Finite Group Databasebeta

← #37545832

Subgroups — #37545832

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#26041986C211yesmaximal, characteristic, complemented
2#26041638C211yesmaximal, characteristic, complemented
3#26041607C211yesmaximal, characteristic, complemented
4#26041595C211yesmaximal, characteristic, complemented
5#26041564C211yesmaximal, characteristic, complemented
6#26041535C211yesmaximal, characteristic, complemented
7#26041506C211yesmaximal, characteristic, complemented
8#26041498C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041435C211yesmaximal, characteristic, complemented
11#26041384C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041280C211yesmaximal, characteristic, complemented
14C2×#19153740C211yesmaximal, characteristic, complemented
15#548×C181C211yesmaximal, characteristic
16C2×#15668439C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C2×#15668433C2211yescharacteristic
20C22×#13089678C2211yescharacteristic
21#19153816C2211yescharacteristic
22#19153807—21no—
23#19153795C2211yescharacteristic
24#19153795C2211yescharacteristic
25#19153793—21no—
26#19153793—21no—
27C4×Dic362C2211yescharacteristic
28D4×Dic181—21no—
29C2×C4⋊Dic181C2211yescharacteristic
30C2×C4⋊Dic181C2211yescharacteristic
31#19153747C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35#19153740C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153739C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153737C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47#19153727C2211yescharacteristic
48C2×#41×C181C2211yescharacteristic
49C22×Q8×C181C2211yescharacteristic
50#124×C181—21no—
51#124×C181C2211yescharacteristic
52C22⋊Q8×C181—21no—
53C22⋊Q8×C181C2211yescharacteristic
54#121×C181—21no—
55#121×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57C4×D4×C181—21no—
58C4×D4×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61C22×Dic181—21no—
62C22×Dic181—21no—
63#15668439—22no—
64#15668439—41no—
65#15668437—41no—
66#15668433—41no—
67C2×#13089678—41no—
68C2×#13089678C2311yescharacteristic
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C4⋊Dic181—21no—
73C4⋊Dic181—21no—
74C4⋊Dic181—21no—
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81C22⋊Dic181—21no—
82C22⋊Dic181—21no—
83C4×Dic181—21no—
84#41×C181—22no—
85#41×C181—41no—
86C2×Q8×C181—21no—
87C2×Q8×C181—22no—
88C2×Q8×C181—41no—
89C2×Q8×C181C2311yescharacteristic
90C2×Q8×C181C2311yescharacteristic
91C2×D4×C181—21no—
92C2×D4×C181—21no—
93C2×D4×C181D411yescharacteristic
94C2×D4×C181C2311yescharacteristic
95C22×C4×C181—21no—
96C22×C4×C181—21no—
97C22×C4×C181C2311yescharacteristic
98C22×C4×C181C2311yescharacteristic
99C4⋊C4×C181—21no—
100C4⋊C4×C181—21no—
101C4⋊C4×C181D411yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C22⋊C4×C181—21no—
104C22⋊C4×C181—21no—
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C22⋊C4×C181D411yescharacteristic
108C22⋊C4×C181D411yescharacteristic
109C42×C181—21no—
110C42×C181C2311yescharacteristic
111Dic362—21no—
112Dic362—21no—
113Dic362—81no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C2×Dic181—81no—
119C2×Dic181—81no—
120#13089678—21no—
121#13089678—21no—
122#13089678—21no—
123#13089678—21no—
124C23×C181—21no—
125C23×C181C2×D411yescharacteristic
126C23×C181C2×D411yescharacteristic
127Q8×C181—21no—
128Q8×C181—21no—
129Q8×C181—21no—
130Q8×C181—42no—
131Q8×C181#4112yes—
132D4×C181—22no—
133D4×C181—21no—
134D4×C181—21no—
135D4×C181—42no—
136D4×C181—42no—
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—41no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181#13712yes—
160C22×C181C22×D411yescharacteristic
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—22no—
164C4×C181—21no—
165C4×C181—21no—
166C4×C181—41no—
167C4×C181—41no—
168C4×C181C2×#4111yescharacteristic
169C4×C181H(5,𝔽2)11yescharacteristic
170Dic181—41no—
171Dic181—41no—
172Dic181—41no—
173Dic181—41no—
174C2×C181—22no—
175C2×C181—41no—
176C2×C181—81no—
177C2×C181#54611yescharacteristic
178C2×C181C2×#13711yescharacteristic
179C2×C181#57711yescharacteristic
180C181#331611yescharacteristic, complemented, Sylow
181#3316—1811nomaximal, Sylow
182#554—1811no—
183#548D18111yescharacteristic
184#489—1811no—
185#483—1811no—
186#481—1811no—
187#475—1811no—
188#470—1811no—
189#465—1811no—
190#464—1811no—
191#452—1811no—
192#451—1811no—
193#441—1811no—
194#435—1811no—
195#417—1811no—
196C2×#103—1811no—
197C2×#41C2×D18111yescharacteristic
198C22×Q8C2×D18111yescharacteristic
199C2×Q16—1811no—
200C2×Q16—1811no—
201C2×SD16—1811no—
202C2×M16—1811no—
203C22×C8—1811no—
204C4⋊Q8—1811no—
205#125—3621no—
206#124—21no—
207#124C2×D18111yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8—3621no—
210C22⋊Q8—3621no—
211C22⋊Q8—1811no—
212C22⋊Q8—1811no—
213C22⋊Q8C2×D18111yescharacteristic
214#121—21no—
215#121C2×D18111yescharacteristic
216C4×Q8—1811no—
217C4×Q8C2×D18111yescharacteristic
218C4×D4—21no—
219C4×D4—3621no—
220C4×D4C2×D18111yescharacteristic
221C2×C4⋊C4—1811no—
222C2×C4⋊C4—1811no—
223#106—1811no—
224#106—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#102—1811no—
235#102—1811no—
236#102—1811no—
237C22⋊C8—1811no—
238C22⋊C8—1811no—
239#97—1811no—
240#41—22no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—3621no—
246C2×Q8—1811no—
247C2×Q8C22×D18111yescharacteristic
248C2×Q8C22×D18111yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊D18111yescharacteristic
252C2×D4C22×D18111yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3621no—
256C22×C4—3621no—
257C22×C4C22×D18111yescharacteristic
258C22×C4C22×D18111yescharacteristic
259Q16—3622no—
260Q16—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—3621no—
271C4⋊C4—3621no—
272C4⋊C4—3621no—
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—21no—
284C22⋊C4—21no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4C22⋊D18111yescharacteristic
288C22⋊C4C22⋊D18111yescharacteristic
289C42—21no—
290C42—3621no—
291C42C22×D18111yescharacteristic
292C23—21no—
293C23C2×C22⋊D18111yescharacteristic
294C23C2×C22⋊D18111yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—3621no—
300Q8—3621no—
301Q8—14481no—
302Q8#1566844812yes—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—42no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—3621no—
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—41no—
338C22—41no—
339C22—41no—
340C22#1915384612yes—
341C22C22×C22⋊D18111yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—7241no—
350C4—7241no—
351C4—7241no—
352C4—7241no—
353C4C2×#1566844811yescharacteristic
354C4#1915386111yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2#2604192311yescharacteristic
359C2C2×#1915384611yescharacteristic
360C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radical#548×C181C2
socleC22×C181C22×D4
hypercenter#548D181
hypocenterC181#3316
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #26041986C2semidirect
2. #26041638C2semidirect
3. #26041607C2semidirect
4. #26041595C2semidirect
5. #26041564C2semidirect
6. #26041535C2semidirect
7. #26041506C2semidirect
8. #26041498C2semidirect
9. #26041441C2semidirect
10. #26041435C2semidirect
11. #26041384C2semidirect
12. #26041351C2semidirect
13. #26041280C2semidirect
14. C2×#19153740C2semidirect
180. C181#3316semidirect

Maximal subgroups

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

subgroupindex
#260419862
#260416382
#260416072
#260415952
#260415642
#260415352
#260415062
#260414982
#260414412
#260414352
#260413842
#260413512
#260412802
C2×#191537402
#548×C1812
#3316181