Finite Group Databasebeta

← #37545290

Subgroups — #37545290

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×#19153845C211yesmaximal, characteristic, complemented
2C2×Q16×D181C211yesmaximal, characteristic, complemented
3#26041944C211yesmaximal, characteristic, complemented
4#26041937C211yesmaximal, characteristic
5#26041534C211yesmaximal, characteristic, complemented
6#26041505C211yesmaximal, characteristic, complemented
7#26041498C211yesmaximal, characteristic, complemented
8#26041486C211yesmaximal, characteristic, complemented
9#26041438C211yesmaximal, characteristic, complemented
10#26041433C211yesmaximal, characteristic, complemented
11#26041433C211yesmaximal, characteristic, complemented
12#26041416C211yesmaximal, characteristic, complemented
13#26041387C211yesmaximal, characteristic, complemented
14#26041352C211yesmaximal, characteristic, complemented
15#497×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic
19C2×Q8×D181C2211yescharacteristic
20#19153845—24no—
21C2×Dic724C2211yescharacteristic
22C2×#15668439C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668439C2211yescharacteristic
25Q16×D181—24no—
26C2×#15668437C2211yescharacteristic
27C2×#15668432C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic
29#19153815—21no—
30#19153802—21no—
31#19153794—21no—
32#19153794—21no—
33#19153793—21no—
34#19153792C2211yescharacteristic
35#19153792C2211yescharacteristic
36#19153792C2211yescharacteristic
37#19153789—21no—
38#19153787C2211yescharacteristic
39C4×Dic362C2211yescharacteristic
40C4×Dic362C2211yescharacteristic
41C4×D724C2211yescharacteristic
42#19153778—21no—
43#19153776—21no—
44#19153748C2211yescharacteristic
45C4⋊#13089678C2211yescharacteristic
46#19153741C2211yescharacteristic
47#19153741C2211yescharacteristic
48#19153740C2211yescharacteristic
49#19153740C2211yescharacteristic
50#19153740C2211yescharacteristic
51#19153737C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153728C2211yescharacteristic
54C2×Q16×C181C2211yescharacteristic
55C2×Q16×C181C2211yescharacteristic
56C2×SD16×C181C2211yescharacteristic
57C2×SD16×C181C2211yescharacteristic
58C4⋊Q8×C181C2211yescharacteristic
59#124×C181C2211yescharacteristic
60#97×C181C2211yescharacteristic
61#15668448—22no—
62#15668448—41no—
63C2×Dic362—21no—
64C2×Dic362—21no—
65C2×Dic362—21no—
66C2×Dic362C2311yescharacteristic
67C2×Dic362C2311yescharacteristic
68Q8×D181—22no—
69Q8×D181—22no—
70Q8×D181—22no—
71Q8×D181—41no—
72Q8×D181—41no—
73Q8×D181—41no—
74C2×D724D411yescharacteristic
75C2×C22⋊D181—21no—
76C2×C22⋊D181—21no—
77C22×Dic181—21no—
78C2×C4×D181—21no—
79C2×C4×D181—21no—
80C2×C4×D181—21no—
81C2×C4×D181C2311yescharacteristic
82C2×C4×D181C2311yescharacteristic
83Dic724D414yescomplemented
84#15668439—22no—
85#15668439—22no—
86#15668439—22no—
87#15668437—22no—
88#15668432—22no—
89C2×#13089678C2311yescharacteristic
90C2×#13089678C2311yescharacteristic
91C8×D181—22no—
92C4⋊Dic181D411yescharacteristic
93C4⋊Dic181C2311yescharacteristic
94C4⋊Dic181C2311yescharacteristic
95#15668428—21no—
96#15668428—21no—
97#15668428—21no—
98#15668428—21no—
99#15668428—21no—
100C22⋊Dic181—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426D411yescharacteristic
105#15668426D411yescharacteristic
106C4×Dic181—21no—
107C4×Dic181—21no—
108C2×Q8×C181C2311yescharacteristic
109C2×Q8×C181C2311yescharacteristic
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181C2311yescharacteristic
112Q16×C181—22no—
113Q16×C181—41no—
114SD16×C181—22no—
115SD16×C181—41no—
116C2×C8×C181C2311yescharacteristic
117C2×C8×C181C2311yescharacteristic
118C4⋊C4×C181—21no—
119C4⋊C4×C181—21no—
120C22⋊C4×C181—21no—
121C22⋊C4×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181C2×D411yescharacteristic
124C22×D181C2×D411yescharacteristic
125Dic362—21no—
126Dic362—21no—
127Dic362—42no—
128Dic362—42no—
129Dic362—42no—
130Dic362C2×D412yes—
131Dic362C2×D412yes—
132D724—22no—
133C22⋊D181—42no—
134C22⋊D181—42no—
135C2×Dic181—21no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181—42no—
140C2×Dic181C2×D411yescharacteristic
141C2×Dic181C2×D411yescharacteristic
142C4×D181—21no—
143C4×D181—21no—
144C4×D181—21no—
145C4×D181—21no—
146C4×D181—42no—
147C4×D181—42no—
148C4×D181—42no—
149#13089678—21no—
150#13089678—21no—
151C23×C181—21no—
152Q8×C181—21no—
153Q8×C181—21no—
154Q8×C181—21no—
155Q8×C181—21no—
156Q8×C181—21no—
157Q8×C181—21no—
158D4×C181—21no—
159D4×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×C181—41no—
169C22×C181—41no—
170C22×C181—41no—
171C22×C181C22×D411yescharacteristic
172C2×D181—21no—
173C2×D181—21no—
174C2×D181—21no—
175C2×D181—21no—
176C2×D181—21no—
177C2×D181#13712yescomplemented
178C4×C181—21no—
179C4×C181—21no—
180C4×C181—41no—
181C4×C181—41no—
182C4×C181—41no—
183C4×C181C22×D411yescharacteristic
184C4×C181H(5,𝔽2)11yescharacteristic
185Dic181—21no—
186Dic181—21no—
187Dic181—22no—
188Dic181—22no—
189Dic181—41no—
190Dic181—41no—
191C2×C181—81no—
192C2×C181D4211yescharacteristic
193C2×C181C2×#13711yescharacteristic
194C2×C181#57811yescharacteristic
195D181—22no—
196D181—41no—
197C181#327011yescharacteristic, complemented, Sylow
198#3270—1811nomaximal, Sylow
199C2×#137—1811no—
200C22×Q16—1811no—
201D4×Q8—1811no—
202#548—1811no—
203#497D18111yescharacteristic, complemented
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×Q8—1811no—
217C22×Q8—1811no—
218#137—3624no—
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×M16—1811no—
230C22×C8—1811no—
231C4⋊Q8—3621no—
232C4⋊Q8C2×D18111yescharacteristic
233#124—3621no—
234#124C2×D18111yescharacteristic
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#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—7241no—
270C2×Q8—7241no—
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—3621no—
280C22×C4—3621no—
281C22×C4—3621no—
282C22×C4—3621no—
283C22×C4—1811no—
284C22×C4—1811no—
285Q16—22no—
286Q16—41no—
287Q16—3622no—
288Q16—3622no—
289Q16—3622no—
290Q16—1814no—
291SD16—22no—
292SD16—41no—
293SD16—3622no—
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⋊C4—1811no—
309C4⋊C4—1811no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3621no—
313C22⋊C4—3621no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C22⋊C4—1811no—
317C22⋊C4—1811no—
318C42—3621no—
319C42—3621no—
320C42C22×D18111yescharacteristic
321C23—21no—
322C23—1811no—
323C23—1811no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—3621no—
331Q8—3621no—
332Q8—1812no—
333Q8—1812no—
334Q8—7242no—
335Q8—7242no—
336Q8—7242no—
337D4—21no—
338D4—21no—
339D4—3622no—
340D4—7242no—
341D4—7242no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—3621no—
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—
365C8D4×D18112yes—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—3621no—
370C22—3621no—
371C22—3621no—
372C22—3621no—
373C22—3621no—
374C22—1812no—
375C22C2×D4×D18111yescharacteristic
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3621no—
382C4—3621no—
383C4—3622no—
384C4—3622no—
385C4—7241no—
386C4—7241no—
387C4C2×D4×D18111yescharacteristic
388C4#1915386111yescharacteristic
389C2—81no—
390C2—3622no—
391C2—7241no—
392C2#2604191411yescharacteristic
393C2#137×D18111yescharacteristic
394C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#497×C181C2
socleC22×C181C22×D4
prime radical#26041937C2
hypercenter#497D181
hypocenterC181#3270
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#19153845C2semidirect
2. C2×Q16×D181C2semidirect
3. #26041944C2semidirect
5. #26041534C2semidirect
6. #26041505C2semidirect
7. #26041498C2semidirect
8. #26041486C2semidirect
9. #26041438C2semidirect
10. #26041433C2semidirect
11. #26041433C2semidirect
12. #26041416C2semidirect
13. #26041387C2semidirect
14. #26041352C2semidirect
15. #497×C181C2semidirect
83. Dic724D4semidirect
177. C2×D181#137semidirect
197. C181#3270semidirect
203. #497D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538452
C2×Q16×D1812
#260419442
#260419372
#260415342
#260415052
#260414982
#260414862
#260414382
#260414332
#260414162
#260413872
#260413522
#497×C1812
#3270181