Finite Group Databasebeta

← #37545285

Subgroups — #37545285

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×Q16×D181C211yesmaximal, characteristic, complemented
2Q8×C22⋊D181C211yesmaximal, characteristic, complemented
3#26041936C211yesmaximal, characteristic
4#26041670C211yesmaximal, characteristic, complemented
5#26041537C211yesmaximal, characteristic, complemented
6#26041505C211yesmaximal, characteristic, complemented
7#26041499C211yesmaximal, characteristic, complemented
8#26041487C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041435C211yesmaximal, characteristic, complemented
11#26041433C211yesmaximal, characteristic, complemented
12#26041416C211yesmaximal, characteristic, complemented
13#26041386C211yesmaximal, characteristic, complemented
14#26041348C211yesmaximal, characteristic, complemented
15C2×#137×C181C211yesmaximal, characteristic, complemented
16C2×Q8×D181C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×Dic724C2211yescharacteristic
19C2×#15668439C2211yescharacteristic
20C2×#15668439C2211yescharacteristic
21C2×#15668439C2211yescharacteristic
22Q16×D181—24no—
23C2×#15668437C2211yescharacteristic
24C2×C8×D181C2211yescharacteristic
25#19153814—21no—
26#19153814C2211yescharacteristic
27#19153803—21no—
28#19153803C2211yescharacteristic
29#19153795—21no—
30#19153795—21no—
31#19153795C2211yescharacteristic
32#19153795C2211yescharacteristic
33#19153792—21no—
34#19153792C2211yescharacteristic
35#19153787—21no—
36#19153787C2211yescharacteristic
37Q8×Dic181C2211yescharacteristic
38Q8×Dic181C2211yescharacteristic
39C4×C22⋊D181—21no—
40C4×C22⋊D181—21no—
41C4×C22⋊D181C2211yescharacteristic
42#19153748C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153741C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153740C2211yescharacteristic
48#19153740C2211yescharacteristic
49#19153737C2211yescharacteristic
50C22⋊#13089678C2211yescharacteristic
51#19153728C2211yescharacteristic
52#19153726C2211yescharacteristic
53C2×#41×C181C2211yescharacteristic
54C22×Q8×C181C2211yescharacteristic
55#137×C181—24no—
56C2×Q16×C181C2211yescharacteristic
57C2×Q16×C181C2211yescharacteristic
58C2×SD16×C181C2211yescharacteristic
59C2×SD16×C181C2211yescharacteristic
60C2×M16×C181C2211yescharacteristic
61C2×Dic362—21no—
62C2×Dic362—21no—
63C2×Dic362C2311yescharacteristic
64C2×Dic362C2311yescharacteristic
65Q8×D181—22no—
66Q8×D181—22no—
67Q8×D181—41no—
68Q8×D181—41no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181D411yescharacteristic
71C2×C4×D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181C2311yescharacteristic
74Dic724—22no—
75#15668439—22no—
76#15668439—22no—
77#15668439—41no—
78#15668437—41no—
79C2×#13089678C2311yescharacteristic
80C2×#13089678C2311yescharacteristic
81C8×D181—22no—
82C4⋊Dic181—21no—
83C4⋊Dic181—21no—
84C4⋊Dic181C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86#15668428—21no—
87#15668428—21no—
88#15668428—21no—
89#15668428—21no—
90#15668428—21no—
91#15668428D411yescharacteristic
92C22⋊Dic181—21no—
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95C22⋊Dic181—21no—
96C22⋊Dic181—21no—
97C22⋊Dic181D411yescharacteristic
98#15668426—21no—
99#15668426D411yescharacteristic
100C4×Dic181—21no—
101C4×Dic181—21no—
102C4×Dic181C2311yescharacteristic
103#41×C181—22no—
104#41×C181—41no—
105C2×Q8×C181—21no—
106C2×Q8×C181—22no—
107C2×Q8×C181—41no—
108C2×Q8×C181C2311yescharacteristic
109C2×Q8×C181C2311yescharacteristic
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181—21no—
112C2×D4×C181C2311yescharacteristic
113C22×C4×C181—21no—
114C22×C4×C181—21no—
115C22×C4×C181C2311yescharacteristic
116Q16×C181—22no—
117Q16×C181D414yescomplemented
118SD16×C181—22no—
119SD16×C181—22no—
120M16×C181—22no—
121C2×C8×C181C2311yescharacteristic
122C2×C8×C181C2311yescharacteristic
123C22×D181C2×D411yescharacteristic
124Dic362—21no—
125Dic362—21no—
126Dic362—21no—
127Dic362—21no—
128Dic362—42no—
129Dic362—42no—
130C22⋊D181—22no—
131C22⋊D181—42no—
132C2×Dic181—21no—
133C2×Dic181—21no—
134C2×Dic181—21no—
135C2×Dic181—21no—
136C2×Dic181C2×D411yescharacteristic
137C2×Dic181C2×D411yescharacteristic
138C2×Dic181C2×D411yescharacteristic
139C4×D181—21no—
140C4×D181—21no—
141C4×D181—42no—
142C4×D181—42no—
143#13089678—21no—
144#13089678—21no—
145C23×C181—21no—
146C23×C181C2×D411yescharacteristic
147Q8×C181—21no—
148Q8×C181—21no—
149Q8×C181—21no—
150Q8×C181—21no—
151Q8×C181—42no—
152Q8×C181C2×D412yes—
153Q8×C181C2×D412yes—
154D4×C181—21no—
155D4×C181—21no—
156D4×C181—42no—
157C2×C4×C181—21no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—42no—
163C2×C4×C181—42no—
164C2×C4×C181C2×D411yescharacteristic
165C2×C4×C181C2411yescharacteristic
166C8×C181—21no—
167C8×C181C2×D412yes—
168C22×C181—21no—
169C22×C181—21no—
170C22×C181—21no—
171C22×C181—41no—
172C22×C181—41no—
173C22×C181—41no—
174C22×C181C22×D411yescharacteristic
175C2×D181—21no—
176C2×D181—21no—
177C2×D181#13712yescomplemented
178C4×C181—21no—
179C4×C181—22no—
180C4×C181—22no—
181C4×C181—41no—
182C4×C181C22×D411yescharacteristic
183C4×C181H(5,𝔽2)11yescharacteristic
184Dic181—21no—
185Dic181—21no—
186Dic181—21no—
187Dic181—41no—
188Dic181—41no—
189Dic181—41no—
190Dic181—41no—
191C2×C181—41no—
192C2×C181—81no—
193C2×C181D4211yescharacteristic
194C2×C181C2×#13711yescharacteristic
195C2×C181#57811yescharacteristic
196D181—22no—
197C181#327011yescharacteristic, complemented, Sylow
198#3270—1811nomaximal, Sylow
199C2×#137D18111yescharacteristic, complemented
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#441—1811no—
213#435—1811no—
214C2×#41C2×D18111yescharacteristic
215C22×Q8—1811no—
216C22×Q8—1811no—
217C22×Q8C2×D18111yescharacteristic
218#137—24no—
219C2×Q16—3624no—
220C2×Q16—1811no—
221C2×Q16—1811no—
222C2×Q16—1811no—
223C2×Q16—1811no—
224C2×Q16C2×D18111yescharacteristic
225C2×Q16C2×D18111yescharacteristic
226C2×SD16—1811no—
227C2×SD16C2×D18111yescharacteristic
228C2×SD16C2×D18111yescharacteristic
229C2×M16C2×D18111yescharacteristic
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×Q8—1811no—
244C4×Q8—1811no—
245C4×D4—3621no—
246C4×D4—3621no—
247C4×D4—1811no—
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—3621no—
265C2×Q8—3621no—
266C2×Q8—3622no—
267C2×Q8—3622no—
268C2×Q8—1811no—
269C2×Q8—1811no—
270C2×Q8—7241no—
271C2×Q8—7241no—
272C2×Q8C22×D18111yescharacteristic
273C2×Q8C22×D18111yescharacteristic
274C2×Q8C22×D18111yescharacteristic
275C2×D4—21no—
276C2×D4—3621no—
277C2×D4—1811no—
278C2×D4C22×D18111yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3621no—
282C22×C4—3621no—
283C22×C4—1811no—
284C22×C4C22×D18111yescharacteristic
285Q16—22no—
286Q16—3622no—
287Q16—3622no—
288Q16—3622no—
289Q16—7241no—
290Q16C22⋊D18114yescomplemented
291SD16—22no—
292SD16—22no—
293SD16—7241no—
294M16—22no—
295C2×C8—3622no—
296C2×C8—1811no—
297C2×C8—1811no—
298C2×C8C22×D18111yescharacteristic
299C2×C8C22×D18111yescharacteristic
300C4⋊C4—3621no—
301C4⋊C4—3621no—
302C4⋊C4—3621no—
303C4⋊C4—3621no—
304C4⋊C4—3621no—
305C4⋊C4—3621no—
306C4⋊C4—3621no—
307C4⋊C4—1811no—
308C4⋊C4—1811no—
309C4⋊C4—1811no—
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—3621no—
319C42—3621no—
320C42—1811no—
321C23—21no—
322C23—1811no—
323C23D4×D18111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—42no—
329Q8—3621no—
330Q8—3621no—
331Q8—3621no—
332Q8—3621no—
333Q8—7242no—
334Q8—7242no—
335Q8C2×C22⋊D18112yes—
336Q8C2×C22⋊D18112yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—3622no—
341D4—7242no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—3621no—
350C2×C4—3621no—
351C2×C4—3621no—
352C2×C4—3621no—
353C2×C4—3621no—
354C2×C4—3621no—
355C2×C4—1811no—
356C2×C4—1811no—
357C2×C4—1811no—
358C2×C4—7242no—
359C2×C4—7242no—
360C2×C4D4×D18111yescharacteristic
361C2×C4C23×D18111yescharacteristic
362C8—21no—
363C8—3621no—
364C8—3621no—
365C8C2×C22⋊D18112yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—3621no—
373C22—3621no—
374C22—1812no—
375C22C2×D4×D18111yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—3621no—
381C4—3621no—
382C4—3621no—
383C4—7241no—
384C4—7241no—
385C4—7241no—
386C4—7241no—
387C4C22×C22⋊D18111yescharacteristic
388C4#1915386111yescharacteristic
389C2—41no—
390C2—81no—
391C2—3622no—
392C2D4×C22⋊D18111yescharacteristic
393C2#137×D18111yescharacteristic
394C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#137×C181C2
socleC22×C181C22×D4
prime radical#26041936C2
hypercenterC2×#137D181
hypocenterC181#3270
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D181C2semidirect
2. Q8×C22⋊D181C2semidirect
4. #26041670C2semidirect
5. #26041537C2semidirect
6. #26041505C2semidirect
7. #26041499C2semidirect
8. #26041487C2semidirect
9. #26041441C2semidirect
10. #26041435C2semidirect
11. #26041433C2semidirect
12. #26041416C2semidirect
13. #26041386C2semidirect
14. #26041348C2semidirect
15. C2×#137×C181C2semidirect
117. Q16×C181D4semidirect
177. C2×D181#137semidirect
197. C181#3270semidirect
199. C2×#137D181semidirect
290. Q16C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1812
Q8×C22⋊D1812
#260419362
#260416702
#260415372
#260415052
#260414992
#260414872
#260414412
#260414352
#260414332
#260414162
#260413862
#260413482
C2×#137×C1812
#3270181