q8×C61—41no—
69#125×C61—22no—
70#125×C61—41no—
71C22⋊Q8×C61—42no—
72#117×C61—21no—
73#117×C61D411yescharacteristic
74C2×C4⋊C4×C61—21no—
75#107×C61—21no—
76#107×C61—22no—
77#107×C61C2312yes—
78#107×C61C2311yescharacteristic
79#107×C61C2311yescharacteristic
80#106×C61—21no—
81#106×C61—22no—
82#97×C61D412yes—
83C4×C8×C61C2312yes—
84C2×Q8×C61—42no—
85C22×C4×C61—41no—
86C22×C4×C61C2×D411yescharacteristic
87C2×C22⋊D61—41no—
88C22×Dic61—41no—
89#11758222—42no—
90#11758222—42no—
91C2×D244—21no—
92C2×C4×D61—41no—
93C2×Dic122—22no—
94C2×Dic122—21no—
95C2×Dic122—42no—
96C2×Dic122—42no—
97C2×Dic122—41no—
98M16×C61SD1614yes—
99C2×C8×C61C2×D411yescharacteristic
100C2×C8×C61C2×D412yes—
101C2×C8×C61#4112yes—
102C2×C8×C61#4112yes—
103C2×C8×C61C2411yescharacteristic
104C4⋊C4×C61—21no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—42no—
109C4⋊C4×C61—42no—
110C4⋊C4×C61—42no—
111C4⋊C4×C61D811yescharacteristic
112C4⋊C4×C61D811yescharacteristic
113C22⋊C4×C61—41no—
114C22⋊C4×C61D812yes—
115C42×C61—41no—
116C42×C61C2×D412yes—
117#11758198—42no—
118#11758198—42no—
119C4⋊Dic61—42no—
120C4⋊Dic61—42no—
121Dic244—22no—
122Dic244—22no—
123Dic244—21no—
124Dic244—21no—
125Dic244—42no—
126Dic244—44no—
127D488—21no—
128D488—21no—
129#11758190—42no—
130#11758190—44no—
131#11758185—21no—
132#11758185—21no—
133#11758185—22no—
134C23×C61C2×D811yescharacteristic
135C22×D61—41no—
136Q8×C61—162no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—161no—
141C2×C4×C61#12412yes—
142C2×C4×C61C2×D812yes—
143C2×C4×C61C2×D812yes—
144C2×C4×C61C2×D811yescharacteristic
145C2×C4×C61C2×SD1612yes—
146C2×C4×C61C22×D411yescharacteristic
147C22⋊D61—82no—
148C2×Dic61—41no—
149C2×Dic61—42no—
150C2×Dic61—82no—
151D244—41no—
152D244—41no—
153C4×D61—82no—
154Dic122—41no—
155Dic122—42no—
156Dic122—42no—
157Dic122—41no—
158Dic122—82no—
159C8×C61#12412yes—
160C8×C61C2×SD1614yes—
161C8×C61C2×#4111yescharacteristic
162C8×C61C2×#4111yescharacteristic
163#33703—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61#48612yes—
167C22×C61C22×D811yescharacteristic
168C2×D61—81no—
169C2×D61—81no—
170C2×D61—81no—
171C4×C61—22no—
172C4×C61—22no—
173C4×C61—82no—
174C4×C61—81no—
175C4×C61—81no—
176C4×C61#48612yes—
177C4×C61C2×#12411yescharacteristic
178C4×C61C22×SD1611yescharacteristic
179Dic61—82no—
180Dic61—81no—
181#3164D6111yescharacteristic, complemented
182#3156—611no—
183#2245—612no—
184#2243—612no—
185#2242—612no—
186#2241—612no—
187#2141—611no—
188#2139—611no—
189#2138—611no—
190C2×#358—611no—
191#2108—611no—
192C2×C61—22no—
193C2×C61C2×#48611yescharacteristic
194C2×C61#341611yescharacteristic
195C2×C61#341711yescharacteristic
196D61—161no—
197C2×#137—1221no—
198C2×#135—611no—
199C22×Q16—611no—
200#544—21no—
201#536—1221no—
202C2×M32—611no—
203C22×C16—611no—
204#501—22no—
205#500C2×D6112yes—
206#499C2×D6112yes—
207#497—1222no—
208#495—612no—
209#494—612no—
210#428—21no—
211#427C2×D6111yescharacteristic
212C2×#107C2×D6111yescharacteristic
213#405C2×D6111yescharacteristic
214#358—612no—
215#358—612no—
216#358—611no—
217#358—611no—
218#358—612no—
219#358—612no—
220#358—612no—
221#357—611no—
222#357—611no—
223#357—612no—
224#346—612no—
225C4×C16—612no—
226C61#3370311yescharacteristic, complemented, Sylow
227C2×#41—1221no—
228C22×Q8—1221no—
229#137—2444no—
230#135—1222no—
231#135—1222no—
232C2×Q16—612no—
233C2×Q16—611no—
234C2×Q16—1222no—
235C2×Q16—1222no—
236C2×Q16—1221no—
237C2×Q16—1222no—
238C2×SD16—1221no—
239C2×SD16—1222no—
240C2×D8—611no—
241C2×M16C22⋊D6111yescharacteristic
242C22×C8C22×D6111yescharacteristic
243C4⋊Q8—22no—
244C4⋊Q8—41no—
245C4⋊Q8—1222no—
246#125—22no—
247#125—41no—
248#124—1222no—
249#123—2442no—
250C22⋊Q8—42no—
251C22⋊Q8—2441no—
252C22⋊Q8—2441no—
253#117—21no—
254#117C22⋊D6111yescharacteristic
255C2×C4⋊C4—21no—
256M32—1222no—
257C2×C16—611no—
258C2×C16—612no—
259C2×C16—611no—
260C2×C16—1222no—
261#107—21no—
262#107—22no—
263#107C22×D6112yes—
264#107C22×D6111yescharacteristic
265#107C22×D6111yescharacteristic
266#106—21no—
267#106—22no—
268#97C22⋊D6112yes—
269C4×C8C22×D6112yes—
270#41—2442no—
271#41—2442no—
272C2×Q8—42no—
273C2×Q8—1222no—
274C2×Q8—1221no—
275C2×Q8—2442no—
276C2×Q8—2442no—
277C2×Q8—2441no—
278C2×D4—1221no—
279C2×D4—2441no—
280C22×C4—41no—
281C22×C4—2441no—
282C22×C4—2441no—
283C22×C4C2×C22⋊D6111yescharacteristic
284Q16—1222no—
285Q16—1222no—
286Q16—1221no—
287Q16—1221no—
288Q16—2442no—
289Q16—2444no—
290SD16—2442no—
291SD16—2444no—
292D8—1221no—
293D8—1221no—
294M16#1175820114yes—
295C2×C8#1175822612yes—
296C2×C8#1175822612yes—
297C2×C8C2×C22⋊D6111yescharacteristic
298C2×C8C2×C22⋊D6112yes—
299C2×C8C23×D6111yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—2442no—
308C4⋊C4—2442no—
309C4⋊C4#1175819911yescharacteristic
310C4⋊C4#1175819911yescharacteristic
311C22⋊C4—41no—
312C22⋊C4—2442no—
313C22⋊C4—2442no—
314C22⋊C4#1175819912yes—
315C42—41no—
316C42C2×C22⋊D6112yes—
317C16—1221no—
318C16—1221no—
319C16—1222no—
320C23—2441no—
321C23C2×#1175819911yescharacteristic
322Q8—162no—
323Q8—4882no—
324Q8—2442no—
325Q8—2441no—
326Q8—2441no—
327Q8—2442no—
328D4—4882no—
329D4—2441no—
330D4—2441no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—161no—
335C2×C4—4882no—
336C2×C4—4882no—
337C2×C4—2442no—
338C2×C4—2441no—
339C2×C4#1451976612yes—
340C2×C4C2×#1175819912yes—
341C2×C4C2×#1175819912yes—
342C2×C4C2×#1175819911yescharacteristic
343C2×C4C2×#1175820112yes—
344C2×C4C22×C22⋊D6111yescharacteristic
345C8#1451976612yes—
346C8C2×#1175820114yes—
347C8C2×#1175822611yescharacteristic
348C8C2×#1175822611yescharacteristic
349C22—21no—
350C22—21no—
351C22—4881no—
352C22—4881no—
353C22—4881no—
354C22#1684011012yes—
355C22C22×#1175819911yescharacteristic
356C4—22no—
357C4—22no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—4881no—
362C4—4882no—
363C4#1684011012yes—
364C4C2×#1451976611yescharacteristic
365C4C22×#1175820111yescharacteristic
366C2—22no—
367C2—9761no—
368C2C2×#1684011011yescharacteristic
369C2#2145896611yescharacteristic
370C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3164C2
socleC22×C61C22×D8
prime radicalC2×#14519799C22
hypercenter#3164D61
hypocenterC61#33703
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #21448066C2semidirect
3. #21448058C2semidirect
4. #21448053C2semidirect
5. #21448048C2semidirect
6. #21447538C2semidirect
9. C2×#16839565C2semidirect
10. #21447416C2semidirect
11. C61×#3164C2semidirect
181. #3164D61semidirect
226. C61#33703semidirect

Maximal subgroups

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

subgroupindex
#214563132
#214480662
#214480582
#214480532
#214480482
#214475382
#214475292
#214475242
C2×#168395652
#214474162
C61×#31642
#3370361