Finite Group Databasebeta

← #37545293

Subgroups — #37545293

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×#19153848C211yesmaximal, characteristic, complemented
2C2×Q16×D181C211yesmaximal, characteristic, complemented
3Q8×D724C211yesmaximal, characteristic, complemented
4#26041941C211yesmaximal, characteristic
5#26041673C211yesmaximal, characteristic, complemented
6#26041538C211yesmaximal, characteristic, complemented
7#26041509C211yesmaximal, characteristic, complemented
8#26041501C211yesmaximal, characteristic, complemented
9#26041489C211yesmaximal, characteristic, complemented
10#26041444C211yesmaximal, characteristic, complemented
11#26041436C211yesmaximal, characteristic, complemented
12#26041436C211yesmaximal, characteristic, complemented
13#26041419C211yesmaximal, characteristic, complemented
14#26041352C211yesmaximal, characteristic, complemented
15#441×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic
19C2×Q8×D181C2211yescharacteristic
20#19153848—24no—
21C2×Dic724C2211yescharacteristic
22C2×Dic724C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668439C2211yescharacteristic
25C2×#15668439C2211yescharacteristic
26Q16×D181—24no—
27C2×#15668438C2211yescharacteristic
28C2×#15668438C2211yescharacteristic
29C2×#15668436C2211yescharacteristic
30C2×#15668432C2211yescharacteristic
31C2×C8×D181C2211yescharacteristic
32#19153816—21no—
33#19153816C2211yescharacteristic
34#19153804—21no—
35#19153804C2211yescharacteristic
36#19153796—21no—
37#19153796—21no—
38#19153796—21no—
39#19153796C2211yescharacteristic
40#19153796C2211yescharacteristic
41#19153796C2211yescharacteristic
42#19153791—21no—
43#19153791C2211yescharacteristic
44C4×D724—21no—
45C4×D724—21no—
46C4×D724C2211yescharacteristic
47C4⋊#13089678C2211yescharacteristic
48#19153739C2211yescharacteristic
49#19153739C2211yescharacteristic
50#19153739C2211yescharacteristic
51#19153736C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153728C2211yescharacteristic
54C2×Q16×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57#106×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59#103×C181C2211yescharacteristic
60#97×C181C2211yescharacteristic
61#15668450—22no—
62#15668450—41no—
63C2×Dic362—21no—
64C2×Dic362—21no—
65C2×Dic362—21no—
66C2×Dic362C2311yescharacteristic
67C2×Dic362C2311yescharacteristic
68C2×Dic362C2311yescharacteristic
69Q8×D181—22no—
70Q8×D181—22no—
71Q8×D181—22no—
72Q8×D181—41no—
73Q8×D181—41no—
74Q8×D181—41no—
75C2×D724—21no—
76C2×D724—21no—
77C2×D724D411yescharacteristic
78C2×D724C2311yescharacteristic
79C2×C4×D181—21no—
80C2×C4×D181—21no—
81C2×C4×D181—21no—
82C2×C4×D181—21no—
83C2×C4×D181C2311yescharacteristic
84C2×C4×D181C2311yescharacteristic
85Dic724—22no—
86Dic724—41no—
87#15668439—22no—
88#15668439—22no—
89#15668439—22no—
90#15668438—22no—
91#15668438—41no—
92#15668436—22no—
93#15668432—22no—
94C2×#13089678C2311yescharacteristic
95C2×#13089678C2311yescharacteristic
96C8×D181—22no—
97C4⋊Dic181—21no—
98C4⋊Dic181—21no—
99C4⋊Dic181—21no—
100C4⋊Dic181—21no—
101C4⋊Dic181—21no—
102C4⋊Dic181D411yescharacteristic
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426—21no—
109#15668426D411yescharacteristic
110#15668426D411yescharacteristic
111C2×Q8×C181C2311yescharacteristic
112C2×Q8×C181C2311yescharacteristic
113Q16×C181D414yescomplemented
114C2×C8×C181C2311yescharacteristic
115C2×C8×C181C2311yescharacteristic
116C4⋊C4×C181—21no—
117C4⋊C4×C181—21no—
118C4⋊C4×C181C2311yescharacteristic
119C4⋊C4×C181C2311yescharacteristic
120C42×C181—21no—
121C42×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181C2×D411yescharacteristic
125C22×D181C2×D411yescharacteristic
126Dic362—21no—
127Dic362—21no—
128Dic362—21no—
129Dic362—21no—
130Dic362—21no—
131Dic362—21no—
132Dic362—42no—
133Dic362—42no—
134Dic362—42no—
135D724—22no—
136D724—21no—
137D724—21no—
138D724—42no—
139D724—42no—
140C2×Dic181—21no—
141C2×Dic181—21no—
142C2×Dic181—21no—
143C2×Dic181C2×D411yescharacteristic
144C2×Dic181C2×D411yescharacteristic
145C4×D181—21no—
146C4×D181—21no—
147C4×D181—21no—
148C4×D181—21no—
149C4×D181—42no—
150C4×D181—42no—
151C4×D181—42no—
152C4×D181—42no—
153#13089678—21no—
154#13089678—21no—
155Q8×C181—21no—
156Q8×C181—21no—
157Q8×C181C2×D412yes—
158Q8×C181C2×D412yes—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181C2×D411yescharacteristic
164C2×C4×C181C2×D411yescharacteristic
165C2×C4×C181C2411yescharacteristic
166C8×C181—21no—
167C8×C181C2×D412yes—
168C22×C181C22×D411yescharacteristic
169C2×D181—21no—
170C2×D181—21no—
171C2×D181—21no—
172C2×D181—21no—
173C2×D181—21no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C2×D181#13712yescomplemented
178C4×C181—22no—
179C4×C181—22no—
180C4×C181—21no—
181C4×C181—21no—
182C4×C181—41no—
183C4×C181—41no—
184C4×C181C22×D411yescharacteristic
185C4×C181H(5,𝔽2)11yescharacteristic
186Dic181—21no—
187Dic181—21no—
188Dic181—41no—
189Dic181—41no—
190Dic181—41no—
191C2×C181D4211yescharacteristic
192C2×C181C2×#13711yescharacteristic
193C2×C181#57811yescharacteristic
194D181—22no—
195D181—41no—
196D181—81no—
197C181#327011yescharacteristic, complemented, Sylow
198#3270—1811nomaximal, Sylow
199C2×#137—1811no—
200C22×Q16—1811no—
201D4×Q8—1811no—
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#441D18111yescharacteristic, complemented
213#435—1811no—
214C2×#41—1811no—
215C22×Q8—1811no—
216C22×Q8—1811no—
217C22×Q8—1811no—
218#137—3624no—
219C2×Q16—3624no—
220C2×Q16—1811no—
221C2×Q16—1811no—
222C2×Q16—1811no—
223C2×Q16—1811no—
224C2×Q16—1811no—
225C2×Q16C2×D18111yescharacteristic
226C2×SD16—1811no—
227C2×SD16—1811no—
228C2×SD16—1811no—
229C2×M16—1811no—
230C22×C8—1811no—
231C4⋊Q8—3621no—
232C4⋊Q8—1811no—
233#124—3621no—
234#124—1811no—
235C22⋊Q8—3621no—
236C22⋊Q8—3621no—
237C22⋊Q8—3621no—
238C22⋊Q8—1811no—
239C22⋊Q8—1811no—
240C22⋊Q8—1811no—
241#121—3621no—
242#121—1811no—
243C4×Q8C2×D18111yescharacteristic
244C4×Q8C2×D18111yescharacteristic
245C4×D4—3621no—
246C4×D4—3621no—
247C4×D4—1811no—
248#106C2×D18111yescharacteristic
249C4⋊C8—1811no—
250#103—1811no—
251#103—1811no—
252#103—1811no—
253#103C2×D18111yescharacteristic
254#103C2×D18111yescharacteristic
255#102—1811no—
256C22⋊C8—1811no—
257C22⋊C8—1811no—
258#97C2×D18111yescharacteristic
259#41—3622no—
260#41—7241no—
261C2×Q8—3621no—
262C2×Q8—3621no—
263C2×Q8—3621no—
264C2×Q8—3622no—
265C2×Q8—3622no—
266C2×Q8—3622no—
267C2×Q8—1811no—
268C2×Q8—1811no—
269C2×Q8—1811no—
270C2×Q8—7241no—
271C2×Q8—7241no—
272C2×Q8—7241no—
273C2×Q8C22×D18111yescharacteristic
274C2×Q8C22×D18111yescharacteristic
275C2×D4—3621no—
276C2×D4—3621no—
277C2×D4—1811no—
278C2×D4—1811no—
279C22×C4—3621no—
280C22×C4—3621no—
281C22×C4—3621no—
282C22×C4—3621no—
283C22×C4—1811no—
284C22×C4—1811no—
285Q16—3622no—
286Q16—3622no—
287Q16—3622no—
288Q16—3622no—
289Q16—7241no—
290Q16D72414yescomplemented
291SD16—3622no—
292SD16—3622no—
293SD16—7241no—
294M16—3622no—
295C2×C8—3622no—
296C2×C8—1811no—
297C2×C8—1811no—
298C2×C8C22×D18111yescharacteristic
299C2×C8C22×D18111yescharacteristic
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—3621no—
311C22⋊C4—3621no—
312C22⋊C4—3621no—
313C22⋊C4—3621no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C22⋊C4—1811no—
317C22⋊C4—1811no—
318C42—21no—
319C42—21no—
320C42C22×D18111yescharacteristic
321C23—3621no—
322C23—1811no—
323C23—1811no—
324Q8—21no—
325Q8—21no—
326Q8—3621no—
327Q8—3621no—
328Q8—3621no—
329Q8—3621no—
330Q8—3621no—
331Q8—3621no—
332Q8—7242no—
333Q8—7242no—
334Q8—7242no—
335Q8C2×D72412yes—
336Q8C2×D72412yes—
337D4—3622no—
338D4—3621no—
339D4—3621no—
340D4—7242no—
341D4—7242no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—3621no—
347C2×C4—3621no—
348C2×C4—3621no—
349C2×C4—3621no—
350C2×C4—3621no—
351C2×C4—3621no—
352C2×C4—3621no—
353C2×C4—1811no—
354C2×C4—1811no—
355C2×C4—7242no—
356C2×C4—7242no—
357C2×C4—7242no—
358C2×C4—7242no—
359C2×C4D4×D18111yescharacteristic
360C2×C4D4×D18111yescharacteristic
361C2×C4C23×D18111yescharacteristic
362C8—21no—
363C8—3621no—
364C8—3621no—
365C8C2×D72412yes—
366C22—3621no—
367C22—3621no—
368C22—3621no—
369C22—3621no—
370C22—3621no—
371C22—1812no—
372C22—7241no—
373C22—7241no—
374C22—7241no—
375C22C2×D4×D18111yescharacteristic
376C4—22no—
377C4—22no—
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—3621no—
383C4—3621no—
384C4—7241no—
385C4—7241no—
386C4—7241no—
387C4C22×D72411yescharacteristic
388C4#1915386211yescharacteristic
389C2—3622no—
390C2—14481no—
391C2—7241no—
392C2D4×D72411yescharacteristic
393C2#137×D18111yescharacteristic
394C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#441×C181C2
socleC22×C181C22×D4
prime radical#26041941C2
hypercenter#441D181
hypocenterC181#3270
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. C2×#19153848C2semidirect
2. C2×Q16×D181C2semidirect
3. Q8×D724C2semidirect
5. #26041673C2semidirect
6. #26041538C2semidirect
7. #26041509C2semidirect
8. #26041501C2semidirect
9. #26041489C2semidirect
10. #26041444C2semidirect
11. #26041436C2semidirect
12. #26041436C2semidirect
13. #26041419C2semidirect
14. #26041352C2semidirect
15. #441×C181C2semidirect
113. Q16×C181D4semidirect
177. C2×D181#137semidirect
197. C181#3270semidirect
212. #441D181semidirect
290. Q16D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538482
C2×Q16×D1812
Q8×D7242
#260419412
#260416732
#260415382
#260415092
#260415012
#260414892
#260414442
#260414362
#260414192
#260413522
#441×C1812
#3270181