Finite Group Databasebeta

← #37545892

Subgroups — #37545892

Order 23168.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041975C211yesmaximal, characteristic
2#26041647C211yesmaximal, characteristic, complemented
3#26041617C211yesmaximal, characteristic, complemented
4#26041612C211yesmaximal, characteristic, complemented
5#26041586C211yesmaximal, characteristic, complemented
6#26041527C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041481C211yesmaximal, characteristic, complemented
9#26041418C211yesmaximal, characteristic, complemented
10#26041418C211yesmaximal, characteristic, complemented
11#26041377C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041288C211yesmaximal, characteristic, complemented
14C2×#19153740C211yesmaximal, characteristic, complemented
15D4×Q8×C181C211yesmaximal, characteristic, complemented
16C2×#15668436C2211yescharacteristic
17C2×#15668436C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19C2×#15668433C2211yescharacteristic
20C22×#13089678C2211yescharacteristic
21#19153811—21no—
22#19153808C2211yescharacteristic
23#19153798—21no—
24#19153790C2211yescharacteristic
25#19153790C2211yescharacteristic
26#19153788—21no—
27C4×D724C2211yescharacteristic
28#19153776—21no—
29#19153775C2211yescharacteristic
30C2×C4⋊Dic181C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153749C2211yescharacteristic
33#19153747C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35#19153740C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153736C2211yescharacteristic
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×D724C2311yescharacteristic
60C2×C22⋊D181—21no—
61C2×C22⋊D181—21no—
62C22×Dic181—21no—
63C2×C4×D181—21no—
64#15668436—22no—
65#15668436—41no—
66#15668436—41no—
67#15668433—41no—
68C2×#13089678—41no—
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C4⋊Dic181—21no—
74C4⋊Dic181C2311yescharacteristic
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79C22⋊Dic181—21no—
80#15668426—21no—
81#15668426—21no—
82#15668426—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
111C22×D181—21no—
112D724—21no—
113D724—21no—
114C22⋊D181—42no—
115C22⋊D181—81no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C2×Dic181—21no—
119C2×Dic181—81no—
120C4×D181—81no—
121#13089678—21no—
122#13089678—21no—
123#13089678—21no—
124#13089678—21no—
125C23×C181C2×D411yescharacteristic
126C23×C181C2×D411yescharacteristic
127Q8×C181—21no—
128Q8×C181—21no—
129Q8×C181—21no—
130Q8×C181—21no—
131Q8×C181—21no—
132Q8×C181—42no—
133Q8×C181—42no—
134Q8×C181#4112yes—
135D4×C181—22no—
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181—21no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—42no—
145C2×C4×C181—42no—
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2×D411yescharacteristic
150C2×C4×C181C2411yescharacteristic
151C22×C181—21no—
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181#13712yes—
157C22×C181C22×D411yescharacteristic
158C2×D181—41no—
159C2×D181—41no—
160C2×D181—41no—
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—22no—
164C4×C181—21no—
165C4×C181—21no—
166C4×C181—41no—
167C4×C181—41no—
168C4×C181—41no—
169C4×C181C2×#4111yescharacteristic
170C4×C181H(5,𝔽2)11yescharacteristic
171Dic181—41no—
172Dic181—41no—
173Dic181—41no—
174C2×C181—22no—
175C2×C181—41no—
176C2×C181#54611yescharacteristic
177C2×C181C2×#13711yescharacteristic
178C2×C181#57711yescharacteristic
179D181—81no—
180C181#332011yescharacteristic, complemented, Sylow
181#3320—1811nomaximal, Sylow
182#553—1811no—
183D4×Q8D18111yescharacteristic, complemented
184#491—1811no—
185#485—1811no—
186#484—1811no—
187#479—1811no—
188#468—1811no—
189#465—1811no—
190#461—1811no—
191#448—1811no—
192#448—1811no—
193#440—1811no—
194#435—1811no—
195#419—1811no—
196C2×#103—1811no—
197C22×Q8C2×D18111yescharacteristic
198C22×Q8C2×D18111yescharacteristic
199C2×SD16—1811no—
200C2×SD16—1811no—
201C2×SD16—1811no—
202C2×M16—1811no—
203C22×C8—1811no—
204C4⋊Q8—21no—
205C4⋊Q8C2×D18111yescharacteristic
206#126—3621no—
207#125—1811no—
208#123—3621no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8C2×D18111yescharacteristic
212C22⋊Q8C2×D18111yescharacteristic
213#121—3621no—
214#121—1811no—
215#121—1811no—
216C4×Q8C2×D18111yescharacteristic
217C4×D4—21no—
218C4×D4—3621no—
219C4×D4—1811no—
220C4×D4C2×D18111yescharacteristic
221#117—1811no—
222C2×C4⋊C4—1811no—
223#107—1811no—
224#107—1811no—
225#106—1811no—
226C4⋊C8—1811no—
227#103—1811no—
228#103—1811no—
229#103—1811no—
230#103—1811no—
231#103—1811no—
232#103—1811no—
233#103—1811no—
234#103—1811no—
235#103—1811no—
236#102—1811no—
237C22⋊C8—1811no—
238C22⋊C8—1811no—
239#97—1811no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C22×D18111yescharacteristic
247C2×Q8C22×D18111yescharacteristic
248C2×Q8C22×D18111yescharacteristic
249C2×D4—3621no—
250C2×D4—3621no—
251C2×D4—1811no—
252C2×D4C22⋊D18111yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3621no—
256C22×C4—3621no—
257C22×C4C22×D18111yescharacteristic
258C22×C4C22×D18111yescharacteristic
259SD16—3622no—
260SD16—7241no—
261SD16—7241no—
262M16—7241no—
263C2×C8—1811no—
264C2×C8—1811no—
265C2×C8—1811no—
266C2×C8—1811no—
267C2×C8—7241no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—3621no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—1811no—
277C4⋊C4—1811no—
278C4⋊C4—1811no—
279C4⋊C4C22⋊D18111yescharacteristic
280C4⋊C4C22×D18111yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4C22⋊D18111yescharacteristic
288C22⋊C4C22⋊D18111yescharacteristic
289C42—21no—
290C42—3621no—
291C42C22×D18111yescharacteristic
292C23—3621no—
293C23C2×C22⋊D18111yescharacteristic
294C23C2×C22⋊D18111yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—42no—
302Q8#1566844812yes—
303D4—22no—
304D4—3621no—
305D4—3621no—
306D4—14481no—
307D4—7242no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—3621no—
319C2×C4—3621no—
320C2×C4—3621no—
321C2×C4—14481no—
322C2×C4—14481no—
323C2×C4C2×C22⋊D18111yescharacteristic
324C2×C4C2×C22⋊D18111yescharacteristic
325C2×C4C2×C22⋊D18111yescharacteristic
326C2×C4C2×C22⋊D18111yescharacteristic
327C2×C4C23×D18111yescharacteristic
328C8—3621no—
329C8—3621no—
330C8—3621no—
331C8—3621no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—7241no—
338C22—7241no—
339C22—7241no—
340C22#1915384712yes—
341C22C22×C22⋊D18111yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—7241no—
351C4—7241no—
352C4—7241no—
353C4C2×#1566844811yescharacteristic
354C4#1915386111yescharacteristic
355C2—22no—
356C2—41no—
357C2—14481no—
358C2#2604192311yescharacteristic
359C2C2×#1915384711yescharacteristic
360C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingD4×Q8×C181C2
socleC22×C181C22×D4
prime radical#26041975C2
hypercenterD4×Q8D181
hypocenterC181#3320
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. #26041612C2semidirect
5. #26041586C2semidirect
6. #26041527C2semidirect
7. #26041510C2semidirect
8. #26041481C2semidirect
9. #26041418C2semidirect
10. #26041418C2semidirect
11. #26041377C2semidirect
12. #26041351C2semidirect
13. #26041288C2semidirect
14. C2×#19153740C2semidirect
15. D4×Q8×C181C2semidirect
180. C181#3320semidirect
183. D4×Q8D181semidirect

Maximal subgroups

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

subgroupindex
#260419752
#260416472
#260416172
#260416122
#260415862
#260415272
#260415102
#260414812
#260414182
#260413772
#260413512
#260412882
C2×#191537402
D4×Q8×C1812
#3320181