Finite Group Databasebeta

← #29835358

Subgroups — #29835358

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#21456358C211yesmaximal, characteristic, complemented
2#21456313C211yesmaximal, characteristic
3#21448065C212yesmaximal, complemented
4#21448056C212yesmaximal, complemented
5#21448051C212yesmaximal, complemented
6#21448047C212yesmaximal, complemented
7#21447540C211yesmaximal, characteristic, complemented
8#21447532C211yesmaximal, characteristic
9#21447527C211yesmaximal, characteristic
10C2×#16839567C211yesmaximal, characteristic, complemented
11C61×#2108C211yesmaximal, characteristic, complemented
12C2×#14519810—21no—
13C2×#14519799C2211yescharacteristic
14C22×Dic244C2211yescharacteristic
15#16840390—21no—
16#16840338—21no—
17#16840171—22no—
18#16840167C2212yes—
19#16840164C2212yes—
20#16840156—22no—
21#16840146C2212yes—
22#16840142C2212yes—
23#16839804—21no—
24#16839800C2211yescharacteristic
25C2×#14519711C2211yescharacteristic
26#16839567C2212yes—
27#16839567C2212yes—
28#16839567C2211yescharacteristic
29#16839567C2211yescharacteristic
30#16839567C2212yes—
31#16839567C2212yes—
32#16839567C2212yes—
33#16839564C2211yescharacteristic
34#16839564C2211yescharacteristic
35#16839564C2212yes—
36C2×M32×C61C2211yescharacteristic
37C22×C16×C61C2211yescharacteristic
38C61×#405C2211yescharacteristic
39C61×#346C2212yes—
40C4×C16×C61C2212yes—
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#14519777—22no—
57#14519776—41no—
58#14519769—22no—
59#14519768—41no—
60#14519765—22no—
61#14519761—42no—
62#14519757—41no—
63#14519757—41no—
64#14519754—42no—
65#14519736—21no—
66C2×C4⋊Dic61—21no—
67#14519711—21no—
68#14519711—22no—
69#14519711C2312yes—
70#14519711C2311yescharacteristic
71#14519711C2311yescharacteristic
72#14519709—21no—
73#14519709—22no—
74C2×M16×C61D411yescharacteristic
75C22×C8×C61C2311yescharacteristic
76#117×C61D411yescharacteristic
77M32×C61—22no—
78C2×C16×C61—22no—
79C2×C16×C61C2311yescharacteristic
80C2×C16×C61C2312yes—
81C2×C16×C61C2311yescharacteristic
82#97×C61D412yes—
83C4×C8×C61C2312yes—
84C22×C4×C61C2×D411yescharacteristic
85C2×C22⋊D61—41no—
86C22×Dic61—41no—
87C22×Dic61—41no—
88#11758222—42no—
89#11758222—42no—
90C2×D244—21no—
91C2×C4×D61—41no—
92C2×Dic122—22no—
93C2×Dic122—21no—
94C2×Dic122—42no—
95C2×Dic122—42no—
96C2×Dic122—41no—
97C2×Dic122—42no—
98M16×C61SD1614yes—
99C2×C8×C61C2×D411yescharacteristic
100C2×C8×C61C2×D412yes—
101C2×C8×C61#4112yes—
102C2×C8×C61#4112yes—
103C2×C8×C61C2411yescharacteristic
104C4⋊C4×C61D811yescharacteristic
105C4⋊C4×C61D811yescharacteristic
106C22⋊C4×C61D812yes—
107C42×C61C2×D412yes—
108C22⋊Dic61—41no—
109#11758198—42no—
110#11758198—42no—
111C4⋊Dic61—21no—
112C4⋊Dic61—22no—
113C4⋊Dic61—21no—
114C4⋊Dic61—41no—
115C4⋊Dic61—42no—
116C4⋊Dic61—42no—
117C4⋊Dic61—42no—
118#11758196—42no—
119#11758196—42no—
120C4×Dic61—41no—
121Dic244—22no—
122Dic244—22no—
123Dic244—21no—
124Dic244—21no—
125Dic244—42no—
126Dic244—44no—
127D488—21no—
128D488—21no—
129#11758190—42no—
130#11758190—44no—
131C16×C61—21no—
132C16×C61—21no—
133C16×C61—22no—
134C23×C61C2×D811yescharacteristic
135C22×D61—41no—
136C2×C4×C61#12412yes—
137C2×C4×C61C2×D812yes—
138C2×C4×C61C2×D812yes—
139C2×C4×C61C2×D811yescharacteristic
140C2×C4×C61C2×SD1612yes—
141C2×C4×C61C22×D411yescharacteristic
142C22⋊D61—82no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—42no—
146C2×Dic61—41no—
147C2×Dic61—42no—
148C2×Dic61—82no—
149C2×Dic61—161no—
150D244—41no—
151D244—41no—
152C4×D61—82no—
153Dic122—41no—
154Dic122—42no—
155Dic122—42no—
156Dic122—41no—
157Dic122—82no—
158Dic122—162no—
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#48612yes—
174C4×C61C2×#12411yescharacteristic
175C4×C61C22×SD1611yescharacteristic
176Dic61—82no—
177Dic61—81no—
178Dic61—81no—
179Dic61—82no—
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×#358—611no—
191#2108D6111yescharacteristic, complemented
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—1221no—
201#536—1221no—
202C2×M32C2×D6111yescharacteristic
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×#107—611no—
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#346C2×D6112yes—
225C4×C16C2×D6112yes—
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—1221no—
235C2×Q16—1222no—
236C2×Q16—1222no—
237C2×Q16—1222no—
238C2×SD16—1221no—
239C2×SD16—1222no—
240C2×D8—611no—
241C2×M16D24411yescharacteristic
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—2441no—
251C22⋊Q8—2441no—
252C22⋊Q8—2442no—
253#117—1221no—
254#117D24411yescharacteristic
255C2×C4⋊C4—1221no—
256M32—22no—
257C2×C16—22no—
258C2×C16C22×D6111yescharacteristic
259C2×C16C22×D6112yes—
260C2×C16C22×D6111yescharacteristic
261#107—612no—
262#107—611no—
263#107—611no—
264#107—1221no—
265#107—1222no—
266#106—1221no—
267#106—1222no—
268#97D24412yes—
269C4×C8C22×D6112yes—
270#41—2442no—
271#41—2442no—
272C2×Q8—1222no—
273C2×Q8—1221no—
274C2×Q8—2442no—
275C2×Q8—2442no—
276C2×Q8—2442no—
277C2×Q8—2441no—
278C2×D4—1221no—
279C2×D4—2441no—
280C22×C4—2441no—
281C22×C4—2441no—
282C22×C4—2441no—
283C22×C4C2×D24411yescharacteristic
284Q16—1222no—
285Q16—1222no—
286Q16—1221no—
287Q16—1221no—
288Q16—2442no—
289Q16—2444no—
290SD16—2442no—
291SD16—2444no—
292D8—1221no—
293D8—1221no—
294M16#1175819014yes—
295C2×C8C2×D24411yescharacteristic
296C2×C8C2×D24412yes—
297C2×C8#1175822212yes—
298C2×C8#1175822212yes—
299C2×C8C23×D6111yescharacteristic
300C4⋊C4—1221no—
301C4⋊C4—1222no—
302C4⋊C4—1221no—
303C4⋊C4—2441no—
304C4⋊C4—2442no—
305C4⋊C4—2442no—
306C4⋊C4—2442no—
307C4⋊C4—2442no—
308C4⋊C4—2442no—
309C4⋊C4D48811yescharacteristic
310C4⋊C4D48811yescharacteristic
311C22⋊C4—2441no—
312C22⋊C4—2442no—
313C22⋊C4—2442no—
314C22⋊C4D48812yes—
315C42—2441no—
316C42C2×D24412yes—
317C16—21no—
318C16—21no—
319C16—22no—
320C23—2441no—
321C23C2×D48811yescharacteristic
322Q8—9762no—
323Q8—4882no—
324Q8—2441no—
325Q8—2442no—
326Q8—2442no—
327Q8—2441no—
328D4—4882no—
329D4—2441no—
330D4—2441no—
331C2×C4—9761no—
332C2×C4—4882no—
333C2×C4—4882no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2442no—
337C2×C4—2442no—
338C2×C4—2441no—
339C2×C4#1451976512yes—
340C2×C4C2×D48812yes—
341C2×C4C2×D48812yes—
342C2×C4C2×D48811yescharacteristic
343C2×C4C2×#1175819012yes—
344C2×C4C22×D24411yescharacteristic
345C8#1451976512yes—
346C8C2×#1175819014yes—
347C8C2×#1175822211yescharacteristic
348C8C2×#1175822211yescharacteristic
349C22—21no—
350C22—21no—
351C22—4881no—
352C22—4881no—
353C22—4881no—
354C22#1684010912yes—
355C22C22×D48811yescharacteristic
356C4—22no—
357C4—22no—
358C4—4882no—
359C4—4881no—
360C4—4881no—
361C4—4881no—
362C4—4882no—
363C4#1684010912yes—
364C4C2×#1451976511yescharacteristic
365C4C22×#1175819011yescharacteristic
366C2—22no—
367C2—9761no—
368C2C2×#1684010911yescharacteristic
369C2#2145896711yescharacteristic
370C2#2145897211yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2108C2
socleC22×C61C22×D8
prime radicalC2×#14519799C22
hypercenter#2108D61
hypocenterC61#33703
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108

Complements

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

NH ≅ G/Ntype
1. #21456358C2semidirect
3. #21448065C2semidirect
4. #21448056C2semidirect
5. #21448051C2semidirect
6. #21448047C2semidirect
7. #21447540C2semidirect
10. C2×#16839567C2semidirect
11. C61×#2108C2semidirect
191. #2108D61semidirect
226. C61#33703semidirect

Maximal subgroups

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

subgroupindex
#214563582
#214563132
#214480652
#214480562
#214480512
#214480472
#214475402
#214475322
#214475272
C2×#168395672
C61×#21082
#3370361