Finite Group Databasebeta

← #36335649

Subgroups — #36335649

Order 21376.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128863C211yesmaximal, characteristic, complemented
2C22×#14581937C211yesmaximal, characteristic, complemented
3#24923915C211yesmaximal, characteristic
4#24923661C211yesmaximal, characteristic, complemented
5#24923517C211yesmaximal, characteristic, complemented
6#24923508C211yesmaximal, characteristic, complemented
7#24923491C211yesmaximal, characteristic, complemented
8#24923471C211yesmaximal, characteristic, complemented
9#24923429C211yesmaximal, characteristic, complemented
10#24923425C211yesmaximal, characteristic, complemented
11#24923419C211yesmaximal, characteristic, complemented
12#24923408C211yesmaximal, characteristic, complemented
13#24923367C211yesmaximal, characteristic, complemented
14#24923341C211yesmaximal, characteristic, complemented
15D4×Q8×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×D668C2211yescharacteristic
18#19128863—24no—
19C2×#14581940C2211yescharacteristic, complemented
20C2×#14581940C2211yescharacteristic, complemented
21C2×#14581940C2211yescharacteristic, complemented
22C2×#14581937—24no—
23C2×#14581937C2211yescharacteristic
24C2×#14581937C2211yescharacteristic
25C2×#14581937C2211yescharacteristic
26C2×#14581937C2211yescharacteristic
27C2×#14581937C2211yescharacteristic
28C2×#14581937C2211yescharacteristic
29C2×#14581934C2211yescharacteristic, complemented
30C22×C167⋊C8C2211yescharacteristic
31#19128820C2211yescharacteristic
32#19128817—21no—
33#19128811C2211yescharacteristic
34#19128806C2211yescharacteristic
35#19128805—21no—
36#19128800—21no—
37C4×D668C2211yescharacteristic
38C22⋊C4×D167—21no—
39#19128765C2211yescharacteristic, complemented
40C4⋊(C167⋊C8)C2211yescharacteristic, complemented
41#19128756C2211yescharacteristic, complemented
42#19128756C2211yescharacteristic, complemented
43#19128756C2211yescharacteristic, complemented
44#19128755C2211yescharacteristic, complemented
45#19128752C2211yescharacteristic
46#19128752C2211yescharacteristic
47C22⋊(C167⋊C8)C2211yescharacteristic
48C22⋊(C167⋊C8)C2211yescharacteristic, complemented
49#19128743C2211yescharacteristic, complemented
50C22×Q8×C167C2211yescharacteristic
51C22×Q8×C167C2211yescharacteristic, complemented
52C4⋊Q8×C167—21no—
53C4⋊Q8×C167C2211yescharacteristic, complemented
54C22⋊Q8×C167—21no—
55C22⋊Q8×C167—21no—
56C22⋊Q8×C167C2211yescharacteristic
57C22⋊Q8×C167C2211yescharacteristic, complemented
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—22no—
76#14581940—41no—
77#14581937—22no—
78#14581937—22no—
79#14581937—22no—
80#14581937—22no—
81#14581937—41no—
82#14581937D414yescomplemented
83#14581934—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—
96C2×Q8×C167—21no—
97C2×Q8×C167—21no—
98C2×Q8×C167—22no—
99C2×Q8×C167—22no—
100C2×Q8×C167—41no—
101C2×Q8×C167—41no—
102C2×Q8×C167C2311yescharacteristic
103C2×Q8×C167C2311yescharacteristic
104C2×Q8×C167C2311yescharacteristic
105C2×D4×C167D411yescharacteristic
106C22×C4×C167—21no—
107C22×C4×C167—21no—
108C22×C4×C167C2311yescharacteristic
109C22×C4×C167C2311yescharacteristic
110C4⋊C4×C167—21no—
111C4⋊C4×C167—21no—
112C4⋊C4×C167—21no—
113C4⋊C4×C167—21no—
114C4⋊C4×C167—21no—
115C4⋊C4×C167D411yescharacteristic
116C4⋊C4×C167C2311yescharacteristic
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×C167C2×D411yescharacteristic
146C23×C167C2×D411yescharacteristic
147Q8×C167—21no—
148Q8×C167—21no—
149Q8×C167—21no—
150Q8×C167—21no—
151Q8×C167—21no—
152Q8×C167—42no—
153Q8×C167—42no—
154Q8×C167C2×D412yes—
155D4×C167—22no—
156C2×C4×C167—21no—
157C2×C4×C167—21no—
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—42no—
165C2×C4×C167—42no—
166C2×C4×C167C2×D411yescharacteristic
167C2×C4×C167C2×D411yescharacteristic
168C2×C4×C167C2×D411yescharacteristic
169C2×C4×C167C2×D411yescharacteristic
170C2×C4×C167C2411yescharacteristic
171C22×C167—21no—
172C22×C167—21no—
173C22×C167—21no—
174C22×C167—21no—
175C22×C167—21no—
176C22×C167#13712yescomplemented
177C22×C167C22×D411yescharacteristic
178C2×D167—22no—
179C2×D167—41no—
180C2×D167—41no—
181C2×D167—41no—
182C2×D167—41no—
183C2×D167—41no—
184C2×D167—84no—
185C4×C167—21no—
186C4×C167—21no—
187C4×C167—22no—
188C4×C167—21no—
189C4×C167—21no—
190C4×C167—41no—
191C4×C167—41no—
192C4×C167—41no—
193C4×C167C22×D411yescharacteristic
194C4×C167H(5,𝔽2)11yescharacteristic
195Dic167—41no—
196Dic167—41no—
197C2×C167—22no—
198C2×C167—41no—
199C2×C167D4211yescharacteristic
200C2×C167C2×#13711yescharacteristic
201C2×C167#57711yescharacteristic
202D167—42no—
203D167—81no—
204C167#326611yescharacteristic, complemented, Sylow
205#3266—1671nomaximal, Sylow
206C2×#137—1671no—
207C22×SD16—1671no—
208D4×Q8D16711yescharacteristic, complemented
209#546—1671no—
210#497—1671no—
211#468—1671no—
212#467—1671no—
213#464—1671no—
214#461—1671no—
215#452—1671no—
216#451—1671no—
217#450—1671no—
218#448—1671no—
219#440—1671no—
220#435—1671no—
221C2×#41—1671no—
222C22×Q8C2×D16711yescharacteristic
223C22×Q8C2×D16711yescharacteristic, complemented
224C22×D4—1671no—
225#137—3344no—
226C2×Q16—1671no—
227C2×Q16—1671no—
228C2×Q16—1671no—
229C2×SD16—3344no—
230C2×SD16—1671no—
231C2×SD16—1671no—
232C2×SD16—1671no—
233C2×SD16—1671no—
234C2×SD16—1671no—
235C2×SD16—1671no—
236C2×M16—1671no—
237C22×C8—1671no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D16711yescharacteristic, complemented
240#124—1671no—
241#123—3341no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1671no—
245C22⋊Q8C2×D16711yescharacteristic
246C22⋊Q8C2×D16711yescharacteristic, complemented
247#121—3341no—
248#121—1671no—
249#120—3341no—
250C4×Q8C2×D16711yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1671no—
253C4×D4C2×D16711yescharacteristic
254C2×C22⋊C4—3341no—
255#107—1671no—
256C4⋊C8—1671no—
257#103—1671no—
258#103—1671no—
259#103—1671no—
260#103—1671no—
261#102—1671no—
262#102—1671no—
263C22⋊C8—1671no—
264C22⋊C8—1671no—
265#97—1671no—
266C24—3341no—
267#41—3342no—
268#41—6681no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1671no—
276C2×Q8C22×D16711yescharacteristic
277C2×Q8C22×D16711yescharacteristic
278C2×Q8C22×D16711yescharacteristic
279C2×D4—3342no—
280C2×D4—3341no—
281C2×D4—3341no—
282C2×D4—3341no—
283C2×D4—6681no—
284C2×D4—1671no—
285C2×D4—1671no—
286C2×D4C22⋊D16711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3341no—
290C22×C4—3341no—
291C22×C4C22×D16711yescharacteristic
292C22×C4C22×D16711yescharacteristic
293Q16—3342no—
294Q16—3342no—
295Q16—6681no—
296SD16—3342no—
297SD16—3342no—
298SD16—3342no—
299SD16—3342no—
300SD16—6681no—
301SD16—1674no—
302M16—3342no—
303C2×C8—3342no—
304C2×C8—1671no—
305C2×C8—1671no—
306C2×C8—1671no—
307C2×C8—1671no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—3341no—
314C4⋊C4—1671no—
315C4⋊C4C22⋊D16711yescharacteristic
316C4⋊C4C22×D16711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—3341no—
320C22⋊C4—3341no—
321C22⋊C4—3341no—
322C22⋊C4—3341no—
323C22⋊C4—3341no—
324C22⋊C4C22⋊D16711yescharacteristic
325C22⋊C4C22⋊D16711yescharacteristic
326C42—21no—
327C42C22×D16711yescharacteristic
328C23—13361no—
329C23—3341no—
330C23—3341no—
331C23—6682no—
332C23—6682no—
333C23C2×C22⋊D16711yescharacteristic
334C23C2×C22⋊D16711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—3341no—
343Q8—3341no—
344Q8D4×D16712yes—
345D4—22no—
346D4—13361no—
347D4—3341no—
348D4—3341no—
349D4—3341no—
350D4—6682no—
351D4—6682no—
352D4—1672no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—13361no—
364C2×C4—3341no—
365C2×C4—3341no—
366C2×C4—6682no—
367C2×C4C2×C22⋊D16711yescharacteristic
368C2×C4C2×C22⋊D16711yescharacteristic
369C2×C4C2×C22⋊D16711yescharacteristic
370C2×C4C2×C22⋊D16711yescharacteristic
371C2×C4C23×D16711yescharacteristic
372C8—3341no—
373C8—3341no—
374C8—3341no—
375C8—1672no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—13364no—
382C22—3342no—
383C22—6681no—
384C22—6681no—
385C22—6681no—
386C22—6681no—
387C22—6681no—
388C22#1912886312yescomplemented
389C22C22×C22⋊D16711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—6681no—
399C4—6681no—
400C4C2×D4×D16711yescharacteristic
401C4#1912887711yescharacteristic
402C2—22no—
403C2—41no—
404C2—13361no—
405C2—6682no—
406C2D4×C22⋊D16711yescharacteristic
407C2C2×#1912886311yescharacteristic
408C2#2492407711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingD4×Q8×C167C2
socleC22×C167C22×D4
prime radical#24923915C2
hypercenterD4×Q8D167
hypocenterC167#3266
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19128863C2semidirect
2. C22×#14581937C2semidirect
4. #24923661C2semidirect
5. #24923517C2semidirect
6. #24923508C2semidirect
7. #24923491C2semidirect
8. #24923471C2semidirect
9. #24923429C2semidirect
10. #24923425C2semidirect
11. #24923419C2semidirect
12. #24923408C2semidirect
13. #24923367C2semidirect
14. #24923341C2semidirect
15. D4×Q8×C167C2semidirect
19. C2×#14581940C22semidirect
20. C2×#14581940C22semidirect
21. C2×#14581940C22semidirect
29. C2×#14581934C22semidirect
39. #19128765C22semidirect
40. C4⋊(C167⋊C8)C22semidirect
41. #19128756C22semidirect
42. #19128756C22semidirect
43. #19128756C22semidirect
44. #19128755C22semidirect
48. C22⋊(C167⋊C8)C22semidirect
49. #19128743C22semidirect
51. C22×Q8×C167C22semidirect
53. C4⋊Q8×C167C22semidirect
57. C22⋊Q8×C167C22semidirect
58. C4×Q8×C167C22semidirect
82. #14581937D4semidirect
176. C22×C167#137semidirect
204. C167#3266semidirect
208. D4×Q8D167semidirect
223. C22×Q8C2×D167semidirect
239. C4⋊Q8C2×D167semidirect
246. C22⋊Q8C2×D167semidirect
250. C4×Q8C2×D167semidirect
388. C22#19128863semidirect

Maximal subgroups

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

subgroupindex
C2×#191288632
C22×#145819372
#249239152
#249236612
#249235172
#249235082
#249234912
#249234712
#249234292
#249234252
#249234192
#249234082
#249233672
#249233412
D4×Q8×C1672
#3266167