Finite Group Databasebeta

← #37546660

Subgroups — #37546660

Order 23168.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041989C211yesmaximal, characteristic
2#26041689C211yesmaximal, characteristic, complemented
3#26041685C211yesmaximal, characteristic, complemented
4#26041678C211yesmaximal, characteristic, complemented
5#26041586C211yesmaximal, characteristic, complemented
6#26041586C211yesmaximal, characteristic, complemented
7#26041576C211yesmaximal, characteristic
8#26041576C211yesmaximal, characteristic
9#26041566C211yesmaximal, characteristic
10#26041564C211yesmaximal, characteristic, complemented
11#26041410C211yesmaximal, characteristic, complemented
12#26041377C211yesmaximal, characteristic
13#26041377C211yesmaximal, characteristic
14C4×#15668436C211yesmaximal, characteristic
15#557×C181C211yesmaximal, characteristic, complemented
16C2×#15668436C2211yescharacteristic
17#19153816C2211yescharacteristic
18#19153808C2211yescharacteristic
19#19153808C2211yescharacteristic
20#19153794—21no—
21#19153794—21no—
22#19153792—21no—
23C4×D724C2211yescharacteristic
24C4×D724C2211yescharacteristic
25C4×D724C2211yescharacteristic
26C4⋊C4×D181—21no—
27#19153749C2211yescharacteristic
28#19153749C2211yescharacteristic
29#19153749C2211yescharacteristic
30#19153749C2211yescharacteristic
31#19153747C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36C4⋊#13089678C2211yescharacteristic
37C4⋊#13089678C2211yescharacteristic
38C4⋊#13089678C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44#19153736C2211yescharacteristic
45#19153727C2211yescharacteristic
46#19153727C2211yescharacteristic
47C4×#13089678C2211yescharacteristic
48C4⋊Q8×C181—21no—
49C4⋊Q8×C181C2211yescharacteristic
50#125×C181—21no—
51#125×C181—21no—
52#125×C181C2211yescharacteristic
53#125×C181C2211yescharacteristic
54C4×Q8×C181—21no—
55C4×Q8×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57C4×Q8×C181C2211yescharacteristic
58C4×Q8×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×D724C2311yescharacteristic
61C2×C4×D181—21no—
62C2×C4×D181—21no—
63C2×C4×D181—21no—
64#15668436Q814yescomplemented
65C2×#13089678C2311yescharacteristic
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C4⋊Dic181—21no—
70C4⋊Dic181C2311yescharacteristic
71C4⋊Dic181C2311yescharacteristic
72C4⋊Dic181C2311yescharacteristic
73#15668428—21no—
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77#15668426—21no—
78#15668426—21no—
79#15668426—21no—
80C2×Q8×C181—21no—
81C2×Q8×C181D411yescharacteristic
82C2×Q8×C181C2311yescharacteristic
83C4⋊C4×C181—21no—
84C4⋊C4×C181—21no—
85C4⋊C4×C181—21no—
86C4⋊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
95C4⋊C4×C181C2311yescharacteristic
96C4⋊C4×C181C2311yescharacteristic
97C42×C181—21no—
98C42×C181—21no—
99C42×C181—21no—
100C42×C181C2311yescharacteristic
101C42×C181C2311yescharacteristic
102C42×C181C2311yescharacteristic
103C22×D181—21no—
104Dic362—81no—
105D724—21no—
106D724C2×Q812yes—
107C2×Dic181—21no—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C4×D181—42no—
111C4×D181—81no—
112C4×D181—81no—
113#13089678—21no—
114#13089678—21no—
115#13089678—21no—
116#13089678C2×Q812yes—
117Q8×C181—21no—
118Q8×C181—41no—
119Q8×C181—42no—
120Q8×C181C2×Q812yes—
121C2×C4×C181—21no—
122C2×C4×C181—21no—
123C2×C4×C181—21no—
124C2×C4×C181—21no—
125C2×C4×C181C2×D411yescharacteristic
126C2×C4×C181C2×D411yescharacteristic
127C2×C4×C181C2×D411yescharacteristic
128C2×C4×C181C2×D411yescharacteristic
129C2×C4×C181C2×D411yescharacteristic
130C2×C4×C181C2×D411yescharacteristic
131C2×C4×C181C2411yescharacteristic
132C22×C181C22×D411yescharacteristic
133C2×D181—22no—
134C2×D181—41no—
135C2×D181—41no—
136C4×C181—22no—
137C4×C181—21no—
138C4×C181—21no—
139C4×C181—21no—
140C4×C181—21no—
141C4×C181—21no—
142C4×C181—21no—
143C4×C181—41no—
144C4×C181—41no—
145C4×C181—41no—
146C4×C181C22×Q811yescharacteristic
147C4×C1812−1+411yescharacteristic
148Dic181—41no—
149Dic181—41no—
150Dic181—41no—
151C2×C181D4×Q811yescharacteristic
152C2×C181#57511yescharacteristic
153C2×C181#57711yescharacteristic
154D181—42no—
155C181#338811yescharacteristic, complemented, Sylow
156#3388—1811nomaximal, Sylow
157#557D18111yescharacteristic, complemented
158#554—1811no—
159#501—1811no—
160#501—1811no—
161#499—1811no—
162#479—1811no—
163#479—1811no—
164#478—1811no—
165#478—1811no—
166#476—1811no—
167#475—1811no—
168#446—1811no—
169#440—1811no—
170#440—1811no—
171C4×SD16—1811no—
172C2×SD16—1811no—
173C4⋊Q8—21no—
174C4⋊Q8—1811no—
175C4⋊Q8C2×D18111yescharacteristic
176#125—21no—
177#125—21no—
178#125—1811no—
179#125—1811no—
180#125C2×D18111yescharacteristic
181#125C2×D18111yescharacteristic
182C22⋊Q8—3621no—
183C22⋊Q8—3621no—
184C22⋊Q8—3621no—
185C4×Q8—21no—
186C4×Q8C2×D18111yescharacteristic
187C4×Q8C2×D18111yescharacteristic
188C4×Q8C2×D18111yescharacteristic
189C4×Q8C2×D18111yescharacteristic
190C4×D4—1811no—
191C4×D4—1811no—
192C4×D4—1811no—
193C2×C4⋊C4—3621no—
194#107—1811no—
195#107—1811no—
196#107—1811no—
197#107—1811no—
198#106—1811no—
199#106—1811no—
200#106—1811no—
201#106—1811no—
202#106—1811no—
203C4⋊C8—1811no—
204C4⋊C8—1811no—
205C4⋊C8—1811no—
206#103—1811no—
207#103—1811no—
208#103—1811no—
209#102—1811no—
210#102—1811no—
211#102—1811no—
212#97—1811no—
213#97—1811no—
214C4×C8—1811no—
215C2×Q8—21no—
216C2×Q8—3621no—
217C2×Q8C22⋊D18111yescharacteristic
218C2×Q8C22×D18111yescharacteristic
219C2×D4—1811no—
220C22×C4—3621no—
221C22×C4—3621no—
222C22×C4—3621no—
223SD16—1814no—
224C2×C8—1811no—
225C2×C8—1811no—
226C2×C8—1811no—
227C2×C8—1811no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
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—1811no—
242C4⋊C4—1811no—
243C4⋊C4—1811no—
244C4⋊C4C22⋊D18111yescharacteristic
245C4⋊C4C22⋊D18111yescharacteristic
246C4⋊C4C22⋊D18111yescharacteristic
247C4⋊C4C22×D18111yescharacteristic
248C4⋊C4C22×D18111yescharacteristic
249C4⋊C4C22×D18111yescharacteristic
250C22⋊C4—3621no—
251C22⋊C4—3621no—
252C22⋊C4—3621no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42C22×D18111yescharacteristic
257C42C22×D18111yescharacteristic
258C42C22×D18111yescharacteristic
259C23—3621no—
260Q8—21no—
261Q8—41no—
262Q8—42no—
263Q8—14481no—
264Q8Q8×D18112yes—
265D4—3621no—
266D4—1812no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—3621no—
272C2×C4—3621no—
273C2×C4—3621no—
274C2×C4—14481no—
275C2×C4—14481no—
276C2×C4—7242no—
277C2×C4C2×C22⋊D18111yescharacteristic
278C2×C4C2×C22⋊D18111yescharacteristic
279C2×C4C2×C22⋊D18111yescharacteristic
280C2×C4C2×C22⋊D18111yescharacteristic
281C2×C4C2×C22⋊D18111yescharacteristic
282C2×C4C2×C22⋊D18111yescharacteristic
283C2×C4C23×D18111yescharacteristic
284C8—3621no—
285C8—3621no—
286C8—3621no—
287C8—1812no—
288C22—3622no—
289C22—7241no—
290C22—7241no—
291C22C22×C22⋊D18111yescharacteristic
292C4—22no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—7241no—
303C4—7241no—
304C4—7241no—
305C4C2×Q8×D18111yescharacteristic
306C4#1915386411yescharacteristic
307C2—7242no—
308C2Q8×C22⋊D18111yescharacteristic
309C2#2604207711yescharacteristic
310C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#557×C181C2
socleC22×C181C22×D4
prime radicalC2×D724C23
hypercenter#557D181
hypocenterC181#3388
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
2. #26041689C2semidirect
3. #26041685C2semidirect
4. #26041678C2semidirect
5. #26041586C2semidirect
6. #26041586C2semidirect
10. #26041564C2semidirect
11. #26041410C2semidirect
15. #557×C181C2semidirect
64. #15668436Q8semidirect
155. C181#3388semidirect
157. #557D181semidirect

Maximal subgroups

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

subgroupindex
#260419892
#260416892
#260416852
#260416782
#260415862
#260415762
#260415662
#260415642
#260414102
#260413772
C4×#156684362
#557×C1812
#3388181