Finite Group Databasebeta

← #37545426

Subgroups — #37545426

Order 23168.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153836C211yesmaximal, characteristic, complemented
2C2×SD16×D181C211yesmaximal, characteristic, complemented
3#26041939C211yesmaximal, characteristic, complemented
4#26041917C211yesmaximal, characteristic, complemented
5#26041664C211yesmaximal, characteristic, complemented
6#26041511C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041504C211yesmaximal, characteristic, complemented
9#26041496C211yesmaximal, characteristic, complemented
10#26041433C211yesmaximal, characteristic, complemented
11#26041426C211yesmaximal, characteristic, complemented
12#26041414C211yesmaximal, characteristic, complemented
13#26041370C211yesmaximal, characteristic, complemented
14#26041342C211yesmaximal, characteristic, complemented
15#452×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic, complemented
17C2×Q8×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19#19153836—24no—
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668438C2211yescharacteristic
22C2×#15668437C2211yescharacteristic
23C2×#15668436C2211yescharacteristic
24C2×#15668436C2211yescharacteristic, complemented
25SD16×D181—24no—
26C2×D1448C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C22×#13089678C2211yescharacteristic, complemented
29C2×C8×D181C2211yescharacteristic
30#19153805C2211yescharacteristic
31#19153805C2211yescharacteristic, complemented
32#19153802—21no—
33#19153799—21no—
34#19153794—21no—
35#19153792C2211yescharacteristic
36#19153791C2211yescharacteristic
37#19153790C2211yescharacteristic
38#19153789—21no—
39#19153788—21no—
40#19153784—21no—
41#19153781C2211yescharacteristic, complemented
42D4×Dic181C2211yescharacteristic
43#19153778—21no—
44C4×C22⋊D181C2211yescharacteristic
45C2×#15668426—21no—
46#19153747C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153740C2211yescharacteristic, complemented
50#19153738C2211yescharacteristic, complemented
51#19153736C2211yescharacteristic
52#19153728C2211yescharacteristic
53C8×Dic181C2211yescharacteristic, complemented
54C2×#41×C181C2211yescharacteristic
55C2×Q16×C181C2211yescharacteristic, complemented
56C2×SD16×C181C2211yescharacteristic
57C22⋊Q8×C181C2211yescharacteristic, complemented
58#103×C181C2211yescharacteristic, complemented
59#102×C181C2211yescharacteristic
60C22⋊C8×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668449—22no—
63#15668449—41no—
64C2×Dic362—21no—
65C2×Dic362C2311yescharacteristic
66Q8×D181—22no—
67Q8×D181—41no—
68C2×D724—21no—
69C2×D724C2311yescharacteristic
70C2×D724C2311yescharacteristic
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181D411yescharacteristic
75D4×D181—22no—
76D4×D181—41no—
77C22×Dic181—21no—
78C2×C4×D181—21no—
79C2×C4×D181—21no—
80C2×C4×D181C2311yescharacteristic
81#15668439—22no—
82#15668438—22no—
83#15668437D414yescomplemented
84#15668436—22no—
85#15668436—22no—
86D1448—41no—
87#15668434—22no—
88C2×#13089678—22no—
89C2×#13089678C2311yescharacteristic
90C2×#13089678C2311yescharacteristic
91C8×D181—22no—
92C4⋊Dic181C2311yescharacteristic
93#15668428—21no—
94#15668428—21no—
95#15668428D411yescharacteristic
96C22⋊Dic181—21no—
97C22⋊Dic181D411yescharacteristic
98#15668426—21no—
99#15668426—21no—
100#15668426—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426D411yescharacteristic
106C4×Dic181—21no—
107C4×Dic181C2311yescharacteristic
108#41×C181—22no—
109#41×C181—41no—
110C2×Q8×C181C2311yescharacteristic
111C2×Q8×C181C2311yescharacteristic
112C2×D4×C181—21no—
113C2×D4×C181C2311yescharacteristic
114C22×C4×C181—21no—
115C22×C4×C181C2311yescharacteristic
116Q16×C181—41no—
117SD16×C181—22no—
118C2×C8×C181C2311yescharacteristic
119C2×C8×C181C2311yescharacteristic
120C4⋊C4×C181—21no—
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—42no—
126C22×D181—42no—
127C22×D181—81no—
128C22×D181C2×D411yescharacteristic
129Dic362—21no—
130Dic362—42no—
131Dic362C2×D412yes—
132D724—21no—
133D724—21no—
134D724—21no—
135D724—21no—
136D724—42no—
137C22⋊D181—22no—
138C22⋊D181—42no—
139C22⋊D181—42no—
140C22⋊D181—81no—
141C2×Dic181—21no—
142C2×Dic181—21no—
143C2×Dic181—81no—
144C2×Dic181C2×D411yescharacteristic
145C2×Dic181C2×D411yescharacteristic
146C2×Dic181C2×D411yescharacteristic
147C4×D181—21no—
148C4×D181—21no—
149C4×D181—42no—
150C4×D181—42no—
151#13089678—21no—
152#13089678C2×D412yes—
153C23×C181—21no—
154C23×C181C2×D411yescharacteristic
155Q8×C181—21no—
156Q8×C181—21no—
157Q8×C181—21no—
158Q8×C181—21no—
159D4×C181—21no—
160D4×C181—42no—
161D4×C181C2×D412yes—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181—42no—
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181—21no—
172C22×C181—21no—
173C22×C181—22no—
174C22×C181—21no—
175C22×C181—21no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181C22×D411yescharacteristic
179C2×D181—21no—
180C2×D181—21no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—41no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—84no—
188C2×D181#13512yescomplemented
189C4×C181—21no—
190C4×C181—41no—
191C4×C181—41no—
192C4×C181—41no—
193C4×C181C22×D411yescharacteristic
194C4×C181H(5,𝔽2)11yescharacteristic
195Dic181—21no—
196Dic181—22no—
197Dic181—21no—
198Dic181—21no—
199Dic181—41no—
200C2×C181—42no—
201C2×C181—41no—
202C2×C181D4211yescharacteristic
203C2×C181C2×#13511yescharacteristic
204C2×C181#57711yescharacteristic
205D181—22no—
206D181—81no—
207D181—81no—
208C181#328211yescharacteristic, complemented, Sylow
209#3282—1811nomaximal, Sylow
210C2×#135—1811no—
211C22×SD16—1811no—
212#548—1811no—
213#546—1811no—
214#495—1811no—
215#465—1811no—
216#465—1811no—
217#464—1811no—
218#463—1811no—
219#452D18111yescharacteristic, complemented
220#451—1811no—
221#449—1811no—
222#447—1811no—
223C4×SD16—1811no—
224C8×D4—1811no—
225C2×#41—1811no—
226C2×#41C2×D18111yescharacteristic
227C22×Q8—1811no—
228C22×D4—1811no—
229#135—3624no—
230C2×Q16—1811no—
231C2×Q16C2×D18111yescharacteristic, complemented
232C2×SD16—3624no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16—1811no—
237C2×SD16C2×D18111yescharacteristic
238C2×D8—1811no—
239C2×D8—1811no—
240C22×C8—1811no—
241C22×C8—1811no—
242#124—3621no—
243#124—1811no—
244#124—1811no—
245#123—3621no—
246C22⋊Q8—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8C2×D18111yescharacteristic, complemented
249#121—3621no—
250#121—3621no—
251#121—1811no—
252#121—1811no—
253#120—3621no—
254C4×Q8—1811no—
255C4×D4—3621no—
256C4×D4—1811no—
257C4×D4—1811no—
258C2×C22⋊C4—3621no—
259#106—1811no—
260C4⋊C8—1811no—
261#103—1811no—
262#103—1811no—
263#103C2×D18111yescharacteristic, complemented
264#102—1811no—
265#102—1811no—
266#102C2×D18111yescharacteristic
267C22⋊C8—1811no—
268C22⋊C8C2×D18111yescharacteristic, complemented
269C4×C8—1811no—
270C24—3621no—
271#41—22no—
272#41—41no—
273#41—3622no—
274#41—7241no—
275C2×Q8—3621no—
276C2×Q8—3622no—
277C2×Q8—1811no—
278C2×Q8—7241no—
279C2×Q8C22×D18111yescharacteristic
280C2×Q8C22×D18111yescharacteristic
281C2×D4—21no—
282C2×D4—3621no—
283C2×D4—3622no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—1811no—
288C2×D4—1811no—
289C2×D4—1811no—
290C2×D4—7241no—
291C2×D4C22×D18111yescharacteristic
292C22×C4—21no—
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—1811no—
297C22×C4C22×D18111yescharacteristic
298Q16—41no—
299Q16—3622no—
300SD16—22no—
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—1814no—
305D8—3622no—
306D8—7241no—
307C2×C8—3622no—
308C2×C8—3622no—
309C2×C8—1811no—
310C2×C8—1811no—
311C2×C8C22×D18111yescharacteristic
312C2×C8C22×D18111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3621no—
315C4⋊C4—3621no—
316C4⋊C4—1811no—
317C4⋊C4—1811no—
318C4⋊C4C22×D18111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—1811no—
329C22⋊C4—1811no—
330C42—3621no—
331C42—1811no—
332C23—21no—
333C23—3621no—
334C23—3621no—
335C23—14481no—
336C23—1811no—
337C23—7242no—
338C23—7242no—
339C23D4×D18111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3621no—
345Q8—1812no—
346Q8—7242no—
347D4—21no—
348D4—42no—
349D4—3622no—
350D4—3621no—
351D4—3621no—
352D4—3621no—
353D4—3621no—
354D4—14481no—
355D4—7242no—
356D4—7242no—
357D4—7242no—
358D4D4×D18112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—3621no—
366C2×C4—3621no—
367C2×C4—3621no—
368C2×C4—3621no—
369C2×C4—14481no—
370C2×C4—1811no—
371C2×C4—1811no—
372C2×C4—1811no—
373C2×C4—7242no—
374C2×C4—7242no—
375C2×C4D4×D18111yescharacteristic
376C2×C4C23×D18111yescharacteristic
377C8—21no—
378C8—21no—
379C8—3621no—
380C8—1812no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—3621no—
388C22—3621no—
389C22—14484no—
390C22—1812no—
391C22—7241no—
392C22—7241no—
393C22—7241no—
394C22—7241no—
395C22—7241no—
396C22—7241no—
397C22C2×D4×D18111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—3621no—
403C4—3622no—
404C4—3621no—
405C4—3621no—
406C4—7241no—
407C4C2×D4×D18111yescharacteristic
408C4#1915386111yescharacteristic
409C2—41no—
410C2—42no—
411C2—3622no—
412C2—14481no—
413C2—14481no—
414C2#2604191111yescharacteristic
415C2#135×D18111yescharacteristic
416C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#452×C181C2
socleC22×C181C22×D4
hypercenter#452D181
hypocenterC181#3282
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#19153836C2semidirect
2. C2×SD16×D181C2semidirect
3. #26041939C2semidirect
4. #26041917C2semidirect
5. #26041664C2semidirect
6. #26041511C2semidirect
7. #26041510C2semidirect
8. #26041504C2semidirect
9. #26041496C2semidirect
10. #26041433C2semidirect
11. #26041426C2semidirect
12. #26041414C2semidirect
13. #26041370C2semidirect
14. #26041342C2semidirect
15. #452×C181C2semidirect
16. C2×#15668449C22semidirect
20. C2×#15668439C22semidirect
24. C2×#15668436C22semidirect
28. C22×#13089678C22semidirect
31. #19153805C22semidirect
41. #19153781C22semidirect
46. #19153747C22semidirect
47. #19153745C22semidirect
48. #19153741C22semidirect
49. #19153740C22semidirect
50. #19153738C22semidirect
53. C8×Dic181C22semidirect
55. C2×Q16×C181C22semidirect
57. C22⋊Q8×C181C22semidirect
58. #103×C181C22semidirect
60. C22⋊C8×C181C22semidirect
83. #15668437D4semidirect
188. C2×D181#135semidirect
208. C181#3282semidirect
219. #452D181semidirect
231. C2×Q16C2×D181semidirect
248. C22⋊Q8C2×D181semidirect
263. #103C2×D181semidirect
268. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538362
C2×SD16×D1812
#260419392
#260419172
#260416642
#260415112
#260415102
#260415042
#260414962
#260414332
#260414262
#260414142
#260413702
#260413422
#452×C1812
#3282181