Finite Group Databasebeta

← #49170592

Subgroups — #49170592

Order 37504.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D1172C211yesmaximal, characteristic, complemented
2#34036477C211yesmaximal, characteristic, complemented
3#34036442C212yesmaximal, complemented
4#34036406C211yesmaximal, characteristic, complemented
5#34036363C212yesmaximal, complemented
6#34036324C211yesmaximal, characteristic, complemented
7#34036262C212yesmaximal, complemented
8C4×#17989787C211yesmaximal, characteristic, complemented
9D4×#14556873C211yesmaximal, characteristic, complemented
10C2×#22627544C212yesmaximal, complemented
11#547×C293C211yesmaximal, characteristic, complemented
12C22×D1172C2212yes—
13C2×D4×D293—21no—
14C2×#17989787C2211yescharacteristic
15C2×#17989787C2212yescomplemented
16C22×#14556873C2212yescomplemented
17#22627621C2211yescharacteristic
18#22627599—21no—
19#22627598C2212yes—
20#22627594—22no—
21C4×D1172C2211yescharacteristic
22#22627557C2212yescomplemented
23#22627557C2211yescharacteristic, complemented
24C4⋊#14556873C2211yescharacteristic, complemented
25#22627545C2211yescharacteristic, complemented
26#22627544C2212yes—
27#22627544C2211yescharacteristic
28#22627544C2212yescomplemented
29#22627544C2212yescomplemented
30#22627544C2212yescomplemented
31C22⋊#14556873C2212yescomplemented
32C4×#14556873C2211yescharacteristic, complemented
33C2×#41×C293—21no—
34C4⋊Q8×C293C2211yescharacteristic, complemented
35#123×C293—22no—
36C22⋊Q8×C293—21no—
37#121×C293C2212yescomplemented
38C4×D4×C293C2211yescharacteristic
39C4×D4×C293C2211yescharacteristic, complemented
40C2×C4⋊C4×C293C2212yescomplemented
41C23×D293—22no—
42C2×D1172—24no—
43C2×D1172—22no—
44C2×D1172—21no—
45C2×D1172—21no—
46C2×D1172—42no—
47C2×D1172C2312yes—
48C2×D1172C2311yescharacteristic
49C2×C22⋊D293—22no—
50D4×D293—44no—
51C2×C4×D293—21no—
52#17989787—22no—
53#17989787—42no—
54C2×#14556873—42no—
55C2×#14556873C2311yescharacteristic
56C2×#14556873C2312yes—
57C2×#14556873C2311yescharacteristic
58C4⋊Dic293C2311yescharacteristic
59#17989779—21no—
60#17989779—22no—
61#41×C293—44no—
62C2×Q8×C293—21no—
63C2×D4×C293—22no—
64C2×D4×C293D411yescharacteristic
65C2×D4×C293C2311yescharacteristic
66C22×C4×C293—22no—
67C22×C4×C293—21no—
68C22×C4×C293C2312yes—
69C4⋊C4×C293—22no—
70C4⋊C4×C293—21no—
71C4⋊C4×C293D411yescharacteristic
72C4⋊C4×C293C2311yescharacteristic
73C4⋊C4×C293C2312yes—
74C22⋊C4×C293—21no—
75C22⋊C4×C293—22no—
76C22⋊C4×C293D412yes—
77C42×C293C2311yescharacteristic
78C22×D293—21no—
79C22×D293—22no—
80C22×D293—44no—
81C22×D293—44no—
82C22×D293—82no—
83D1172—22no—
84D1172—21no—
85D1172—22no—
86D1172—44no—
87D1172—42no—
88D1172—81no—
89D1172#4112yescomplemented
90C22⋊D293—44no—
91C2×Dic293—21no—
92C4×D293—42no—
93#14556873—21no—
94#14556873—21no—
95#14556873—22no—
96C23×C293—21no—
97C23×C293C2×D412yes—
98Q8×C293—42no—
99D4×C293—21no—
100D4×C293—21no—
101D4×C293—44no—
102D4×C293D814yescomplemented
103C2×C4×C293—22no—
104C2×C4×C293—22no—
105C2×C4×C293—21no—
106C2×C4×C293—22no—
107C2×C4×C293—44no—
108C2×C4×C293—42no—
109C2×C4×C293C2×D411yescharacteristic
110C2×C4×C293C2×D412yes—
111C2×C4×C293C2×D411yescharacteristic
112C2×C4×C293C2411yescharacteristic
113C22×C293—22no—
114C22×C293—22no—
115C22×C293—41no—
116C22×C293—41no—
117C22×C293—41no—
118C22×C293C2×D814yes—
119C22×C293C22×D411yescharacteristic
120C2×D293—22no—
121C2×D293—41no—
122C2×D293—41no—
123C2×D293—42no—
124C2×D293—42no—
125C2×D293—42no—
126C2×D293—88no—
127C4×C293—22no—
128C4×C293—21no—
129C4×C293—41no—
130C4×C293—42no—
131C4×C293C2×D812yes—
132C4×C293C2×#4111yescharacteristic
133C4×C293H(5,𝔽2)11yescharacteristic
134Dic293—41no—
135C2×C293—24no—
136C2×C293—81no—
137C2×C293#54611yescharacteristic
138C2×C293C22×D811yescharacteristic
139C2×C293#57711yescharacteristic
140D293—42no—
141D293—82no—
142C293#329411yescharacteristic, complemented, Sylow
143#3294—2931nomaximal, Sylow
144#547D29311yescharacteristic, complemented
145D42—2931no—
146#486—2931no—
147#480—2932no—
148#474—2931no—
149#466—2932no—
150#459—2931no—
151#447—2932no—
152C4×D8—2931no—
153C8×D4—2931no—
154C2×#102—2932no—
155C2×#41—21no—
156C22×D4—2932no—
157C22×D4—5861no—
158C2×D8—2931no—
159C2×D8—2932no—
160C22×C8—2932no—
161C4⋊Q8C2×D29311yescharacteristic, complemented
162#127—2931no—
163#123—22no—
164C22⋊Q8—21no—
165#121—2932no—
166#121—5861no—
167#121C2×D29312yescomplemented
168#120—5862no—
169C4×D4—2931no—
170C4×D4C2×D29311yescharacteristic
171C4×D4C2×D29311yescharacteristic, complemented
172C2×C4⋊C4C2×D29312yescomplemented
173#107—2931no—
174#107—2932no—
175C4⋊C8—2931no—
176#102—2931no—
177#102—2931no—
178#102—2932no—
179#102—2932no—
180#102—2932no—
181#102—2932no—
182C22⋊C8—2932no—
183C4×C8—2931no—
184C24—5862no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—2931no—
189C2×D4—2932no—
190C2×D4—5864no—
191C2×D4—5862no—
192C2×D4—5861no—
193C2×D4—5862no—
194C2×D4—5861no—
195C2×D4—11722no—
196C2×D4—11724no—
197C2×D4C22⋊D29311yescharacteristic
198C2×D4C22×D29311yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—5861no—
202C22×C4C22×D29312yes—
203D8—5862no—
204D8—11722no—
205C2×C8—2931no—
206C2×C8—2931no—
207C2×C8—2932no—
208C2×C8—11722no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—2931no—
212C4⋊C4C22⋊D29311yescharacteristic
213C4⋊C4C22×D29311yescharacteristic
214C4⋊C4C22×D29312yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—5861no—
218C22⋊C4—5862no—
219C22⋊C4C22⋊D29312yes—
220C42C22×D29311yescharacteristic
221C23—21no—
222C23—23442no—
223C23—5861no—
224C23—5862no—
225C23—11724no—
226C23—11724no—
227C23C2×C22⋊D29312yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—2932no—
233D4—23441no—
234D4—5862no—
235D4—5861no—
236D4—5862no—
237D4—11724no—
238D4—11724no—
239D4—11722no—
240D4#1798978714yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—5861no—
248C2×C4—11722no—
249C2×C4C2×C22⋊D29311yescharacteristic
250C2×C4C2×C22⋊D29312yes—
251C2×C4C2×C22⋊D29311yescharacteristic
252C2×C4C23×D29311yescharacteristic
253C8—5861no—
254C8—5861no—
255C8—5862no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—23448no—
262C22—5862no—
263C22—11722no—
264C22—11722no—
265C22—11722no—
266C22—11721no—
267C22—11721no—
268C22C2×#1798978714yes—
269C22C22×C22⋊D29311yescharacteristic
270C4—22no—
271C4—21no—
272C4—41no—
273C4—42no—
274C4—11721no—
275C4C2×#1798978712yes—
276C4#41×D29311yescharacteristic
277C4#2262767011yescharacteristic
278C2—24no—
279C2—81no—
280C2—23442no—
281C2—11722no—
282C2#3403676711yescharacteristic
283C2C22×#1798978711yescharacteristic
284C2#3403693711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#547×C293C2
socleC22×C293C22×D4
hypercenter#547D293
hypocenterC293#3294
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D1172C2semidirect
2. #34036477C2semidirect
3. #34036442C2semidirect
4. #34036406C2semidirect
5. #34036363C2semidirect
6. #34036324C2semidirect
7. #34036262C2semidirect
8. C4×#17989787C2semidirect
9. D4×#14556873C2semidirect
10. C2×#22627544C2semidirect
11. #547×C293C2semidirect
15. C2×#17989787C22semidirect
16. C22×#14556873C22semidirect
22. #22627557C22semidirect
23. #22627557C22semidirect
24. C4⋊#14556873C22semidirect
25. #22627545C22semidirect
28. #22627544C22semidirect
29. #22627544C22semidirect
30. #22627544C22semidirect
31. C22⋊#14556873C22semidirect
32. C4×#14556873C22semidirect
34. C4⋊Q8×C293C22semidirect
37. #121×C293C22semidirect
39. C4×D4×C293C22semidirect
40. C2×C4⋊C4×C293C22semidirect
89. D1172#41semidirect
102. D4×C293D8semidirect
142. C293#3294semidirect
144. #547D293semidirect
161. C4⋊Q8C2×D293semidirect
167. #121C2×D293semidirect
171. C4×D4C2×D293semidirect
172. C2×C4⋊C4C2×D293semidirect
240. D4#17989787semidirect

Maximal subgroups

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

subgroupindex
D4×D11722
#340364772
#340364422
#340364062
#340363632
#340363242
#340362622
C4×#179897872
D4×#145568732
C2×#226275442
#547×C2932
#3294293