Finite Group Databasebeta

← #37546557

Subgroups — #37546557

Order 23168.

332 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042007C211yesmaximal, characteristic, complemented
2#26041647C211yesmaximal, characteristic, complemented
3#26041638C211yesmaximal, characteristic
4#26041633C211yesmaximal, characteristic
5#26041586C211yesmaximal, characteristic, complemented
6#26041579C211yesmaximal, characteristic
7#26041574C211yesmaximal, characteristic, complemented
8#26041498C211yesmaximal, characteristic
9#26041498C211yesmaximal, characteristic
10#26041486C211yesmaximal, characteristic, complemented
11#26041410C211yesmaximal, characteristic, complemented
12#26041384C211yesmaximal, characteristic, complemented
13#26041379C211yesmaximal, characteristic
14C4×#15668439C211yesmaximal, characteristic, complemented
15#556×C181C211yesmaximal, characteristic
16C2×#15668439C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19#19153816C2211yescharacteristic
20#19153814—21no—
21#19153808C2211yescharacteristic
22#19153808C2211yescharacteristic
23#19153806—21no—
24#19153806—21no—
25C4×Dic362C2211yescharacteristic
26C4×Dic362C2211yescharacteristic
27C4×Dic362C2211yescharacteristic
28#19153781—21no—
29#19153749C2211yescharacteristic
30#19153749C2211yescharacteristic
31#19153747C2211yescharacteristic
32C4⋊#13089678C2211yescharacteristic
33C4⋊#13089678C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35#19153740C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153739C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153737C2211yescharacteristic
45#19153727C2211yescharacteristic
46#19153727C2211yescharacteristic
47C4×#13089678C2211yescharacteristic
48C4⋊Q8×C181C2211yescharacteristic
49#126×C181—21no—
50#126×C181—21no—
51#124×C181C2211yescharacteristic
52#124×C181C2211yescharacteristic
53C22⋊Q8×C181—21no—
54C4×Q8×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57C4×D4×C181C2211yescharacteristic
58#117×C181—21no—
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61#15668439—22no—
62#15668439—22no—
63#15668437—22no—
64C2×#13089678C2311yescharacteristic
65C2×#13089678C2311yescharacteristic
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C4⋊Dic181—21no—
69C4⋊Dic181C2311yescharacteristic
70C4⋊Dic181C2311yescharacteristic
71C4⋊Dic181C2311yescharacteristic
72#15668428—21no—
73#15668428—21no—
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79C4×Dic181—21no—
80C4×Dic181—21no—
81C4×Dic181—21no—
82C2×Q8×C181D411yescharacteristic
83C2×Q8×C181C2311yescharacteristic
84C2×Q8×C181C2311yescharacteristic
85C2×D4×C181C2311yescharacteristic
86C22×C4×C181—21no—
87C4⋊C4×C181—21no—
88C4⋊C4×C181—21no—
89C4⋊C4×C181—21no—
90C4⋊C4×C181—21no—
91C4⋊C4×C181D411yescharacteristic
92C4⋊C4×C181D411yescharacteristic
93C4⋊C4×C181D411yescharacteristic
94C4⋊C4×C181C2311yescharacteristic
95C22⋊C4×C181—21no—
96C22⋊C4×C181—21no—
97C22⋊C4×C181—21no—
98C22⋊C4×C181—21no—
99C22⋊C4×C181—21no—
100C42×C181—21no—
101C42×C181—21no—
102C42×C181C2311yescharacteristic
103C42×C181C2311yescharacteristic
104C42×C181C2311yescharacteristic
105Dic362—21no—
106Dic362—42no—
107Dic362#4112yes—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112#13089678—21no—
113#13089678—21no—
114#13089678—21no—
115#13089678—21no—
116C23×C181—21no—
117Q8×C181—21no—
118Q8×C181—21no—
119Q8×C181—21no—
120Q8×C181—21no—
121Q8×C181—41no—
122D4×C181—21no—
123D4×C181—21no—
124C2×C4×C181—21no—
125C2×C4×C181—21no—
126C2×C4×C181—21no—
127C2×C4×C181—42no—
128C2×C4×C181C2×D411yescharacteristic
129C2×C4×C181C2×D411yescharacteristic
130C2×C4×C181C2×D411yescharacteristic
131C2×C4×C181C2×D411yescharacteristic
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2411yescharacteristic
135C22×C181—41no—
136C22×C181—41no—
137C22×C181—41no—
138C22×C181C22×D411yescharacteristic
139C4×C181—21no—
140C4×C181—21no—
141C4×C181—21no—
142C4×C181—21no—
143C4×C181—21no—
144C4×C181—21no—
145C4×C181—41no—
146C4×C181—41no—
147C4×C181—41no—
148C4×C181C2×#4111yescharacteristic
149C4×C1812−1+411yescharacteristic
150Dic181—22no—
151Dic181—41no—
152Dic181—41no—
153Dic181—41no—
154C2×C181—81no—
155C2×C181#54811yescharacteristic
156C2×C181#57511yescharacteristic
157C2×C181#57811yescharacteristic
158C181#337711yescharacteristic, complemented, Sylow
159#3377—1811nomaximal, Sylow
160#557—1811no—
161#556D18111yescharacteristic
162#491—1811no—
163#489—1811no—
164#488—1811no—
165#479—1811no—
166#478—1811no—
167#477—1811no—
168#464—1811no—
169#464—1811no—
170#462—1811no—
171#446—1811no—
172#441—1811no—
173#440—1811no—
174C4×Q16—1811no—
175C2×Q16—1811no—
176C2×Q16—1811no—
177C2×SD16—1811no—
178C4⋊Q8—3621no—
179C4⋊Q8—1811no—
180C4⋊Q8C2×D18111yescharacteristic
181#126—21no—
182#126—21no—
183#125—3621no—
184#125—3621no—
185#125—1811no—
186#125—1811no—
187#124C2×D18111yescharacteristic
188#124C2×D18111yescharacteristic
189C22⋊Q8—21no—
190C4×Q8—3621no—
191C4×Q8—1811no—
192C4×Q8—1811no—
193C4×Q8—1811no—
194C4×Q8C2×D18111yescharacteristic
195C4×Q8C2×D18111yescharacteristic
196C4×Q8C2×D18111yescharacteristic
197C4×D4C2×D18111yescharacteristic
198#117—21no—
199#107—1811no—
200#107—1811no—
201#106—1811no—
202C4⋊C8—1811no—
203C4⋊C8—1811no—
204C4⋊C8—1811no—
205#103—1811no—
206#103—1811no—
207#103—1811no—
208#103—1811no—
209#103—1811no—
210#103—1811no—
211#103—1811no—
212#102—1811no—
213#102—1811no—
214#102—1811no—
215#97—1811no—
216#97—1811no—
217C4×C8—1811no—
218C2×Q8—3621no—
219C2×Q8—1811no—
220C2×Q8C22⋊D18111yescharacteristic
221C2×Q8C22×D18111yescharacteristic
222C2×Q8C22×D18111yescharacteristic
223C2×D4C22×D18111yescharacteristic
224C22×C4—21no—
225Q16—3622no—
226Q16—3622no—
227SD16—3622no—
228C2×C8—1811no—
229C2×C8—1811no—
230C2×C8—1811no—
231C2×C8—1811no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—3621no—
237C4⋊C4—3621no—
238C4⋊C4—3621no—
239C4⋊C4—3621no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—3621no—
243C4⋊C4—3621no—
244C4⋊C4—1811no—
245C4⋊C4—1811no—
246C4⋊C4—1811no—
247C4⋊C4C22⋊D18111yescharacteristic
248C4⋊C4C22⋊D18111yescharacteristic
249C4⋊C4C22⋊D18111yescharacteristic
250C4⋊C4C22×D18111yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C42—21no—
257C42—21no—
258C42—3621no—
259C42—3621no—
260C42—3621no—
261C42C22×D18111yescharacteristic
262C42C22×D18111yescharacteristic
263C42C22×D18111yescharacteristic
264C23—21no—
265Q8—21no—
266Q8—21no—
267Q8—21no—
268Q8—21no—
269Q8—41no—
270Q8—3621no—
271Q8—1812no—
272Q8—7242no—
273D4—21no—
274D4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—3621no—
280C2×C4—3621no—
281C2×C4—3621no—
282C2×C4—3621no—
283C2×C4C2×C22⋊D18111yescharacteristic
284C2×C4C2×C22⋊D18111yescharacteristic
285C2×C4C2×C22⋊D18111yescharacteristic
286C2×C4C2×C22⋊D18111yescharacteristic
287C2×C4C2×C22⋊D18111yescharacteristic
288C2×C4C2×C22⋊D18111yescharacteristic
289C2×C4C23×D18111yescharacteristic
290C8—3621no—
291C8—3621no—
292C8—3621no—
293C8—3621no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22C22×C22⋊D18111yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—3622no—
308C4—7241no—
309C4—7241no—
310C4—7241no—
311C4#41×D18111yescharacteristic
312C4#1915386311yescharacteristic
313C2—81no—
314C2#2604193611yescharacteristic
315C2#2604207711yescharacteristic
316C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#556×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181C23
hypercenter#556D181
hypocenterC181#3377
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #26042007C2semidirect
2. #26041647C2semidirect
5. #26041586C2semidirect
7. #26041574C2semidirect
10. #26041486C2semidirect
11. #26041410C2semidirect
12. #26041384C2semidirect
14. C4×#15668439C2semidirect
158. C181#3377semidirect

Maximal subgroups

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

subgroupindex
#260420072
#260416472
#260416382
#260416332
#260415862
#260415792
#260415742
#260414982
#260414862
#260414102
#260413842
#260413792
C4×#156684392
#556×C1812
#3377181