Finite Group Databasebeta

← #29835354

Subgroups — #29835354

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#21456355C211yesmaximal, characteristic, complemented
2#21456308C211yesmaximal, characteristic
3#21448063C212yesmaximal, complemented
4#21448057C212yesmaximal, complemented
5#21448052C212yesmaximal, complemented
6#21448045C212yesmaximal, complemented
7#21447539C211yesmaximal, characteristic, complemented
8#21447529C211yesmaximal, characteristic
9#21447527C211yesmaximal, characteristic
10#21447415C211yesmaximal, characteristic, complemented
11C2×C61×#358C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13C2×#14519799C2211yescharacteristic
14#16840393—21no—
15#16840336—21no—
16C2×#14519717C2211yescharacteristic
17#16840170—22no—
18#16840166C2212yes—
19#16840163C2212yes—
20#16840155—22no—
21#16840147C2212yes—
22#16840140C2212yes—
23#16839802—21no—
24#16839800C2211yescharacteristic
25#16839717C2211yescharacteristic
26#16839567C2211yescharacteristic
27#16839567C2211yescharacteristic
28#16839566C2212yes—
29#16839566C2212yes—
30#16839565C2212yes—
31#16839564C2211yescharacteristic
32#16839564C2211yescharacteristic
33#16839562C2212yes—
34#16839537C2212yes—
35C16×Dic61C2212yes—
36C22×Q16×C61C2211yescharacteristic
37C22×C16×C61C2211yescharacteristic
38C2×#107×C61C2211yescharacteristic
39C61×#358C2212yes—
40C61×#358C2212yes—
41C2×#11758222—21no—
42#14519808—44no—
43#14519799—22no—
44#14519799—22no—
45C2×Dic244C2311yescharacteristic
46C2×#11758202—22no—
47C2×#11758190—21no—
48C2×#11758201—22no—
49C2×D488C2311yescharacteristic
50C2×#11758194D411yescharacteristic
51#14519776—22no—
52#14519776—41no—
53#14519775—22no—
54#14519768—22no—
55#14519768—41no—
56#14519766—22no—
57#14519760—42no—
58#14519756—42no—
59#14519753—41no—
60#14519753—41no—
61#14519736—21no—
62#14519736D411yescharacteristic
63#14519717—22no—
64C2×#11758185C2312yes—
65#14519711C2311yescharacteristic
66#14519711C2311yescharacteristic
67#14519710—22no—
68#14519709—21no—
69#14519708—22no—
70#14519687D412yes—
71C8×Dic61C2312yes—
72C22×Q8×C61—21no—
73C2×Q16×C61—21no—
74C2×Q16×C61—22no—
75C2×Q16×C61—22no—
76C2×Q16×C61C2312yes—
77C22×C8×C61C2311yescharacteristic
78C2×C4⋊C4×C61—21no—
79C2×C16×C61—22no—
80C2×C16×C61C2311yescharacteristic
81C2×C16×C61C2311yescharacteristic
82#107×C61—21no—
83#107×C61C2312yes—
84C2×Q8×C61—22no—
85C2×Q8×C61—41no—
86C2×Q8×C61—42no—
87C2×Q8×C61—42no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61C2×D411yescharacteristic
91C2×C22⋊D61—41no—
92#11758222—42no—
93#11758222—42no—
94C2×D244—21no—
95C2×C4×D61—41no—
96C2×Dic122—21no—
97C2×Dic122—42no—
98Q16×C61—22no—
99Q16×C61—22no—
100Q16×C61—42no—
101C2×C8×C61C2×D411yescharacteristic
102C2×C8×C61#4112yes—
103C2×C8×C61#4112yes—
104C2×C8×C61C2411yescharacteristic
105C4⋊C4×C61—22no—
106C4⋊C4×C61—41no—
107C22⋊Dic61—41no—
108C22⋊Dic61D812yes—
109#11758202—44no—
110#11758201—44no—
111#11758198—42no—
112#11758198—42no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—42no—
116C4⋊Dic61D811yescharacteristic
117C4⋊Dic61D811yescharacteristic
118#11758196—42no—
119#11758196—42no—
120#11758196—42no—
121#11758196—42no—
122C4×Dic61—41no—
123C4×Dic61C2×D412yes—
124#11758194SD1614yes—
125C2×#92418C2×D412yes—
126Dic244—21no—
127Dic244—21no—
128D488—21no—
129D488—21no—
130#11758190—42no—
131C16×C61—21no—
132C16×C61—21no—
133#11758185—22no—
134C23×C61C2×D811yescharacteristic
135C22×D61—41no—
136Q8×C61—42no—
137Q8×C61—42no—
138Q8×C61—82no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—82no—
142C2×C4×C61—161no—
143C2×C4×C61#12412yes—
144C2×C4×C61C2×D811yescharacteristic
145C2×C4×C61C2×SD1612yes—
146C2×C4×C61C22×D411yescharacteristic
147C22⋊D61—82no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61C2×D812yes—
152C2×Dic61C2×D812yes—
153D244—41no—
154D244—41no—
155C4×D61—82no—
156Dic122—41no—
157Dic122—41no—
158Dic122—162no—
159C8×C61#12412yes—
160C8×C61C2×#4111yescharacteristic
161C8×C61C2×#4111yescharacteristic
162#92418C2×SD1614yes—
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—82no—
172C4×C61—82no—
173C4×C61#48612yes—
174C4×C61C2×#12411yescharacteristic
175C4×C61C22×SD1611yescharacteristic
176Dic61—22no—
177Dic61—22no—
178Dic61—81no—
179Dic61—81no—
180Dic61—81no—
181#3164—611no—
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×#358D6111yescharacteristic, complemented
191#2108—611no—
192C2×C61—22no—
193C2×C61C2×#48611yescharacteristic
194C2×C61#341611yescharacteristic
195C2×C61#341711yescharacteristic
196D61—161no—
197C2×#137—1221no—
198C2×#135—611no—
199C22×Q16C2×D6111yescharacteristic
200#544—1221no—
201#536—1221no—
202C2×M32—611no—
203C22×C16C2×D6111yescharacteristic
204#501—1222no—
205#500—612no—
206#499—612no—
207#497—1222no—
208#495—612no—
209#494—612no—
210#428—1221no—
211#427—611no—
212C2×#107C2×D6111yescharacteristic
213#405—611no—
214#358—612no—
215#358—612no—
216#358—611no—
217#358—611no—
218#358—612no—
219#358C2×D6112yes—
220#358C2×D6112yes—
221#357—611no—
222#357—611no—
223#357—612no—
224#346—612no—
225C4×C16—612no—
226C61#3370311yescharacteristic, complemented, Sylow
227C2×#41—1221no—
228C22×Q8—21no—
229#137—2444no—
230#135—1222no—
231#135—1222no—
232C2×Q16—22no—
233C2×Q16—21no—
234C2×Q16—22no—
235C2×Q16—611no—
236C2×Q16—1222no—
237C2×Q16C22×D6112yes—
238C2×SD16—1222no—
239C2×SD16—1221no—
240C2×D8—611no—
241C2×M16—611no—
242C22×C8C22×D6111yescharacteristic
243C4⋊Q8—1222no—
244C4⋊Q8—1222no—
245C4⋊Q8—2441no—
246#125—1222no—
247#125—2441no—
248#124—1222no—
249#123—2442no—
250C22⋊Q8—2442no—
251C22⋊Q8—2441no—
252C22⋊Q8—2441no—
253#117—611no—
254#117—1221no—
255C2×C4⋊C4—21no—
256M32—1222no—
257C2×C16—22no—
258C2×C16—612no—
259C2×C16C22×D6111yescharacteristic
260C2×C16C22×D6111yescharacteristic
261#107—21no—
262#107—611no—
263#107—611no—
264#107—1222no—
265#107C22×D6112yes—
266#106—1222no—
267#106—1221no—
268#97—612no—
269C4×C8—612no—
270#41—2442no—
271#41—2442no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—1221no—
277C2×Q8—2442no—
278C2×D4—1221no—
279C2×D4—2441no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—2441no—
283C22×C4D4×D6111yescharacteristic
284Q16—22no—
285Q16—22no—
286Q16—42no—
287Q16—1221no—
288Q16—1221no—
289Q16—2444no—
290SD16—2444no—
291SD16—2442no—
292D8—1221no—
293D8—1221no—
294M16—614no—
295C2×C8—612no—
296C2×C8#1175822212yes—
297C2×C8D4×D6111yescharacteristic
298C2×C8#1175822412yes—
299C2×C8C23×D6111yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—611no—
303C4⋊C4—611no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4—2442no—
307C4⋊C4—2442no—
308C4⋊C4—2442no—
309C4⋊C4—2442no—
310C4⋊C4—2442no—
311C22⋊C4—612no—
312C22⋊C4—2442no—
313C22⋊C4—2442no—
314C22⋊C4—2441no—
315C42—612no—
316C42—2441no—
317C16—21no—
318C16—21no—
319C16—1222no—
320C23—2441no—
321C23D8×D6111yescharacteristic
322Q8—42no—
323Q8—42no—
324Q8—82no—
325Q8—9762no—
326Q8—2441no—
327Q8—2441no—
328D4—4882no—
329D4—2441no—
330D4—2441no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—161no—
335C2×C4—612no—
336C2×C4—612no—
337C2×C4—4882no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4#1451976312yes—
342C2×C4D8×D6111yescharacteristic
343C2×C4SD16×D6112yes—
344C2×C4C2×D4×D6111yescharacteristic
345C8—614no—
346C8#1451976312yes—
347C8C2×#1175822411yescharacteristic
348C8C2×#1175822211yescharacteristic
349C22—21no—
350C22—21no—
351C22—4881no—
352C22—4881no—
353C22—4881no—
354C22#1684010712yes—
355C22C2×D8×D6111yescharacteristic
356C4—82no—
357C4—82no—
358C4—1222no—
359C4—1222no—
360C4—4881no—
361C4—4881no—
362C4—4881no—
363C4#1684010712yes—
364C4C2×#1451976311yescharacteristic
365C4C2×SD16×D6111yescharacteristic
366C2—22no—
367C2—9761no—
368C2C2×#1684010711yescharacteristic
369C2#2145896111yescharacteristic
370C2#2145896811yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#358C2
socleC22×C61C22×D8
prime radicalC2×#14519799C22
hypercenterC2×#358D61
hypocenterC61#33703
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
1. #21456355C2semidirect
3. #21448063C2semidirect
4. #21448057C2semidirect
5. #21448052C2semidirect
6. #21448045C2semidirect
7. #21447539C2semidirect
10. #21447415C2semidirect
11. C2×C61×#358C2semidirect
190. C2×#358D61semidirect
226. C61#33703semidirect

Maximal subgroups

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

subgroupindex
#214563552
#214563082
#214480632
#214480572
#214480522
#214480452
#214475392
#214475292
#214475272
#214474152
C2×C61×#3582
#3370361