Finite Group Databasebeta

← #37545772

Subgroups — #37545772

Order 23168.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041969C211yesmaximal, characteristic
2#26041647C211yesmaximal, characteristic, complemented
3#26041617C211yesmaximal, characteristic, complemented
4#26041607C211yesmaximal, characteristic, complemented
5#26041586C211yesmaximal, characteristic, complemented
6#26041535C211yesmaximal, characteristic, complemented
7#26041518C211yesmaximal, characteristic, complemented
8#26041486C211yesmaximal, characteristic, complemented
9#26041435C211yesmaximal, characteristic, complemented
10#26041435C211yesmaximal, characteristic, complemented
11#26041384C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic
13#26041288C211yesmaximal, characteristic, complemented
14C2×#19153740C211yesmaximal, characteristic, complemented
15D4×Q8×C181C211yesmaximal, characteristic
16C2×#15668439C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668439C2211yescharacteristic
19C2×#15668433C2211yescharacteristic
20C22×#13089678C2211yescharacteristic
21#19153809—21no—
22#19153808C2211yescharacteristic
23#19153806—21no—
24#19153797—21no—
25#19153795C2211yescharacteristic
26#19153795C2211yescharacteristic
27C4×Dic362C2211yescharacteristic
28#19153775C2211yescharacteristic
29#19153772—21no—
30C2×C4⋊Dic181C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35#19153740C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153739C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47#19153727C2211yescharacteristic
48C22×Q8×C181C2211yescharacteristic
49C22×Q8×C181C2211yescharacteristic
50C4⋊Q8×C181—21no—
51C4⋊Q8×C181C2211yescharacteristic
52C22⋊Q8×C181—21no—
53C22⋊Q8×C181—21no—
54C22⋊Q8×C181C2211yescharacteristic
55C22⋊Q8×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57C4×D4×C181—21no—
58C4×D4×C181C2211yescharacteristic
59C2×Dic362C2311yescharacteristic
60C22×Dic181—21no—
61#15668439—22no—
62#15668439—41no—
63#15668439—41no—
64#15668433—41no—
65C2×#13089678—41no—
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C2×#13089678C2311yescharacteristic
70C4⋊Dic181—21no—
71C4⋊Dic181C2311yescharacteristic
72C4⋊Dic181C2311yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79C22⋊Dic181—21no—
80C22⋊Dic181—21no—
81C22⋊Dic181—21no—
82C4×Dic181—21no—
83C4×Dic181—21no—
84C2×Q8×C181—21no—
85C2×Q8×C181—21no—
86C2×Q8×C181—22no—
87C2×Q8×C181—22no—
88C2×Q8×C181—41no—
89C2×Q8×C181—41no—
90C2×Q8×C181C2311yescharacteristic
91C2×Q8×C181C2311yescharacteristic
92C2×Q8×C181C2311yescharacteristic
93C2×D4×C181D411yescharacteristic
94C22×C4×C181—21no—
95C22×C4×C181—21no—
96C22×C4×C181C2311yescharacteristic
97C22×C4×C181C2311yescharacteristic
98C4⋊C4×C181—21no—
99C4⋊C4×C181—21no—
100C4⋊C4×C181—21no—
101C4⋊C4×C181—21no—
102C4⋊C4×C181—21no—
103C4⋊C4×C181D411yescharacteristic
104C4⋊C4×C181C2311yescharacteristic
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C22⋊C4×C181D411yescharacteristic
108C22⋊C4×C181D411yescharacteristic
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111Dic362—21no—
112Dic362—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—42no—
118#13089678—21no—
119#13089678—21no—
120#13089678—21no—
121#13089678—21no—
122C23×C181C2×D411yescharacteristic
123C23×C181C2×D411yescharacteristic
124Q8×C181—21no—
125Q8×C181—21no—
126Q8×C181—21no—
127Q8×C181—21no—
128Q8×C181—21no—
129Q8×C181—42no—
130Q8×C181—42no—
131Q8×C181#4112yes—
132D4×C181—22no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181—21no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—42no—
142C2×C4×C181—42no—
143C2×C4×C181C2×D411yescharacteristic
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2×D411yescharacteristic
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2411yescharacteristic
148C22×C181—21no—
149C22×C181—21no—
150C22×C181—21no—
151C22×C181—21no—
152C22×C181—21no—
153C22×C181#13712yes—
154C22×C181C22×D411yescharacteristic
155C4×C181—21no—
156C4×C181—21no—
157C4×C181—22no—
158C4×C181—21no—
159C4×C181—21no—
160C4×C181—41no—
161C4×C181—41no—
162C4×C181—41no—
163C4×C181C2×#4111yescharacteristic
164C4×C181H(5,𝔽2)11yescharacteristic
165Dic181—41no—
166Dic181—41no—
167Dic181—41no—
168Dic181—41no—
169C2×C181—22no—
170C2×C181—41no—
171C2×C181#54611yescharacteristic
172C2×C181C2×#13711yescharacteristic
173C2×C181#57811yescharacteristic
174C181#331211yescharacteristic, complemented, Sylow
175#3312—1811nomaximal, Sylow
176#552—1811no—
177D4×Q8D18111yescharacteristic
178#491—1811no—
179#485—1811no—
180#483—1811no—
181#479—1811no—
182#470—1811no—
183#467—1811no—
184#462—1811no—
185#451—1811no—
186#451—1811no—
187#441—1811no—
188#435—1811no—
189#419—1811no—
190C2×#103—1811no—
191C22×Q8C2×D18111yescharacteristic
192C22×Q8C2×D18111yescharacteristic
193C2×Q16—1811no—
194C2×Q16—1811no—
195C2×Q16—1811no—
196C2×M16—1811no—
197C22×C8—1811no—
198C4⋊Q8—21no—
199C4⋊Q8C2×D18111yescharacteristic
200#126—3621no—
201#125—3621no—
202#125—1811no—
203#123—3621no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—1811no—
207C22⋊Q8—1811no—
208C22⋊Q8C2×D18111yescharacteristic
209C22⋊Q8C2×D18111yescharacteristic
210C4×Q8—1811no—
211C4×Q8C2×D18111yescharacteristic
212C4×D4—21no—
213C4×D4C2×D18111yescharacteristic
214#117—3621no—
215#117—1811no—
216C2×C4⋊C4—1811no—
217#107—1811no—
218#106—1811no—
219#106—1811no—
220C4⋊C8—1811no—
221#103—1811no—
222#103—1811no—
223#103—1811no—
224#103—1811no—
225#103—1811no—
226#103—1811no—
227#103—1811no—
228#103—1811no—
229#103—1811no—
230#103—1811no—
231C22⋊C8—1811no—
232C22⋊C8—1811no—
233#97—1811no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—1811no—
241C2×Q8C22×D18111yescharacteristic
242C2×Q8C22×D18111yescharacteristic
243C2×Q8C22×D18111yescharacteristic
244C2×D4C22⋊D18111yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3621no—
248C22×C4C22×D18111yescharacteristic
249C22×C4C22×D18111yescharacteristic
250Q16—3622no—
251Q16—7241no—
252Q16—7241no—
253M16—7241no—
254C2×C8—1811no—
255C2×C8—1811no—
256C2×C8—1811no—
257C2×C8—1811no—
258C2×C8—7241no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—3621no—
265C4⋊C4—3621no—
266C4⋊C4—3621no—
267C4⋊C4—3621no—
268C4⋊C4—3621no—
269C4⋊C4—3621no—
270C4⋊C4—1811no—
271C4⋊C4—1811no—
272C4⋊C4—1811no—
273C4⋊C4C22⋊D18111yescharacteristic
274C4⋊C4C22×D18111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3621no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4C22⋊D18111yescharacteristic
281C22⋊C4C22⋊D18111yescharacteristic
282C42—21no—
283C42—3621no—
284C42—3621no—
285C42C22×D18111yescharacteristic
286C23C2×C22⋊D18111yescharacteristic
287C23C2×C22⋊D18111yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—42no—
295Q8—3621no—
296Q8—3621no—
297Q8#1566844812yes—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—3621no—
310C2×C4—3621no—
311C2×C4—3621no—
312C2×C4—3621no—
313C2×C4—7242no—
314C2×C4C2×C22⋊D18111yescharacteristic
315C2×C4C2×C22⋊D18111yescharacteristic
316C2×C4C2×C22⋊D18111yescharacteristic
317C2×C4C2×C22⋊D18111yescharacteristic
318C2×C4C23×D18111yescharacteristic
319C8—3621no—
320C8—3621no—
321C8—3621no—
322C8—3621no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1915384712yes—
329C22C22×C22⋊D18111yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—7241no—
339C4—7241no—
340C4—7241no—
341C4—7241no—
342C4C2×#1566844811yescharacteristic
343C4#1915386111yescharacteristic
344C2—22no—
345C2—41no—
346C2#2604192311yescharacteristic
347C2C2×#1915384711yescharacteristic
348C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingD4×Q8×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181D4
hypercenterD4×Q8D181
hypocenterC181#3312
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #26041647C2semidirect
3. #26041617C2semidirect
4. #26041607C2semidirect
5. #26041586C2semidirect
6. #26041535C2semidirect
7. #26041518C2semidirect
8. #26041486C2semidirect
9. #26041435C2semidirect
10. #26041435C2semidirect
11. #26041384C2semidirect
13. #26041288C2semidirect
14. C2×#19153740C2semidirect
174. C181#3312semidirect

Maximal subgroups

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

subgroupindex
#260419692
#260416472
#260416172
#260416072
#260415862
#260415352
#260415182
#260414862
#260414352
#260413842
#260413512
#260412882
C2×#191537402
D4×Q8×C1812
#3312181