Finite Group Databasebeta

← #37545736

Subgroups — #37545736

Order 23168.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D724C211yesmaximal, characteristic
2#26041645C211yesmaximal, characteristic, complemented
3#26041603C211yesmaximal, characteristic, complemented
4#26041592C211yesmaximal, characteristic, complemented
5#26041576C211yesmaximal, characteristic, complemented
6#26041527C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041479C211yesmaximal, characteristic, complemented
9#26041429C211yesmaximal, characteristic, complemented
10#26041429C211yesmaximal, characteristic, complemented
11#26041377C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041283C211yesmaximal, characteristic, complemented
14C2×#19153736C211yesmaximal, characteristic, complemented
15#552×C181C211yesmaximal, characteristic, complemented
16C22×D724C2211yescharacteristic
17C22×D724C2211yescharacteristic
18C2×D4×D181—21no—
19C2×#15668436C2211yescharacteristic
20C2×#15668436C2211yescharacteristic, complemented
21C2×#15668436C2211yescharacteristic, complemented
22C2×#15668433C2211yescharacteristic, complemented
23C22×#13089678C2211yescharacteristic, complemented
24#19153813C2211yescharacteristic
25#19153791—21no—
26#19153790C2211yescharacteristic
27#19153790C2211yescharacteristic
28#19153786—21no—
29#19153786—21no—
30C4×D724C2211yescharacteristic
31#19153749C2211yescharacteristic, complemented
32#19153749C2211yescharacteristic, complemented
33#19153747C2211yescharacteristic, complemented
34C4⋊#13089678C2211yescharacteristic, complemented
35#19153740C2211yescharacteristic, complemented
36#19153736C2211yescharacteristic
37#19153736C2211yescharacteristic
38#19153736C2211yescharacteristic
39#19153736C2211yescharacteristic, complemented
40#19153736C2211yescharacteristic, complemented
41#19153736C2211yescharacteristic, complemented
42#19153736C2211yescharacteristic, complemented
43#19153736C2211yescharacteristic, complemented
44#19153736C2211yescharacteristic, complemented
45C22⋊#13089678C2211yescharacteristic, complemented
46C22⋊#13089678C2211yescharacteristic, complemented
47#19153727C2211yescharacteristic, complemented
48#126×C181—21no—
49#125×C181—21no—
50#125×C181C2211yescharacteristic, complemented
51#123×C181—21no—
52C22⋊Q8×C181C2211yescharacteristic, complemented
53C22⋊Q8×C181C2211yescharacteristic, complemented
54C4×Q8×C181C2211yescharacteristic, complemented
55C4×D4×C181C2211yescharacteristic
56#117×C181—21no—
57#117×C181C2211yescharacteristic, complemented
58C2×C4⋊C4×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60C23×D181—21no—
61C2×D724—22no—
62C2×D724—22no—
63C2×D724—21no—
64C2×D724—21no—
65C2×D724—21no—
66C2×D724—21no—
67C2×D724—41no—
68C2×D724—41no—
69C2×D724C2311yescharacteristic
70C2×D724C2311yescharacteristic
71C2×D724C2311yescharacteristic
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74D4×D181—44no—
75C2×C4×D181—21no—
76#15668436—22no—
77#15668436—41no—
78#15668436—41no—
79#15668433—41no—
80C2×#13089678—41no—
81C2×#13089678C2311yescharacteristic
82C2×#13089678C2311yescharacteristic
83C2×#13089678C2311yescharacteristic
84C2×#13089678C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89C2×Q8×C181C2311yescharacteristic
90C2×D4×C181D411yescharacteristic
91C22×C4×C181—21no—
92C22×C4×C181C2311yescharacteristic
93C22×C4×C181C2311yescharacteristic
94C4⋊C4×C181—21no—
95C4⋊C4×C181—21no—
96C4⋊C4×C181—21no—
97C4⋊C4×C181—21no—
98C4⋊C4×C181—21no—
99C4⋊C4×C181—21no—
100C4⋊C4×C181D411yescharacteristic
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C22⋊C4×C181—21no—
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C22⋊C4×C181D411yescharacteristic
108C22⋊C4×C181D411yescharacteristic
109C42×C181—21no—
110C42×C181—21no—
111C42×C181C2311yescharacteristic
112C22×D181—21no—
113C22×D181—21no—
114C22×D181—21no—
115C22×D181—42no—
116C22×D181—42no—
117C22×D181—42no—
118C22×D181—42no—
119C22×D181—81no—
120C22×D181—81no—
121D724—21no—
122D724—21no—
123D724—21no—
124D724—21no—
125D724—21no—
126D724—42no—
127D724—42no—
128D724—42no—
129D724—81no—
130D724#4112yes—
131C22⋊D181—42no—
132C22⋊D181—42no—
133C2×Dic181—21no—
134C4×D181—42no—
135#13089678—21no—
136#13089678—21no—
137#13089678—21no—
138#13089678—21no—
139C23×C181C2×D411yescharacteristic
140C23×C181C2×D411yescharacteristic
141Q8×C181—21no—
142Q8×C181—21no—
143D4×C181—22no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—42no—
153C2×C4×C181C2×D411yescharacteristic
154C2×C4×C181C2×D411yescharacteristic
155C2×C4×C181C2×D411yescharacteristic
156C2×C4×C181C2×D411yescharacteristic
157C2×C4×C181C2411yescharacteristic
158C22×C181—21no—
159C22×C181—21no—
160C22×C181—21no—
161C22×C181—21no—
162C22×C181—21no—
163C22×C181AGL(1,ℤ/8)12yes—
164C22×C181C22×D411yescharacteristic
165C2×D181—22no—
166C2×D181—41no—
167C2×D181—41no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—41no—
172C2×D181—41no—
173C2×D181—41no—
174C2×D181—84no—
175C2×D181—84no—
176C4×C181—21no—
177C4×C181—21no—
178C4×C181—21no—
179C4×C181—21no—
180C4×C181—41no—
181C4×C181—41no—
182C4×C181—41no—
183C4×C181—41no—
184C4×C181C2×#4111yescharacteristic
185C4×C181H(5,𝔽2)11yescharacteristic
186Dic181—41no—
187C2×C181—22no—
188C2×C181—41no—
189C2×C181#54611yescharacteristic
190C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C181#57711yescharacteristic
192D181—42no—
193D181—81no—
194D181—81no—
195C181#331011yescharacteristic, complemented, Sylow
196#3310—1811nomaximal, Sylow
197#552D18111yescharacteristic, complemented
198D42—1811no—
199#490—1811no—
200#482—1811no—
201#480—1811no—
202#478—1811no—
203#468—1811no—
204#465—1811no—
205#460—1811no—
206#450—1811no—
207#450—1811no—
208#440—1811no—
209#435—1811no—
210#418—1811no—
211C2×#102—1811no—
212C22×D4—3621no—
213C22×D4—1811no—
214C22×D4—1811no—
215C2×SD16—1811no—
216C2×SD16—1811no—
217C2×SD16—1811no—
218C2×M16—1811no—
219C22×C8—1811no—
220#127—1811no—
221#126—21no—
222#125—21no—
223#125C2×D18111yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D18111yescharacteristic, complemented
226C22⋊Q8C2×D18111yescharacteristic, complemented
227#121—3621no—
228#121—1811no—
229#121—1811no—
230#120—3621no—
231#120—3621no—
232C4×Q8C2×D18111yescharacteristic, complemented
233C4×D4—1811no—
234C4×D4C2×D18111yescharacteristic
235#117—21no—
236#117C2×D18111yescharacteristic, complemented
237C2×C4⋊C4C2×D18111yescharacteristic, complemented
238#107—1811no—
239#107—1811no—
240#106—1811no—
241C4⋊C8—1811no—
242#103—1811no—
243#102—1811no—
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102—1811no—
248#102—1811no—
249#102—1811no—
250#102—1811no—
251#102—1811no—
252C22⋊C8—1811no—
253C22⋊C8—1811no—
254#97—1811no—
255C24—3621no—
256C24—3621no—
257C2×Q8C22×D18111yescharacteristic
258C2×D4—3622no—
259C2×D4—3622no—
260C2×D4—3621no—
261C2×D4—3621no—
262C2×D4—3621no—
263C2×D4—3621no—
264C2×D4—3621no—
265C2×D4—3621no—
266C2×D4—1811no—
267C2×D4—1811no—
268C2×D4—1811no—
269C2×D4—7241no—
270C2×D4—7241no—
271C2×D4—7244no—
272C2×D4C22⋊D18111yescharacteristic
273C22×C4—21no—
274C22×C4—3621no—
275C22×C4C22×D18111yescharacteristic
276C22×C4C22×D18111yescharacteristic
277SD16—3622no—
278SD16—7241no—
279SD16—7241no—
280M16—7241no—
281C2×C8—1811no—
282C2×C8—1811no—
283C2×C8—1811no—
284C2×C8—1811no—
285C2×C8—7241no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—1811no—
293C4⋊C4C22⋊D18111yescharacteristic
294C4⋊C4C22×D18111yescharacteristic
295C4⋊C4C22×D18111yescharacteristic
296C4⋊C4C22×D18111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3621no—
301C22⋊C4—3621no—
302C22⋊C4—3621no—
303C22⋊C4C22⋊D18111yescharacteristic
304C22⋊C4C22⋊D18111yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D18111yescharacteristic
308C23—3621no—
309C23—3621no—
310C23—3621no—
311C23—14481no—
312C23—14481no—
313C23—7242no—
314C23—7242no—
315C23—7242no—
316C23—7242no—
317C23C2×C22⋊D18111yescharacteristic
318C23C2×C22⋊D18111yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—3621no—
323D4—3621no—
324D4—3621no—
325D4—3621no—
326D4—3621no—
327D4—14481no—
328D4—1812no—
329D4—7242no—
330D4—7242no—
331D4—7242no—
332D4—7242no—
333D4—7242no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3621no—
344C2×C4—7242no—
345C2×C4C2×C22⋊D18111yescharacteristic
346C2×C4C2×C22⋊D18111yescharacteristic
347C2×C4C2×C22⋊D18111yescharacteristic
348C2×C4C2×C22⋊D18111yescharacteristic
349C2×C4C23×D18111yescharacteristic
350C8—3621no—
351C8—3621no—
352C8—3621no—
353C8—3621no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—3622no—
360C22—14484no—
361C22—14484no—
362C22—7241no—
363C22—7241no—
364C22—7241no—
365C22—7241no—
366C22—7241no—
367C22—7241no—
368C22—7241no—
369C22—7241no—
370C22#1915384112yes—
371C22C22×C22⋊D18111yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—7241no—
381C4#41×D18111yescharacteristic
382C4#1915386211yescharacteristic
383C2—22no—
384C2—41no—
385C2—14481no—
386C2—14481no—
387C2—7242no—
388C2#2604191711yescharacteristic
389C2C2×#1915384111yescharacteristic
390C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#552×C181C2
socleC22×C181C22×D4
prime radicalD4×D724C2
hypercenter#552D181
hypocenterC181#3310
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #26041645C2semidirect
3. #26041603C2semidirect
4. #26041592C2semidirect
5. #26041576C2semidirect
6. #26041527C2semidirect
7. #26041510C2semidirect
8. #26041479C2semidirect
9. #26041429C2semidirect
10. #26041429C2semidirect
11. #26041377C2semidirect
12. #26041351C2semidirect
13. #26041283C2semidirect
14. C2×#19153736C2semidirect
15. #552×C181C2semidirect
20. C2×#15668436C22semidirect
21. C2×#15668436C22semidirect
22. C2×#15668433C22semidirect
23. C22×#13089678C22semidirect
31. #19153749C22semidirect
32. #19153749C22semidirect
33. #19153747C22semidirect
34. C4⋊#13089678C22semidirect
35. #19153740C22semidirect
39. #19153736C22semidirect
40. #19153736C22semidirect
41. #19153736C22semidirect
42. #19153736C22semidirect
43. #19153736C22semidirect
44. #19153736C22semidirect
45. C22⋊#13089678C22semidirect
46. C22⋊#13089678C22semidirect
47. #19153727C22semidirect
50. #125×C181C22semidirect
52. C22⋊Q8×C181C22semidirect
53. C22⋊Q8×C181C22semidirect
54. C4×Q8×C181C22semidirect
57. #117×C181C22semidirect
58. C2×C4⋊C4×C181C22semidirect
195. C181#3310semidirect
197. #552D181semidirect
223. #125C2×D181semidirect
225. C22⋊Q8C2×D181semidirect
226. C22⋊Q8C2×D181semidirect
232. C4×Q8C2×D181semidirect
236. #117C2×D181semidirect
237. C2×C4⋊C4C2×D181semidirect

Maximal subgroups

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

subgroupindex
D4×D7242
#260416452
#260416032
#260415922
#260415762
#260415272
#260415102
#260414792
#260414292
#260413772
#260413512
#260412832
C2×#191537362
#552×C1812
#3310181