Finite Group Databasebeta

← #37545287

Subgroups — #37545287

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×#19153846C211yesmaximal, characteristic, complemented
2C22×#15668439C211yesmaximal, characteristic, complemented
3D4×Dic362C211yesmaximal, characteristic, complemented
4#26041674C211yesmaximal, characteristic, complemented
5#26041535C211yesmaximal, characteristic, complemented
6#26041506C211yesmaximal, characteristic, complemented
7#26041498C211yesmaximal, characteristic, complemented
8#26041489C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041435C211yesmaximal, characteristic, complemented
11#26041434C211yesmaximal, characteristic, complemented
12#26041417C211yesmaximal, characteristic, complemented
13#26041384C211yesmaximal, characteristic, complemented
14#26041351C211yesmaximal, characteristic, complemented
15#548×C181C211yesmaximal, characteristic
16C22×Dic362C2211yescharacteristic
17C22×Dic362C2211yescharacteristic
18#19153846—24no—
19C2×#15668439—24no—
20C2×#15668439C2211yescharacteristic
21C2×#15668439C2211yescharacteristic
22C2×#15668439C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668439C2211yescharacteristic
25C2×#15668439C2211yescharacteristic
26C2×#15668437C2211yescharacteristic
27C2×#15668437C2211yescharacteristic
28C2×#15668437C2211yescharacteristic
29C2×#15668433C2211yescharacteristic
30C22×#13089678C2211yescharacteristic
31#19153816C2211yescharacteristic
32#19153815—21no—
33#19153795C2211yescharacteristic
34#19153795C2211yescharacteristic
35#19153793—21no—
36#19153793—21no—
37C4×Dic362C2211yescharacteristic
38D4×Dic181—21no—
39#19153747C2211yescharacteristic
40C4⋊#13089678C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153739C2211yescharacteristic
44#19153739C2211yescharacteristic
45#19153739C2211yescharacteristic
46#19153737C2211yescharacteristic
47C22⋊#13089678C2211yescharacteristic
48C22⋊#13089678C2211yescharacteristic
49#19153727C2211yescharacteristic
50C2×#41×C181C2211yescharacteristic
51C22×Q8×C181C2211yescharacteristic
52#124×C181—21no—
53#124×C181C2211yescharacteristic
54C22⋊Q8×C181—21no—
55C22⋊Q8×C181C2211yescharacteristic
56#121×C181—21no—
57#121×C181C2211yescharacteristic
58C4×Q8×C181C2211yescharacteristic
59C4×D4×C181—21no—
60C4×D4×C181C2211yescharacteristic
61C2×Dic362—22no—
62C2×Dic362—22no—
63C2×Dic362—21no—
64C2×Dic362—21no—
65C2×Dic362—41no—
66C2×Dic362—41no—
67C2×Dic362C2311yescharacteristic
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70C22×Dic181—21no—
71C22×Dic181—21no—
72#15668439—22no—
73#15668439—22no—
74#15668439—22no—
75#15668439—22no—
76#15668439—41no—
77#15668439D414yescomplemented
78#15668437—22no—
79#15668437—22no—
80#15668437—41no—
81#15668433—22no—
82C2×#13089678—22no—
83C2×#13089678C2311yescharacteristic
84C2×#13089678C2311yescharacteristic
85C2×#13089678C2311yescharacteristic
86C2×#13089678C2311yescharacteristic
87C4⋊Dic181—21no—
88C4⋊Dic181—21no—
89C4⋊Dic181C2311yescharacteristic
90#15668428—21no—
91#15668428—21no—
92#15668428—21no—
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95C4×Dic181—21no—
96#41×C181—22no—
97#41×C181—41no—
98C2×Q8×C181—21no—
99C2×Q8×C181—22no—
100C2×Q8×C181—41no—
101C2×Q8×C181C2311yescharacteristic
102C2×Q8×C181C2311yescharacteristic
103C2×D4×C181—21no—
104C2×D4×C181—21no—
105C2×D4×C181D411yescharacteristic
106C2×D4×C181C2311yescharacteristic
107C22×C4×C181—21no—
108C22×C4×C181—21no—
109C22×C4×C181C2311yescharacteristic
110C22×C4×C181C2311yescharacteristic
111C4⋊C4×C181—21no—
112C4⋊C4×C181—21no—
113C4⋊C4×C181D411yescharacteristic
114C4⋊C4×C181C2311yescharacteristic
115C22⋊C4×C181—21no—
116C22⋊C4×C181—21no—
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C22⋊C4×C181D411yescharacteristic
120C22⋊C4×C181D411yescharacteristic
121C42×C181—21no—
122C42×C181C2311yescharacteristic
123Dic362—21no—
124Dic362—21no—
125Dic362—21no—
126Dic362—21no—
127Dic362—21no—
128Dic362—42no—
129Dic362—42no—
130Dic362C2×D412yes—
131C2×Dic181—21no—
132C2×Dic181—21no—
133C2×Dic181—21no—
134C2×Dic181—21no—
135C2×Dic181—42no—
136C2×Dic181—42no—
137#13089678—21no—
138#13089678—21no—
139#13089678—21no—
140#13089678C2×D412yes—
141C23×C181—21no—
142C23×C181C2×D411yescharacteristic
143C23×C181C2×D411yescharacteristic
144Q8×C181—21no—
145Q8×C181—21no—
146Q8×C181—21no—
147Q8×C181—42no—
148Q8×C181C2×D412yes—
149D4×C181—22no—
150D4×C181—21no—
151D4×C181—21no—
152D4×C181—42no—
153D4×C181—42no—
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—41no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181#13712yescomplemented
177C22×C181C22×D411yescharacteristic
178C4×C181—21no—
179C4×C181—21no—
180C4×C181—22no—
181C4×C181—21no—
182C4×C181—21no—
183C4×C181—41no—
184C4×C181—41no—
185C4×C181C22×D411yescharacteristic
186C4×C181H(5,𝔽2)11yescharacteristic
187Dic181—22no—
188Dic181—41no—
189Dic181—41no—
190Dic181—41no—
191C2×C181—22no—
192C2×C181—41no—
193C2×C181—81no—
194C2×C181D4211yescharacteristic
195C2×C181C2×#13711yescharacteristic
196C2×C181#57811yescharacteristic
197C181#327011yescharacteristic, complemented, Sylow
198#3270—1811nomaximal, Sylow
199C2×#137—1811no—
200C22×Q16—1811no—
201D4×Q8—1811no—
202#548D18111yescharacteristic
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×#41C2×D18111yescharacteristic
215C22×Q8—1811no—
216C22×Q8—1811no—
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—3621no—
232C4⋊Q8—1811no—
233#124—21no—
234#124C2×D18111yescharacteristic
235C22⋊Q8—21no—
236C22⋊Q8—3621no—
237C22⋊Q8—3621no—
238C22⋊Q8—1811no—
239C22⋊Q8—1811no—
240C22⋊Q8C2×D18111yescharacteristic
241#121—21no—
242#121C2×D18111yescharacteristic
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—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—3622no—
265C2×Q8—3622no—
266C2×Q8—3621no—
267C2×Q8—3621no—
268C2×Q8—1811no—
269C2×Q8—1811no—
270C2×Q8—1811no—
271C2×Q8—7241no—
272C2×Q8—7241no—
273C2×Q8C22×D18111yescharacteristic
274C2×Q8C22×D18111yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C22⋊D18111yescharacteristic
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—3621no—
303C4⋊C4—3621no—
304C4⋊C4—3621no—
305C4⋊C4—3621no—
306C4⋊C4—3621no—
307C4⋊C4—1811no—
308C4⋊C4C22⋊D18111yescharacteristic
309C4⋊C4C22×D18111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C22⋊C4C22⋊D18111yescharacteristic
317C22⋊C4C22⋊D18111yescharacteristic
318C42—21no—
319C42—3621no—
320C42C22×D18111yescharacteristic
321C23—21no—
322C23C2×C22⋊D18111yescharacteristic
323C23C2×C22⋊D18111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—3621no—
329Q8—3621no—
330Q8—3621no—
331Q8—3621no—
332Q8—3621no—
333Q8—1812no—
334Q8—7242no—
335Q8—7242no—
336Q8D4×D18112yes—
337D4—22no—
338D4—21no—
339D4—21no—
340D4—42no—
341D4—42no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—3621no—
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—41no—
372C22—41no—
373C22—41no—
374C22#1915384612yescomplemented
375C22C22×C22⋊D18111yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—3622no—
384C4—7241no—
385C4—7241no—
386C4—7241no—
387C4C2×D4×D18111yescharacteristic
388C4#1915386111yescharacteristic
389C2—22no—
390C2—41no—
391C2—81no—
392C2D4×C22⋊D18111yescharacteristic
393C2C2×#1915384611yescharacteristic
394C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radical#548×C181C2
socleC22×C181C22×D4
hypercenter#548D181
hypocenterC181#3270
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. C2×#19153846C2semidirect
2. C22×#15668439C2semidirect
3. D4×Dic362C2semidirect
4. #26041674C2semidirect
5. #26041535C2semidirect
6. #26041506C2semidirect
7. #26041498C2semidirect
8. #26041489C2semidirect
9. #26041441C2semidirect
10. #26041435C2semidirect
11. #26041434C2semidirect
12. #26041417C2semidirect
13. #26041384C2semidirect
14. #26041351C2semidirect
77. #15668439D4semidirect
176. C22×C181#137semidirect
197. C181#3270semidirect
374. C22#19153846semidirect

Maximal subgroups

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

subgroupindex
C2×#191538462
C22×#156684392
D4×Dic3622
#260416742
#260415352
#260415062
#260414982
#260414892
#260414412
#260414352
#260414342
#260414172
#260413842
#260413512
#548×C1812
#3270181