Finite Group Databasebeta

← #37545292

Subgroups — #37545292

Order 23168.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153847C211yesmaximal, characteristic, complemented
2C22×#15668439C211yesmaximal, characteristic, complemented
3#26041938C211yesmaximal, characteristic
4#26041671C211yesmaximal, characteristic, complemented
5#26041535C211yesmaximal, characteristic, complemented
6#26041510C211yesmaximal, characteristic, complemented
7#26041501C211yesmaximal, characteristic, complemented
8#26041486C211yesmaximal, characteristic, complemented
9#26041439C211yesmaximal, characteristic, complemented
10#26041435C211yesmaximal, characteristic, complemented
11#26041434C211yesmaximal, characteristic, complemented
12#26041418C211yesmaximal, characteristic, complemented
13#26041384C211yesmaximal, characteristic, complemented
14#26041351C211yesmaximal, characteristic, complemented
15D4×Q8×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×Dic362C2211yescharacteristic
18#19153847—24no—
19C2×#15668439—24no—
20C2×#15668439C2211yescharacteristic
21C2×#15668439C2211yescharacteristic
22C2×#15668439C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668439C2211yescharacteristic
25C2×#15668439C2211yescharacteristic
26C2×#15668436C2211yescharacteristic
27C2×#15668436C2211yescharacteristic
28C2×#15668436C2211yescharacteristic
29C2×#15668433C2211yescharacteristic
30C22×#13089678C2211yescharacteristic
31#19153804C2211yescharacteristic
32#19153802—21no—
33#19153795C2211yescharacteristic
34#19153794—21no—
35#19153790C2211yescharacteristic
36#19153788—21no—
37C4×Dic362C2211yescharacteristic
38#19153776—21no—
39#19153747C2211yescharacteristic
40C4⋊#13089678C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153739C2211yescharacteristic
45#19153739C2211yescharacteristic
46#19153736C2211yescharacteristic
47C22⋊#13089678C2211yescharacteristic
48C22⋊#13089678C2211yescharacteristic
49#19153727C2211yescharacteristic
50C22×Q8×C181C2211yescharacteristic
51C22×Q8×C181C2211yescharacteristic
52C4⋊Q8×C181—21no—
53C4⋊Q8×C181C2211yescharacteristic
54C22⋊Q8×C181—21no—
55C22⋊Q8×C181—21no—
56C22⋊Q8×C181C2211yescharacteristic
57C22⋊Q8×C181C2211yescharacteristic
58C4×Q8×C181C2211yescharacteristic
59C4×D4×C181—21no—
60C4×D4×C181C2211yescharacteristic
61#15668449—22no—
62#15668449—41no—
63C2×Dic362—22no—
64C2×Dic362—21no—
65C2×Dic362—41no—
66C2×Dic362C2311yescharacteristic
67C2×Dic362C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C22×Dic181—21no—
72C2×C4×D181—21no—
73#15668439—22no—
74#15668439—22no—
75#15668439—22no—
76#15668439—22no—
77#15668439—41no—
78#15668439D414yescomplemented
79#15668436—22no—
80#15668436—22no—
81#15668436—41no—
82#15668433—22no—
83C2×#13089678—22no—
84C2×#13089678C2311yescharacteristic
85C2×#13089678C2311yescharacteristic
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C4⋊Dic181C2311yescharacteristic
89#15668428—21no—
90#15668428—21no—
91C22⋊Dic181—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95C4×Dic181—21no—
96C2×Q8×C181—21no—
97C2×Q8×C181—21no—
98C2×Q8×C181—22no—
99C2×Q8×C181—22no—
100C2×Q8×C181—41no—
101C2×Q8×C181—41no—
102C2×Q8×C181C2311yescharacteristic
103C2×Q8×C181C2311yescharacteristic
104C2×Q8×C181C2311yescharacteristic
105C2×D4×C181D411yescharacteristic
106C22×C4×C181—21no—
107C22×C4×C181—21no—
108C22×C4×C181C2311yescharacteristic
109C22×C4×C181C2311yescharacteristic
110C4⋊C4×C181—21no—
111C4⋊C4×C181—21no—
112C4⋊C4×C181—21no—
113C4⋊C4×C181—21no—
114C4⋊C4×C181—21no—
115C4⋊C4×C181D411yescharacteristic
116C4⋊C4×C181C2311yescharacteristic
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C22⋊C4×C181D411yescharacteristic
120C22⋊C4×C181D411yescharacteristic
121C42×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124Dic362—21no—
125Dic362—21no—
126Dic362—21no—
127Dic362—42no—
128Dic362C2×D412yes—
129D724—21no—
130D724—21no—
131C22⋊D181—42no—
132C22⋊D181—42no—
133C2×Dic181—21no—
134C2×Dic181—21no—
135C2×Dic181—21no—
136C2×Dic181—42no—
137C4×D181—42no—
138#13089678—21no—
139#13089678—21no—
140#13089678—21no—
141#13089678C2×D412yes—
142C23×C181C2×D411yescharacteristic
143C23×C181C2×D411yescharacteristic
144Q8×C181—21no—
145Q8×C181—21no—
146Q8×C181—21no—
147Q8×C181—21no—
148Q8×C181—21no—
149Q8×C181—42no—
150Q8×C181—42no—
151Q8×C181C2×D412yes—
152D4×C181—22no—
153C2×C4×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181—21no—
157C2×C4×C181—21no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—42no—
162C2×C4×C181—42no—
163C2×C4×C181C2×D411yescharacteristic
164C2×C4×C181C2×D411yescharacteristic
165C2×C4×C181C2×D411yescharacteristic
166C2×C4×C181C2×D411yescharacteristic
167C2×C4×C181C2411yescharacteristic
168C22×C181—21no—
169C22×C181—21no—
170C22×C181—21no—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181#13712yescomplemented
174C22×C181C22×D411yescharacteristic
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—41no—
178C4×C181—21no—
179C4×C181—21no—
180C4×C181—22no—
181C4×C181—21no—
182C4×C181—21no—
183C4×C181—41no—
184C4×C181—41no—
185C4×C181—41no—
186C4×C181C22×D411yescharacteristic
187C4×C181H(5,𝔽2)11yescharacteristic
188Dic181—22no—
189Dic181—41no—
190Dic181—41no—
191C2×C181—22no—
192C2×C181—41no—
193C2×C181D4211yescharacteristic
194C2×C181C2×#13711yescharacteristic
195C2×C181#57811yescharacteristic
196D181—81no—
197C181#327011yescharacteristic, complemented, Sylow
198#3270—1811nomaximal, Sylow
199C2×#137—1811no—
200C22×Q16—1811no—
201D4×Q8D18111yescharacteristic, complemented
202#548—1811no—
203#497—1811no—
204#470—1811no—
205#465—1811no—
206#464—1811no—
207#462—1811no—
208#452—1811no—
209#451—1811no—
210#451—1811no—
211#448—1811no—
212#441—1811no—
213#435—1811no—
214C2×#41—1811no—
215C22×Q8—1811no—
216C22×Q8C2×D18111yescharacteristic
217C22×Q8C2×D18111yescharacteristic
218#137—3624no—
219C2×Q16—3624no—
220C2×Q16—1811no—
221C2×Q16—1811no—
222C2×Q16—1811no—
223C2×Q16—1811no—
224C2×Q16—1811no—
225C2×Q16—1811no—
226C2×SD16—1811no—
227C2×SD16—1811no—
228C2×SD16—1811no—
229C2×M16—1811no—
230C22×C8—1811no—
231C4⋊Q8—21no—
232C4⋊Q8C2×D18111yescharacteristic
233#124—3621no—
234#124—1811no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—3621no—
238C22⋊Q8—1811no—
239C22⋊Q8C2×D18111yescharacteristic
240C22⋊Q8C2×D18111yescharacteristic
241#121—3621no—
242#121—1811no—
243C4×Q8—1811no—
244C4×Q8C2×D18111yescharacteristic
245C4×D4—21no—
246C4×D4—3621no—
247C4×D4C2×D18111yescharacteristic
248#106—1811no—
249C4⋊C8—1811no—
250#103—1811no—
251#103—1811no—
252#103—1811no—
253#103—1811no—
254#103—1811no—
255#102—1811no—
256C22⋊C8—1811no—
257C22⋊C8—1811no—
258#97—1811no—
259#41—3622no—
260#41—7241no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—3622no—
268C2×Q8—3621no—
269C2×Q8—1811no—
270C2×Q8—1811no—
271C2×Q8—7241no—
272C2×Q8C22×D18111yescharacteristic
273C2×Q8C22×D18111yescharacteristic
274C2×Q8C22×D18111yescharacteristic
275C2×D4—3621no—
276C2×D4—3621no—
277C2×D4—1811no—
278C2×D4C22⋊D18111yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3621no—
282C22×C4—3621no—
283C22×C4C22×D18111yescharacteristic
284C22×C4C22×D18111yescharacteristic
285Q16—3622no—
286Q16—3622no—
287Q16—3622no—
288Q16—3622no—
289Q16—1814no—
290Q16—7241no—
291SD16—3622no—
292SD16—3622no—
293SD16—7241no—
294M16—3622no—
295C2×C8—3622no—
296C2×C8—1811no—
297C2×C8—1811no—
298C2×C8—1811no—
299C2×C8—1811no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—3621no—
306C4⋊C4—3621no—
307C4⋊C4—1811no—
308C4⋊C4C22⋊D18111yescharacteristic
309C4⋊C4C22×D18111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3621no—
313C22⋊C4—3621no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C22⋊C4C22⋊D18111yescharacteristic
317C22⋊C4C22⋊D18111yescharacteristic
318C42—21no—
319C42—3621no—
320C42C22×D18111yescharacteristic
321C23—3621no—
322C23C2×C22⋊D18111yescharacteristic
323C23C2×C22⋊D18111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—42no—
331Q8—3621no—
332Q8—3621no—
333Q8—3621no—
334Q8—1812no—
335Q8—7242no—
336Q8D4×D18112yes—
337D4—22no—
338D4—3621no—
339D4—3621no—
340D4—7242no—
341D4—7242no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—3621no—
353C2×C4—3621no—
354C2×C4—3621no—
355C2×C4—7242no—
356C2×C4—7242no—
357C2×C4C2×C22⋊D18111yescharacteristic
358C2×C4C2×C22⋊D18111yescharacteristic
359C2×C4C2×C22⋊D18111yescharacteristic
360C2×C4C2×C22⋊D18111yescharacteristic
361C2×C4C23×D18111yescharacteristic
362C8—3621no—
363C8—3621no—
364C8—3621no—
365C8—1812no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—7241no—
372C22—7241no—
373C22—7241no—
374C22#1915384712yescomplemented
375C22C22×C22⋊D18111yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3622no—
385C4—7241no—
386C4—7241no—
387C4C2×D4×D18111yescharacteristic
388C4#1915386111yescharacteristic
389C2—22no—
390C2—41no—
391C2—14481no—
392C2D4×C22⋊D18111yescharacteristic
393C2C2×#1915384711yescharacteristic
394C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingD4×Q8×C181C2
socleC22×C181C22×D4
prime radical#26041938C2
hypercenterD4×Q8D181
hypocenterC181#3270
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
1. C2×#19153847C2semidirect
2. C22×#15668439C2semidirect
4. #26041671C2semidirect
5. #26041535C2semidirect
6. #26041510C2semidirect
7. #26041501C2semidirect
8. #26041486C2semidirect
9. #26041439C2semidirect
10. #26041435C2semidirect
11. #26041434C2semidirect
12. #26041418C2semidirect
13. #26041384C2semidirect
14. #26041351C2semidirect
15. D4×Q8×C181C2semidirect
78. #15668439D4semidirect
173. C22×C181#137semidirect
197. C181#3270semidirect
201. D4×Q8D181semidirect
374. C22#19153847semidirect

Maximal subgroups

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

subgroupindex
C2×#191538472
C22×#156684392
#260419382
#260416712
#260415352
#260415102
#260415012
#260414862
#260414392
#260414352
#260414342
#260414182
#260413842
#260413512
D4×Q8×C1812
#3270181