Finite Group Databasebeta

← #37545623

Subgroups — #37545623

Order 23168.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041975C211yesmaximal, characteristic, complemented
2#26041633C211yesmaximal, characteristic, complemented
3#26041617C211yesmaximal, characteristic, complemented
4#26041595C211yesmaximal, characteristic, complemented
5#26041586C211yesmaximal, characteristic, complemented
6#26041513C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041491C211yesmaximal, characteristic, complemented
9#26041424C211yesmaximal, characteristic, complemented
10#26041418C211yesmaximal, characteristic, complemented
11C4×#15668436C211yesmaximal, characteristic, complemented
12D4×#13089678C211yesmaximal, characteristic, complemented
13#26041275C211yesmaximal, characteristic, complemented
14C2×#19153740C211yesmaximal, characteristic, complemented
15#548×C181C211yesmaximal, characteristic, complemented
16C2×#15668436C2211yescharacteristic
17C2×#15668436C2211yescharacteristic
18C2×#15668434C2211yescharacteristic
19C22×#13089678C2211yescharacteristic
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#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35#19153740C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153737C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153736C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47C4×#13089678C2211yescharacteristic
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×D724C2311yescharacteristic
60C2×C22⋊D181—21no—
61C2×C22⋊D181—21no—
62C22×Dic181—21no—
63C2×C4×D181—21no—
64#15668436—22no—
65#15668436—41no—
66#15668434—41no—
67C2×#13089678—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—
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
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×C181—21no—
126C23×C181C2×D411yescharacteristic
127C23×C181C2×D411yescharacteristic
128Q8×C181—21no—
129Q8×C181—21no—
130Q8×C181—21no—
131Q8×C181—42no—
132Q8×C181#4112yes—
133D4×C181—22no—
134D4×C181—21no—
135D4×C181—21no—
136D4×C181—42no—
137D4×C181—42no—
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—21no—
145C2×C4×C181—42no—
146C2×C4×C181—42no—
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2×D411yescharacteristic
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2411yescharacteristic
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181#13512yes—
161C22×C181C22×D411yescharacteristic
162C2×D181—41no—
163C2×D181—41no—
164C2×D181—41no—
165C4×C181—21no—
166C4×C181—21no—
167C4×C181—22no—
168C4×C181—21no—
169C4×C181—21no—
170C4×C181—41no—
171C4×C181—41no—
172C4×C181C2×#4111yescharacteristic
173C4×C181H(5,𝔽2)11yescharacteristic
174Dic181—41no—
175Dic181—41no—
176Dic181—41no—
177C2×C181—22no—
178C2×C181—41no—
179C2×C181—81no—
180C2×C181#54611yescharacteristic
181C2×C181C2×#13511yescharacteristic
182C2×C181#57711yescharacteristic
183D181—81no—
184C181#330111yescharacteristic, complemented, Sylow
185#3301—1811nomaximal, Sylow
186#553—1811no—
187#548D18111yescharacteristic, complemented
188#488—1811no—
189#485—1811no—
190#481—1811no—
191#479—1811no—
192#466—1811no—
193#465—1811no—
194#463—1811no—
195#449—1811no—
196#448—1811no—
197C4×SD16—1811no—
198C8×D4—1811no—
199#416—1811no—
200C2×#103—1811no—
201C2×#41C2×D18111yescharacteristic
202C22×Q8C2×D18111yescharacteristic
203C2×SD16—1811no—
204C2×SD16—1811no—
205C2×D8—1811no—
206C22×C8—1811no—
207C22×C8—1811no—
208#126—3621no—
209#125—1811no—
210#124—21no—
211#124C2×D18111yescharacteristic
212#123—3621no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D18111yescharacteristic
215#121—21no—
216#121—3621no—
217#121—1811no—
218#121—1811no—
219#121C2×D18111yescharacteristic
220C4×Q8C2×D18111yescharacteristic
221C4×D4—21no—
222C4×D4—3621no—
223C4×D4—1811no—
224C4×D4C2×D18111yescharacteristic
225#117—1811no—
226C2×C4⋊C4—1811no—
227#107—1811no—
228#106—1811no—
229#106—1811no—
230C4⋊C8—1811no—
231#103—1811no—
232#103—1811no—
233#103—1811no—
234#103—1811no—
235#103—1811no—
236#103—1811no—
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102—1811no—
241C22⋊C8—1811no—
242C22⋊C8—1811no—
243C4×C8—1811no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D18111yescharacteristic
250C2×Q8C22×D18111yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—3621no—
254C2×D4—3621no—
255C2×D4—1811no—
256C2×D4C22⋊D18111yescharacteristic
257C2×D4C22×D18111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3621no—
261C22×C4—3621no—
262C22×C4C22×D18111yescharacteristic
263C22×C4C22×D18111yescharacteristic
264SD16—3622no—
265SD16—7241no—
266D8—7241no—
267C2×C8—1811no—
268C2×C8—1811no—
269C2×C8—1811no—
270C2×C8—1811no—
271C2×C8—7241no—
272C2×C8—7241no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—3621no—
276C4⋊C4—3621no—
277C4⋊C4—3621no—
278C4⋊C4—1811no—
279C4⋊C4—1811no—
280C4⋊C4—1811no—
281C4⋊C4C22⋊D18111yescharacteristic
282C4⋊C4C22×D18111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4C22⋊D18111yescharacteristic
292C22⋊C4C22⋊D18111yescharacteristic
293C42—21no—
294C42—3621no—
295C42C22×D18111yescharacteristic
296C23—21no—
297C23—3621no—
298C23C2×C22⋊D18111yescharacteristic
299C23C2×C22⋊D18111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1566844812yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—3621no—
311D4—3621no—
312D4—14481no—
313D4—7242no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—14481no—
327C2×C4—14481no—
328C2×C4C2×C22⋊D18111yescharacteristic
329C2×C4C2×C22⋊D18111yescharacteristic
330C2×C4C2×C22⋊D18111yescharacteristic
331C2×C4C2×C22⋊D18111yescharacteristic
332C2×C4C23×D18111yescharacteristic
333C8—3621no—
334C8—3621no—
335C8—3621no—
336C8—3621no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—7241no—
346C22—7241no—
347C22—7241no—
348C22#1915383612yes—
349C22C22×C22⋊D18111yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—7241no—
358C4—7241no—
359C4—7241no—
360C4C2×#1566844811yescharacteristic
361C4#1915386111yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—14481no—
366C2#2604192311yescharacteristic
367C2C2×#1915383611yescharacteristic
368C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#548×C181C2
socleC22×C181C22×D4
hypercenter#548D181
hypocenterC181#3301
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. #26041975C2semidirect
2. #26041633C2semidirect
3. #26041617C2semidirect
4. #26041595C2semidirect
5. #26041586C2semidirect
6. #26041513C2semidirect
7. #26041510C2semidirect
8. #26041491C2semidirect
9. #26041424C2semidirect
10. #26041418C2semidirect
11. C4×#15668436C2semidirect
12. D4×#13089678C2semidirect
13. #26041275C2semidirect
14. C2×#19153740C2semidirect
15. #548×C181C2semidirect
184. C181#3301semidirect
187. #548D181semidirect

Maximal subgroups

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

subgroupindex
#260419752
#260416332
#260416172
#260415952
#260415862
#260415132
#260415102
#260414912
#260414242
#260414182
C4×#156684362
D4×#130896782
#260412752
C2×#191537402
#548×C1812
#3301181