Finite Group Databasebeta

← #37545458

Subgroups — #37545458

Order 23168.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153836C211yesmaximal, characteristic, complemented
2C22×#15668439C211yesmaximal, characteristic, complemented
3#26041938C211yesmaximal, characteristic, complemented
4#26041661C211yesmaximal, characteristic, complemented
5#26041518C211yesmaximal, characteristic, complemented
6#26041513C211yesmaximal, characteristic, complemented
7#26041501C211yesmaximal, characteristic, complemented
8#26041498C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041439C211yesmaximal, characteristic, complemented
11#26041418C211yesmaximal, characteristic, complemented
12#26041417C211yesmaximal, characteristic, complemented
13C4×#15668439C211yesmaximal, characteristic, complemented
14D4×#13089678C211yesmaximal, characteristic, complemented
15#548×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×Dic362C2211yescharacteristic
18#19153836—24no—
19C2×#15668439—24no—
20C2×#15668439C2211yescharacteristic
21C2×#15668439C2211yescharacteristic
22C2×#15668439C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668437C2211yescharacteristic
25C2×#15668437C2211yescharacteristic
26C2×#15668436C2211yescharacteristic
27C2×#15668436C2211yescharacteristic
28C2×#15668434C2211yescharacteristic
29C22×#13089678C2211yescharacteristic
30C22×#13089678C2211yescharacteristic
31#19153804C2211yescharacteristic
32#19153802—21no—
33#19153795C2211yescharacteristic
34#19153794—21no—
35#19153790C2211yescharacteristic
36#19153788—21no—
37C4×Dic362C2211yescharacteristic
38#19153776—21no—
39#19153749C2211yescharacteristic
40C4⋊#13089678C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153739C2211yescharacteristic
44#19153739C2211yescharacteristic
45#19153737C2211yescharacteristic
46#19153736C2211yescharacteristic
47C22⋊#13089678C2211yescharacteristic
48C22⋊#13089678C2211yescharacteristic
49C4×#13089678C2211yescharacteristic
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
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#15668439D414yescomplemented
77#15668437—22no—
78#15668437—41no—
79#15668436—22no—
80#15668436—41no—
81#15668434—22no—
82C2×#13089678—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—
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
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×C181—21no—
143C23×C181C2×D411yescharacteristic
144C23×C181C2×D411yescharacteristic
145Q8×C181—21no—
146Q8×C181—21no—
147Q8×C181—21no—
148Q8×C181—42no—
149Q8×C181C2×D412yes—
150D4×C181—22no—
151D4×C181—21no—
152D4×C181—21no—
153D4×C181—42no—
154D4×C181—42no—
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—21no—
162C2×C4×C181—42no—
163C2×C4×C181—42no—
164C2×C4×C181C2×D411yescharacteristic
165C2×C4×C181C2×D411yescharacteristic
166C2×C4×C181C2×D411yescharacteristic
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2411yescharacteristic
169C22×C181—21no—
170C22×C181—21no—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181#13512yescomplemented
178C22×C181C22×D411yescharacteristic
179C2×D181—41no—
180C2×D181—41no—
181C2×D181—41no—
182C4×C181—21no—
183C4×C181—21no—
184C4×C181—22no—
185C4×C181—21no—
186C4×C181—21no—
187C4×C181—41no—
188C4×C181—41no—
189C4×C181C22×D411yescharacteristic
190C4×C181H(5,𝔽2)11yescharacteristic
191Dic181—22no—
192Dic181—41no—
193Dic181—41no—
194C2×C181—22no—
195C2×C181—41no—
196C2×C181—81no—
197C2×C181D4211yescharacteristic
198C2×C181C2×#13511yescharacteristic
199C2×C181#57811yescharacteristic
200D181—81no—
201C181#328511yescharacteristic, complemented, Sylow
202#3285—1811nomaximal, Sylow
203C2×#135—1811no—
204C22×Q16—1811no—
205#548—1811no—
206#548D18111yescharacteristic, complemented
207#495—1811no—
208#467—1811no—
209#466—1811no—
210#464—1811no—
211#464—1811no—
212#452—1811no—
213#452—1811no—
214#448—1811no—
215#448—1811no—
216C4×Q16—1811no—
217C8×D4—1811no—
218C2×#41—1811no—
219C2×#41C2×D18111yescharacteristic
220C22×Q8—1811no—
221C22×Q8C2×D18111yescharacteristic
222#135—3624no—
223C2×Q16—3624no—
224C2×Q16—1811no—
225C2×Q16—1811no—
226C2×Q16—1811no—
227C2×Q16—1811no—
228C2×SD16—1811no—
229C2×SD16—1811no—
230C2×SD16—1811no—
231C2×SD16—1811no—
232C2×D8—1811no—
233C22×C8—1811no—
234C22×C8—1811no—
235#124—21no—
236#124—3621no—
237#124—1811no—
238#124C2×D18111yescharacteristic
239C22⋊Q8—21no—
240C22⋊Q8—3621no—
241C22⋊Q8—1811no—
242C22⋊Q8C2×D18111yescharacteristic
243#121—21no—
244#121—3621no—
245#121—1811no—
246#121C2×D18111yescharacteristic
247C4×Q8—1811no—
248C4×Q8C2×D18111yescharacteristic
249C4×D4—21no—
250C4×D4—3621no—
251C4×D4C2×D18111yescharacteristic
252#107—1811no—
253C4⋊C8—1811no—
254#103—1811no—
255#103—1811no—
256#103—1811no—
257#103—1811no—
258#102—1811no—
259#102—1811no—
260C22⋊C8—1811no—
261C22⋊C8—1811no—
262C4×C8—1811no—
263#41—22no—
264#41—41no—
265#41—3622no—
266#41—7241no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—3622no—
271C2×Q8—3621no—
272C2×Q8—1811no—
273C2×Q8—1811no—
274C2×Q8—7241no—
275C2×Q8C22×D18111yescharacteristic
276C2×Q8C22×D18111yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—3621no—
280C2×D4—3621no—
281C2×D4—1811no—
282C2×D4C22⋊D18111yescharacteristic
283C2×D4C22×D18111yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—3621no—
287C22×C4—3621no—
288C22×C4C22×D18111yescharacteristic
289C22×C4C22×D18111yescharacteristic
290Q16—3622no—
291Q16—3622no—
292Q16—3622no—
293Q16—1814no—
294SD16—3622no—
295SD16—3622no—
296SD16—7241no—
297SD16—7241no—
298D8—3622no—
299C2×C8—3622no—
300C2×C8—3622no—
301C2×C8—1811no—
302C2×C8—1811no—
303C2×C8—1811no—
304C2×C8—1811no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—3621no—
308C4⋊C4—3621no—
309C4⋊C4—1811no—
310C4⋊C4C22⋊D18111yescharacteristic
311C4⋊C4C22×D18111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—3621no—
317C22⋊C4—3621no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4C22⋊D18111yescharacteristic
321C22⋊C4C22⋊D18111yescharacteristic
322C42—21no—
323C42—3621no—
324C42C22×D18111yescharacteristic
325C23—21no—
326C23—3621no—
327C23C2×C22⋊D18111yescharacteristic
328C23C2×C22⋊D18111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—3621no—
334Q8—3621no—
335Q8—3621no—
336Q8—1812no—
337Q8—7242no—
338Q8D4×D18112yes—
339D4—22no—
340D4—21no—
341D4—21no—
342D4—42no—
343D4—42no—
344D4—3621no—
345D4—3621no—
346D4—7242no—
347D4—7242no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—3621no—
358C2×C4—3621no—
359C2×C4—3621no—
360C2×C4—7242no—
361C2×C4—7242no—
362C2×C4C2×C22⋊D18111yescharacteristic
363C2×C4C2×C22⋊D18111yescharacteristic
364C2×C4C2×C22⋊D18111yescharacteristic
365C2×C4C2×C22⋊D18111yescharacteristic
366C2×C4C23×D18111yescharacteristic
367C8—3621no—
368C8—3621no—
369C8—3621no—
370C8—1812no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—7241no—
380C22—7241no—
381C22—7241no—
382C22#1915383612yescomplemented
383C22C22×C22⋊D18111yescharacteristic
384C4—21no—
385C4—21no—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—3622no—
392C4—7241no—
393C4—7241no—
394C4C2×D4×D18111yescharacteristic
395C4#1915386111yescharacteristic
396C2—22no—
397C2—41no—
398C2—81no—
399C2—14481no—
400C2D4×C22⋊D18111yescharacteristic
401C2C2×#1915383611yescharacteristic
402C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#548×C181C2
socleC22×C181C22×D4
hypercenter#548D181
hypocenterC181#3285
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×#19153836C2semidirect
2. C22×#15668439C2semidirect
3. #26041938C2semidirect
4. #26041661C2semidirect
5. #26041518C2semidirect
6. #26041513C2semidirect
7. #26041501C2semidirect
8. #26041498C2semidirect
9. #26041441C2semidirect
10. #26041439C2semidirect
11. #26041418C2semidirect
12. #26041417C2semidirect
13. C4×#15668439C2semidirect
14. D4×#13089678C2semidirect
15. #548×C181C2semidirect
76. #15668439D4semidirect
177. C22×C181#135semidirect
201. C181#3285semidirect
206. #548D181semidirect
382. C22#19153836semidirect

Maximal subgroups

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

subgroupindex
C2×#191538362
C22×#156684392
#260419382
#260416612
#260415182
#260415132
#260415012
#260414982
#260414412
#260414392
#260414182
#260414172
C4×#156684392
D4×#130896782
#548×C1812
#3285181