Finite Group Databasebeta

← #29593710

Subgroups — #29593710

Order 15616.

312 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448044C211yesmaximal, characteristic, complemented
2#21447854C211yesmaximal, characteristic
3#21445179C211yesmaximal, characteristic, complemented
4#21443877C211yesmaximal, characteristic, complemented
5Q8⋊#11758185C211yesmaximal, characteristic, complemented
6#21443780C211yesmaximal, characteristic, complemented
7C61×#3380C211yesmaximal, characteristic, complemented
8C2×#14519718—21no—
9#16840139C2211yescharacteristic
10#16839962—21no—
11C4⋊#11758185C2211yescharacteristic
12#16839562—22no—
13#16839562—21no—
14C4×#11758185C2211yescharacteristic
15#16839515C2211yescharacteristic
16#16839505—21no—
17C61×#554—21no—
18C61×#500C2211yescharacteristic
19C61×#499—21no—
20C61×#478—22no—
21C61×#476—21no—
22C8×Q8×C61C2211yescharacteristic
23C4×D8×C61—21no—
24C4×D8×C61C2211yescharacteristic
25C2×#11758201—41no—
26C2×D488—21no—
27C2×D488—21no—
28#14519774—21no—
29#14519718—42no—
30C2×#11758185—21no—
31C2×#11758185—21no—
32C4⋊#92418—21no—
33#14519697—41no—
34C2×D8×C61D411yescharacteristic
35C4⋊Q8×C61—21no—
36#125×C61—22no—
37C22⋊Q8×C61—42no—
38C22⋊Q8×C61—41no—
39C4×Q8×C61D411yescharacteristic
40C4×D4×C61—22no—
41C4×D4×C61—21no—
42C2×C4⋊C4×C61—41no—
43#107×C61—21no—
44#107×C61D411yescharacteristic
45#107×C61D412yes—
46#106×C61—22no—
47C4⋊C8×C61—21no—
48C4⋊C8×C61D411yescharacteristic
49#102×C61—22no—
50#102×C61—21no—
51C4×C8×C61—21no—
52C4×C8×C61C2311yescharacteristic
53C2×Q8×C61—41no—
54C2×Q8×C61D811yescharacteristic
55C2×D4×C61—21no—
56C22×C4×C61—42no—
57C22×C4×C61—41no—
58C2×D244—41no—
59C2×D244—41no—
60C2×D244—41no—
61D8×C61—21no—
62D8×C61SD1612yescomplemented
63C2×C8×C61—21no—
64C2×C8×C61C2×D411yescharacteristic
65C2×C8×C61C2×D411yescharacteristic
66C4⋊C4×C61—21no—
67C4⋊C4×C61—21no—
68C4⋊C4×C61—22no—
69C4⋊C4×C61—42no—
70C4⋊C4×C61—42no—
71C4⋊C4×C61—41no—
72C4⋊C4×C61D811yescharacteristic
73C22⋊C4×C61—42no—
74C22⋊C4×C61—41no—
75C42×C61—21no—
76C42×C61C2×D411yescharacteristic
77#11758201—82no—
78C2×#92418—41no—
79D488—42no—
80D488—41no—
81D488—41no—
82#11758185—41no—
83#11758185—41no—
84C23×C61—41no—
85C22×D61—81no—
86Q8×C61—21no—
87Q8×C61—21no—
88Q8×C61—161no—
89D4×C61—22no—
90D4×C61—41no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—41no—
94C2×C4×C61—42no—
95C2×C4×C61—81no—
96C2×C4×C61—81no—
97C2×C4×C61—162no—
98C2×C4×C61#12011yescharacteristic
99C2×C4×C61C2×D811yescharacteristic
100C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
101D244—81no—
102D244—81no—
103D244—81no—
104D244—81no—
105D244—161no—
106C8×C61—21no—
107C8×C61—41no—
108C8×C61C2×SD1611yescharacteristic
109C8×C61#13711yescharacteristic
110#92418—81no—
111#12355—611nomaximal, Sylow
112C22×C61—42no—
113C22×C61—81no—
114C22×C61—81no—
115C22×C61#44711yescharacteristic
116C2×D61—161no—
117C2×D61—161no—
118C2×D61—161no—
119C4×C61—21no—
120C4×C61—41no—
121C4×C61—41no—
122C4×C61—81no—
123C4×C61—82no—
124C4×C61#44811yescharacteristic
125C4×C61#45411yescharacteristic
126C4×C61#50811yescharacteristic
127C4×C61#50911yescharacteristic
128#3380D6111yescharacteristic, complemented
129#2240—611no—
130#2206—611no—
131#1668—611no—
132#1365—611no—
133Q8⋊C16—611no—
134#1331—611no—
135C2×C61—82no—
136C2×C61#162711yescharacteristic
137C2×C61#218611yescharacteristic
138C2×C61#219211yescharacteristic
139D61—321no—
140#554—21no—
141C2×D16—1221no—
142#500C2×D6111yescharacteristic
143#499—21no—
144#493—611no—
145#478—22no—
146#476—21no—
147#460—1221no—
148C8×Q8C2×D6111yescharacteristic
149C4×D8—21no—
150C4×D8C2×D6111yescharacteristic
151C4⋊C16—611no—
152#357—1221no—
153#357—1222no—
154C4×C16—611no—
155#335—611no—
156#331—1221no—
157C61#1235511yescharacteristic, complemented, Sylow
158C2×SD16—2441no—
159C2×D8—1221no—
160C2×D8—1221no—
161C2×D8C22⋊D6111yescharacteristic
162C4⋊Q8—21no—
163#127—1221no—
164#125—22no—
165C22⋊Q8—42no—
166C22⋊Q8—41no—
167C4×Q8C22⋊D6111yescharacteristic
168C4×D4—22no—
169C4×D4—21no—
170C2×C4⋊C4—41no—
171D16—2442no—
172C2×C16—1221no—
173C2×C16—1221no—
174#107—21no—
175#107C22⋊D6111yescharacteristic
176#107C22⋊D6112yes—
177#106—22no—
178C4⋊C8—21no—
179C4⋊C8—1221no—
180C4⋊C8C22⋊D6111yescharacteristic
181#102—22no—
182#102—21no—
183#102—2441no—
184C4×C8—21no—
185C4×C8C22×D6111yescharacteristic
186C2×Q8—41no—
187C2×Q8#1175819911yescharacteristic
188C2×D4—21no—
189C2×D4—2441no—
190C2×D4—2441no—
191C2×D4—2441no—
192C22×C4—42no—
193C22×C4—41no—
194SD16—4882no—
195D8—21no—
196D8—2442no—
197D8—2441no—
198D8—2441no—
199D8#1175820112yescomplemented
200C2×C8—21no—
201C2×C8—2441no—
202C2×C8C2×C22⋊D6111yescharacteristic
203C2×C8C2×C22⋊D6111yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4#1175819911yescharacteristic
211C22⋊C4—42no—
212C22⋊C4—41no—
213C42—21no—
214C42C2×C22⋊D6111yescharacteristic
215C16—2441no—
216C16—2441no—
217C23—41no—
218C23—4881no—
219Q8—21no—
220Q8—21no—
221Q8—161no—
222D4—22no—
223D4—41no—
224D4—9761no—
225D4—4881no—
226D4—4881no—
227D4—4881no—
228D4—4881no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—162no—
236C2×C4#1451974611yescharacteristic
237C2×C4C2×#1175819911yescharacteristic
238C2×C4#1451980311yescharacteristic
239C8—21no—
240C8—41no—
241C8—4881no—
242C8C2×#1175820111yescharacteristic
243C8#1451980811yescharacteristic
244C22—42no—
245C22—81no—
246C22—81no—
247C22—9761no—
248C22—9761no—
249C22—9761no—
250C22#1683989611yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—81no—
255C4—82no—
256C4#1683990111yescharacteristic
257C4#1683993511yescharacteristic
258C4#1684019211yescharacteristic
259C4#1684019711yescharacteristic
260C2—82no—
261C2—19521no—
262C2#2144492211yescharacteristic
263C2#2144774211yescharacteristic
264C2#2144777611yescharacteristic
subgroupG/H
centerC22#16839896
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#3380C2
socleC22×C61#447
prime radical#21447854C2
hypercenter#3380D61
hypocenterC61#12355
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #21448044C2semidirect
3. #21445179C2semidirect
4. #21443877C2semidirect
5. Q8⋊#11758185C2semidirect
6. #21443780C2semidirect
7. C61×#3380C2semidirect
62. D8×C61SD16semidirect
128. #3380D61semidirect
157. C61#12355semidirect
199. D8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214480442
#214478542
#214451792
#214438772
Q8⋊#117581852
#214437802
C61×#33802
#1235561