Finite Group Databasebeta

← #37545307

Subgroups — #37545307

Order 23168.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153847C211yesmaximal, characteristic, complemented
2C2×#19153836C211yesmaximal, characteristic, complemented
3#26041950C211yesmaximal, characteristic, complemented
4#26041669C211yesmaximal, characteristic, complemented
5#26041530C211yesmaximal, characteristic, complemented
6#26041510C211yesmaximal, characteristic, complemented
7#26041498C211yesmaximal, characteristic, complemented
8#26041479C211yesmaximal, characteristic, complemented
9#26041439C211yesmaximal, characteristic, complemented
10#26041424C211yesmaximal, characteristic, complemented
11#26041422C211yesmaximal, characteristic, complemented
12#26041418C211yesmaximal, characteristic, complemented
13#26041384C211yesmaximal, characteristic, complemented
14#26041351C211yesmaximal, characteristic, complemented
15#548×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×#15668449C2211yescharacteristic
18#19153847—24no—
19#19153836—24no—
20C2×#15668439C2211yescharacteristic
21C2×#15668439C2211yescharacteristic
22C2×#15668437C2211yescharacteristic
23C2×#15668436C2211yescharacteristic
24C2×#15668436C2211yescharacteristic
25C2×#15668436C2211yescharacteristic
26C2×#15668436C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C2×#15668434C2211yescharacteristic
29C2×#15668433C2211yescharacteristic
30C22×#13089678C2211yescharacteristic
31#19153813C2211yescharacteristic
32#19153812—21no—
33#19153790C2211yescharacteristic
34#19153790C2211yescharacteristic
35#19153789—21no—
36#19153789—21no—
37C4×Dic362C2211yescharacteristic
38#19153778—21no—
39#19153747C2211yescharacteristic
40C4⋊#13089678C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153739C2211yescharacteristic
44#19153737C2211yescharacteristic
45#19153736C2211yescharacteristic
46#19153736C2211yescharacteristic
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
61#15668449—22no—
62#15668449—22no—
63#15668449—41no—
64#15668449—41no—
65C2×Dic362C2311yescharacteristic
66C2×D724—21no—
67C2×D724—21no—
68C2×D724C2311yescharacteristic
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C4×D181—21no—
75C2×C4×D181—21no—
76#15668439—22no—
77#15668439D414yescomplemented
78#15668437—22no—
79#15668436—22no—
80#15668436—22no—
81#15668436—22no—
82#15668436—41no—
83#15668434—22no—
84#15668434—41no—
85#15668433—22no—
86C2×#13089678—22no—
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C2×#13089678C2311yescharacteristic
90C2×#13089678C2311yescharacteristic
91C4⋊Dic181C2311yescharacteristic
92#15668428—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—
124C22×D181—21no—
125Dic362—21no—
126Dic362C2×D412yes—
127D724—21no—
128D724—21no—
129D724—21no—
130D724—21no—
131D724—42no—
132D724—81no—
133C22⋊D181—42no—
134C22⋊D181—42no—
135C22⋊D181—81no—
136C22⋊D181—81no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C4×D181—42no—
140C4×D181—42no—
141#13089678—21no—
142#13089678—21no—
143#13089678—21no—
144#13089678C2×D412yes—
145C23×C181—21no—
146C23×C181C2×D411yescharacteristic
147C23×C181C2×D411yescharacteristic
148Q8×C181—21no—
149Q8×C181—21no—
150Q8×C181—21no—
151Q8×C181—42no—
152Q8×C181C2×D412yes—
153D4×C181—21no—
154D4×C181—21no—
155D4×C181—41no—
156D4×C181—42no—
157D4×C181—42no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—42no—
166C2×C4×C181—42no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2×D411yescharacteristic
170C2×C4×C181C2×D411yescharacteristic
171C2×C4×C181C2411yescharacteristic
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—21no—
175C22×C181—21no—
176C22×C181—21no—
177C22×C181—21no—
178C22×C181—41no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181C22×D411yescharacteristic
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—41no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—41no—
188C4×C181—21no—
189C4×C181—21no—
190C4×C181—22no—
191C4×C181—21no—
192C4×C181—21no—
193C4×C181—41no—
194C4×C181—41no—
195C4×C181C22×D411yescharacteristic
196C4×C181H(5,𝔽2)11yescharacteristic
197Dic181—22no—
198Dic181—41no—
199C2×C181—41no—
200C2×C181—41no—
201C2×C181—81no—
202C2×C181D4211yescharacteristic
203C2×C181#57511yescharacteristic
204C2×C181#57711yescharacteristic
205D181—81no—
206D181—81no—
207C181#327111yescharacteristic, complemented, Sylow
208#3271—1811nomaximal, Sylow
209C2×#137—1811no—
210C2×#135—1811no—
211#550—1811no—
212#548D18111yescharacteristic, complemented
213#496—1811no—
214#469—1811no—
215#465—1811no—
216#464—1811no—
217#460—1811no—
218#452—1811no—
219#449—1811no—
220#449—1811no—
221#448—1811no—
222#441—1811no—
223#435—1811no—
224C2×#41—1811no—
225C2×#41—1811no—
226C2×#41C2×D18111yescharacteristic
227C22×Q8C2×D18111yescharacteristic
228#137—3624no—
229#135—3624no—
230C2×Q16—1811no—
231C2×Q16—1811no—
232C2×SD16—1811no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16—1811no—
237C2×D8—1811no—
238C2×D8—1811no—
239C2×M16—1811no—
240C22×C8—1811no—
241#127—3621no—
242#127—1811no—
243#124—21no—
244#124C2×D18111yescharacteristic
245C22⋊Q8—21no—
246C22⋊Q8C2×D18111yescharacteristic
247#121—21no—
248#121—3621no—
249#121—3621no—
250#121—1811no—
251#121—1811no—
252#121C2×D18111yescharacteristic
253C4×Q8—1811no—
254C4×Q8C2×D18111yescharacteristic
255C4×D4—21no—
256C4×D4—3621no—
257C4×D4C2×D18111yescharacteristic
258#106—1811no—
259C4⋊C8—1811no—
260#103—1811no—
261#103—1811no—
262#103—1811no—
263#102—1811no—
264#102—1811no—
265#102—1811no—
266C22⋊C8—1811no—
267C22⋊C8—1811no—
268#97—1811no—
269#41—22no—
270#41—41no—
271#41—3622no—
272#41—3622no—
273#41—7241no—
274#41—7241no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—1811no—
279C2×Q8C22×D18111yescharacteristic
280C2×Q8C22×D18111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—3621no—
288C2×D4—3621no—
289C2×D4—1811no—
290C2×D4—1811no—
291C2×D4C22⋊D18111yescharacteristic
292C2×D4C22×D18111yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—3621no—
296C22×C4—3621no—
297C22×C4C22×D18111yescharacteristic
298C22×C4C22×D18111yescharacteristic
299Q16—3622no—
300Q16—1814no—
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—3622no—
305SD16—7241no—
306D8—3622no—
307D8—7241no—
308M16—3622no—
309C2×C8—3622no—
310C2×C8—1811no—
311C2×C8—1811no—
312C2×C8—1811no—
313C2×C8—1811no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—3621no—
317C4⋊C4—1811no—
318C4⋊C4C22⋊D18111yescharacteristic
319C4⋊C4C22×D18111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4C22⋊D18111yescharacteristic
327C22⋊C4C22⋊D18111yescharacteristic
328C42—21no—
329C42—3621no—
330C42C22×D18111yescharacteristic
331C23—21no—
332C23—3621no—
333C23—3621no—
334C23C2×C22⋊D18111yescharacteristic
335C23C2×C22⋊D18111yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—42no—
340Q8—3621no—
341Q8—1812no—
342Q8D4×D18112yes—
343D4—21no—
344D4—21no—
345D4—41no—
346D4—42no—
347D4—42no—
348D4—3621no—
349D4—3621no—
350D4—3621no—
351D4—3621no—
352D4—14481no—
353D4—14481no—
354D4—14481no—
355D4—7242no—
356D4—7242no—
357D4—7242no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—3621no—
368C2×C4—3621no—
369C2×C4—7242no—
370C2×C4—7242no—
371C2×C4C2×C22⋊D18111yescharacteristic
372C2×C4C2×C22⋊D18111yescharacteristic
373C2×C4C2×C22⋊D18111yescharacteristic
374C2×C4C2×C22⋊D18111yescharacteristic
375C2×C4C23×D18111yescharacteristic
376C8—3621no—
377C8—3621no—
378C8—3621no—
379C8—1812no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—7241no—
390C22—7241no—
391C22—7241no—
392C22—7241no—
393C22—7241no—
394C22—7241no—
395C22C22×C22⋊D18111yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—41no—
402C4—41no—
403C4—3622no—
404C4—7241no—
405C4C2×D4×D18111yescharacteristic
406C4#1915386111yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—14481no—
411C2—14481no—
412C2D4×C22⋊D18111yescharacteristic
413C2#2604207711yescharacteristic
414C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#548×C181C2
socleC22×C181C22×D4
hypercenter#548D181
hypocenterC181#3271
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×#19153847C2semidirect
2. C2×#19153836C2semidirect
3. #26041950C2semidirect
4. #26041669C2semidirect
5. #26041530C2semidirect
6. #26041510C2semidirect
7. #26041498C2semidirect
8. #26041479C2semidirect
9. #26041439C2semidirect
10. #26041424C2semidirect
11. #26041422C2semidirect
12. #26041418C2semidirect
13. #26041384C2semidirect
14. #26041351C2semidirect
15. #548×C181C2semidirect
77. #15668439D4semidirect
207. C181#3271semidirect
212. #548D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538472
C2×#191538362
#260419502
#260416692
#260415302
#260415102
#260414982
#260414792
#260414392
#260414242
#260414222
#260414182
#260413842
#260413512
#548×C1812
#3271181