Finite Group Databasebeta

← #36335846

Subgroups — #36335846

Order 21376.

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×#19128852C211yesmaximal, characteristic, complemented
2C22×#14581937C211yesmaximal, characteristic, complemented
3#24923915C211yesmaximal, characteristic, complemented
4#24923651C211yesmaximal, characteristic, complemented
5#24923500C211yesmaximal, characteristic, complemented
6#24923496C211yesmaximal, characteristic, complemented
7#24923491C211yesmaximal, characteristic, complemented
8#24923481C211yesmaximal, characteristic, complemented
9#24923429C211yesmaximal, characteristic, complemented
10#24923425C211yesmaximal, characteristic, complemented
11#24923414C211yesmaximal, characteristic, complemented
12#24923402C211yesmaximal, characteristic, complemented
13C4×#14581937C211yesmaximal, characteristic, complemented
14D4×C167⋊C8C211yesmaximal, characteristic, complemented
15#548×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×D668C2211yescharacteristic
18#19128852—24no—
19C2×#14581940C2211yescharacteristic, complemented
20C2×#14581940C2211yescharacteristic, complemented
21C2×#14581938C2211yescharacteristic, complemented
22C2×#14581937—24no—
23C2×#14581937C2211yescharacteristic
24C2×#14581937C2211yescharacteristic
25C2×#14581937C2211yescharacteristic
26C2×#14581937C2211yescharacteristic
27C2×#14581935C2211yescharacteristic
28C2×#14581935C2211yescharacteristic
29C22×C167⋊C8C2211yescharacteristic
30C22×C167⋊C8C2211yescharacteristic, complemented
31#19128820C2211yescharacteristic
32#19128817—21no—
33#19128811C2211yescharacteristic
34#19128806C2211yescharacteristic
35#19128805—21no—
36#19128800—21no—
37C4×D668C2211yescharacteristic
38C22⋊C4×D167—21no—
39#19128763C2211yescharacteristic, complemented
40C4⋊(C167⋊C8)C2211yescharacteristic, complemented
41#19128756C2211yescharacteristic, complemented
42#19128756C2211yescharacteristic, complemented
43#19128755C2211yescharacteristic, complemented
44#19128753C2211yescharacteristic, complemented
45#19128752C2211yescharacteristic
46#19128752C2211yescharacteristic
47C22⋊(C167⋊C8)C2211yescharacteristic
48C22⋊(C167⋊C8)C2211yescharacteristic, complemented
49C4×C167⋊C8C2211yescharacteristic, complemented
50C2×#41×C167C2211yescharacteristic, complemented
51C22×Q8×C167C2211yescharacteristic
52#124×C167—21no—
53#124×C167C2211yescharacteristic, complemented
54C22⋊Q8×C167—21no—
55C22⋊Q8×C167C2211yescharacteristic, complemented
56#121×C167—21no—
57#121×C167C2211yescharacteristic
58C4×Q8×C167C2211yescharacteristic, complemented
59C4×D4×C167—21no—
60C4×D4×C167C2211yescharacteristic
61C23×D167—21no—
62#14581950—22no—
63#14581950—41no—
64C2×Dic334C2311yescharacteristic
65C2×D668—22no—
66C2×D668—21no—
67C2×D668—41no—
68C2×D668C2311yescharacteristic
69C2×D668C2311yescharacteristic
70C2×C22⋊D167—21no—
71C2×C22⋊D167—21no—
72C2×C4×D167—21no—
73C2×C4×D167—21no—
74#14581940—22no—
75#14581940—41no—
76#14581938—22no—
77#14581937—22no—
78#14581937—22no—
79#14581937—22no—
80#14581937D414yescomplemented
81#14581935—22no—
82#14581935—41no—
83C2×C167⋊C8—22no—
84C2×C167⋊C8—22no—
85C2×C167⋊C8C2311yescharacteristic
86C2×C167⋊C8C2311yescharacteristic
87C2×C167⋊C8C2311yescharacteristic
88C2×C167⋊C8C2311yescharacteristic
89C4⋊Dic167C2311yescharacteristic
90Dic167⋊C4—21no—
91C22⋊Dic167—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95(C2×D167)⋊C4—21no—
96#41×C167—22no—
97#41×C167—41no—
98C2×Q8×C167—21no—
99C2×Q8×C167—22no—
100C2×Q8×C167—41no—
101C2×Q8×C167C2311yescharacteristic
102C2×Q8×C167C2311yescharacteristic
103C2×D4×C167—21no—
104C2×D4×C167—21no—
105C2×D4×C167D411yescharacteristic
106C2×D4×C167C2311yescharacteristic
107C22×C4×C167—21no—
108C22×C4×C167—21no—
109C22×C4×C167C2311yescharacteristic
110C22×C4×C167C2311yescharacteristic
111C4⋊C4×C167—21no—
112C4⋊C4×C167—21no—
113C4⋊C4×C167D411yescharacteristic
114C4⋊C4×C167C2311yescharacteristic
115C22⋊C4×C167—21no—
116C22⋊C4×C167—21no—
117C22⋊C4×C167—21no—
118C22⋊C4×C167—21no—
119C22⋊C4×C167D411yescharacteristic
120C22⋊C4×C167D411yescharacteristic
121C42×C167—21no—
122C42×C167C2311yescharacteristic
123C22×D167—21no—
124C22×D167—21no—
125C22×D167—42no—
126C22×D167—42no—
127C22×D167—81no—
128Dic334—21no—
129Dic334—21no—
130D668—21no—
131D668—21no—
132D668—21no—
133D668—42no—
134D668C2×D412yes—
135C22⋊D167—42no—
136C22⋊D167—81no—
137C2×Dic167—21no—
138C2×Dic167—21no—
139C4×D167—42no—
140C4×D167—81no—
141C167⋊C8—21no—
142C167⋊C8—21no—
143C167⋊C8—21no—
144C167⋊C8C2×D412yes—
145C23×C167—21no—
146C23×C167C2×D411yescharacteristic
147C23×C167C2×D411yescharacteristic
148Q8×C167—21no—
149Q8×C167—21no—
150Q8×C167—21no—
151Q8×C167—42no—
152Q8×C167C2×D412yes—
153D4×C167—22no—
154D4×C167—21no—
155D4×C167—21no—
156D4×C167—42no—
157D4×C167—42no—
158C2×C4×C167—21no—
159C2×C4×C167—21no—
160C2×C4×C167—21no—
161C2×C4×C167—21no—
162C2×C4×C167—21no—
163C2×C4×C167—21no—
164C2×C4×C167—21no—
165C2×C4×C167—42no—
166C2×C4×C167—42no—
167C2×C4×C167C2×D411yescharacteristic
168C2×C4×C167C2×D411yescharacteristic
169C2×C4×C167C2×D411yescharacteristic
170C2×C4×C167C2×D411yescharacteristic
171C2×C4×C167C2411yescharacteristic
172C22×C167—21no—
173C22×C167—21no—
174C22×C167—21no—
175C22×C167—21no—
176C22×C167—21no—
177C22×C167—41no—
178C22×C167—41no—
179C22×C167—41no—
180C22×C167#13512yescomplemented
181C22×C167C22×D411yescharacteristic
182C2×D167—22no—
183C2×D167—41no—
184C2×D167—41no—
185C2×D167—41no—
186C2×D167—41no—
187C2×D167—41no—
188C2×D167—84no—
189C4×C167—21no—
190C4×C167—21no—
191C4×C167—22no—
192C4×C167—21no—
193C4×C167—21no—
194C4×C167—41no—
195C4×C167—41no—
196C4×C167C22×D411yescharacteristic
197C4×C167H(5,𝔽2)11yescharacteristic
198Dic167—41no—
199Dic167—41no—
200C2×C167—22no—
201C2×C167—41no—
202C2×C167—81no—
203C2×C167D4211yescharacteristic
204C2×C167C2×#13511yescharacteristic
205C2×C167#57711yescharacteristic
206D167—42no—
207D167—81no—
208C167#328211yescharacteristic, complemented, Sylow
209#3282—1671nomaximal, Sylow
210C2×#135—1671no—
211C22×SD16—1671no—
212#548D16711yescharacteristic, complemented
213#546—1671no—
214#495—1671no—
215#465—1671no—
216#465—1671no—
217#464—1671no—
218#463—1671no—
219#452—1671no—
220#451—1671no—
221#449—1671no—
222#447—1671no—
223C4×SD16—1671no—
224C8×D4—1671no—
225C2×#41—1671no—
226C2×#41C2×D16711yescharacteristic, complemented
227C22×Q8C2×D16711yescharacteristic
228C22×D4—1671no—
229#135—3344no—
230C2×Q16—1671no—
231C2×Q16—1671no—
232C2×SD16—3344no—
233C2×SD16—1671no—
234C2×SD16—1671no—
235C2×SD16—1671no—
236C2×SD16—1671no—
237C2×SD16—1671no—
238C2×D8—1671no—
239C2×D8—1671no—
240C22×C8—1671no—
241C22×C8—1671no—
242#124—21no—
243#124—1671no—
244#124C2×D16711yescharacteristic, complemented
245#123—3341no—
246C22⋊Q8—21no—
247C22⋊Q8—1671no—
248C22⋊Q8C2×D16711yescharacteristic, complemented
249#121—21no—
250#121—3341no—
251#121—1671no—
252#121C2×D16711yescharacteristic
253#120—3341no—
254C4×Q8C2×D16711yescharacteristic, complemented
255C4×D4—21no—
256C4×D4—1671no—
257C4×D4C2×D16711yescharacteristic
258C2×C22⋊C4—3341no—
259#106—1671no—
260C4⋊C8—1671no—
261#103—1671no—
262#103—1671no—
263#103—1671no—
264#102—1671no—
265#102—1671no—
266#102—1671no—
267C22⋊C8—1671no—
268C22⋊C8—1671no—
269C4×C8—1671no—
270C24—3341no—
271#41—22no—
272#41—41no—
273#41—3342no—
274#41—6681no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—1671no—
279C2×Q8C22×D16711yescharacteristic
280C2×Q8C22×D16711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3342no—
284C2×D4—3341no—
285C2×D4—3341no—
286C2×D4—3341no—
287C2×D4—6681no—
288C2×D4—1671no—
289C2×D4—1671no—
290C2×D4C22⋊D16711yescharacteristic
291C2×D4C22×D16711yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—3341no—
295C22×C4—3341no—
296C22×C4C22×D16711yescharacteristic
297C22×C4C22×D16711yescharacteristic
298Q16—3342no—
299Q16—6681no—
300SD16—3342no—
301SD16—3342no—
302SD16—3342no—
303SD16—3342no—
304SD16—1674no—
305D8—3342no—
306D8—6681no—
307C2×C8—3342no—
308C2×C8—3342no—
309C2×C8—1671no—
310C2×C8—1671no—
311C2×C8—1671no—
312C2×C8—1671no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—3341no—
316C4⋊C4—1671no—
317C4⋊C4C22⋊D16711yescharacteristic
318C4⋊C4C22×D16711yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3341no—
324C22⋊C4—3341no—
325C22⋊C4—3341no—
326C22⋊C4—3341no—
327C22⋊C4—3341no—
328C22⋊C4C22⋊D16711yescharacteristic
329C22⋊C4C22⋊D16711yescharacteristic
330C42—21no—
331C42C22×D16711yescharacteristic
332C23—21no—
333C23—13361no—
334C23—3341no—
335C23—3341no—
336C23—6682no—
337C23—6682no—
338C23C2×C22⋊D16711yescharacteristic
339C23C2×C22⋊D16711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—42no—
344Q8—3341no—
345Q8—3341no—
346Q8D4×D16712yes—
347D4—22no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—13361no—
353D4—3341no—
354D4—3341no—
355D4—3341no—
356D4—6682no—
357D4—6682no—
358D4—1672no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—13361no—
369C2×C4—3341no—
370C2×C4—3341no—
371C2×C4—6682no—
372C2×C4C2×C22⋊D16711yescharacteristic
373C2×C4C2×C22⋊D16711yescharacteristic
374C2×C4C2×C22⋊D16711yescharacteristic
375C2×C4C2×C22⋊D16711yescharacteristic
376C2×C4C23×D16711yescharacteristic
377C8—3341no—
378C8—3341no—
379C8—3341no—
380C8—1672no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—13364no—
390C22—3342no—
391C22—6681no—
392C22—6681no—
393C22—6681no—
394C22—6681no—
395C22—6681no—
396C22#1912885212yescomplemented
397C22C22×C22⋊D16711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—6681no—
406C4—6681no—
407C4C2×D4×D16711yescharacteristic
408C4#1912887711yescharacteristic
409C2—22no—
410C2—41no—
411C2—81no—
412C2—13361no—
413C2—6682no—
414C2D4×C22⋊D16711yescharacteristic
415C2C2×#1912885211yescharacteristic
416C2#2492407711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#548×C167C2
socleC22×C167C22×D4
hypercenter#548D167
hypocenterC167#3282
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19128852C2semidirect
2. C22×#14581937C2semidirect
3. #24923915C2semidirect
4. #24923651C2semidirect
5. #24923500C2semidirect
6. #24923496C2semidirect
7. #24923491C2semidirect
8. #24923481C2semidirect
9. #24923429C2semidirect
10. #24923425C2semidirect
11. #24923414C2semidirect
12. #24923402C2semidirect
13. C4×#14581937C2semidirect
14. D4×C167⋊C8C2semidirect
15. #548×C167C2semidirect
19. C2×#14581940C22semidirect
20. C2×#14581940C22semidirect
21. C2×#14581938C22semidirect
30. C22×C167⋊C8C22semidirect
39. #19128763C22semidirect
40. C4⋊(C167⋊C8)C22semidirect
41. #19128756C22semidirect
42. #19128756C22semidirect
43. #19128755C22semidirect
44. #19128753C22semidirect
48. C22⋊(C167⋊C8)C22semidirect
49. C4×C167⋊C8C22semidirect
50. C2×#41×C167C22semidirect
53. #124×C167C22semidirect
55. C22⋊Q8×C167C22semidirect
58. C4×Q8×C167C22semidirect
80. #14581937D4semidirect
180. C22×C167#135semidirect
208. C167#3282semidirect
212. #548D167semidirect
226. C2×#41C2×D167semidirect
244. #124C2×D167semidirect
248. C22⋊Q8C2×D167semidirect
254. C4×Q8C2×D167semidirect
396. C22#19128852semidirect

Maximal subgroups

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

subgroupindex
C2×#191288522
C22×#145819372
#249239152
#249236512
#249235002
#249234962
#249234912
#249234812
#249234292
#249234252
#249234142
#249234022
C4×#145819372
D4×C167⋊C82
#548×C1672
#3282167