Finite Group Databasebeta

← #36336161

Subgroups — #36336161

Order 21376.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923959C211yesmaximal, characteristic, complemented
2D4×D668C211yesmaximal, characteristic
3#24923634C211yesmaximal, characteristic, complemented
4#24923594C211yesmaximal, characteristic, complemented
5#24923583C211yesmaximal, characteristic, complemented
6#24923570C211yesmaximal, characteristic, complemented
7#24923515C211yesmaximal, characteristic, complemented
8#24923498C211yesmaximal, characteristic, complemented
9#24923470C211yesmaximal, characteristic, complemented
10#24923422C211yesmaximal, characteristic, complemented
11#24923422C211yesmaximal, characteristic, complemented
12#24923371C211yesmaximal, characteristic, complemented
13#24923275C211yesmaximal, characteristic, complemented
14C2×#19128754C211yesmaximal, characteristic, complemented
15#435×C167C211yesmaximal, characteristic, complemented
16C22×D668C2211yescharacteristic
17C22×D668C2211yescharacteristic
18C2×D4×D167—21no—
19C2×#14581939C2211yescharacteristic
20C2×#14581939C2211yescharacteristic, complemented
21C2×#14581939C2211yescharacteristic, complemented
22#19128829C2211yescharacteristic
23#19128825—21no—
24#19128824C2211yescharacteristic, complemented
25#19128822—21no—
26#19128813—21no—
27#19128811C2211yescharacteristic, complemented
28#19128811C2211yescharacteristic, complemented
29#19128807—21no—
30#19128806C2211yescharacteristic
31#19128806C2211yescharacteristic
32#19128802—21no—
33#19128802—21no—
34C4×Dic334C2211yescharacteristic, complemented
35C4×D668C2211yescharacteristic
36#19128791C2211yescharacteristic, complemented
37#19128788—21no—
38C2×C4⋊Dic167C2211yescharacteristic, complemented
39#19128766C2211yescharacteristic, complemented
40#19128766C2211yescharacteristic, complemented
41#19128764C2211yescharacteristic, complemented
42#19128757C2211yescharacteristic, complemented
43#19128754C2211yescharacteristic
44#19128754C2211yescharacteristic
45#19128754C2211yescharacteristic
46#19128754C2211yescharacteristic, complemented
47#19128754C2211yescharacteristic, complemented
48#19128754C2211yescharacteristic, complemented
49#19128754C2211yescharacteristic, complemented
50#19128754C2211yescharacteristic, complemented
51#19128754C2211yescharacteristic, complemented
52C2×M16×C167C2211yescharacteristic, complemented
53C22×C8×C167C2211yescharacteristic, complemented
54C4×D4×C167C2211yescharacteristic
55C4⋊C8×C167C2211yescharacteristic, complemented
56C22⋊C8×C167C2211yescharacteristic, complemented
57C22⋊C8×C167C2211yescharacteristic, complemented
58#97×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C23×D167—21no—
61C2×Dic334C2311yescharacteristic
62C2×D668—22no—
63C2×D668—22no—
64C2×D668—21no—
65C2×D668—21no—
66C2×D668—21no—
67C2×D668—21no—
68C2×D668—41no—
69C2×D668—41no—
70C2×D668C2311yescharacteristic
71C2×D668C2311yescharacteristic
72C2×D668C2311yescharacteristic
73C2×C22⋊D167—21no—
74C2×C22⋊D167—21no—
75D4×D167—44no—
76C22×Dic167—21no—
77C2×C4×D167—21no—
78#14581939—22no—
79#14581939—41no—
80#14581939—41no—
81C4⋊Dic167—21no—
82C4⋊Dic167C2311yescharacteristic
83C4⋊Dic167C2311yescharacteristic
84C4⋊Dic167C2311yescharacteristic
85C4⋊Dic167C2311yescharacteristic
86Dic167⋊C4—21no—
87Dic167⋊C4—21no—
88Dic167⋊C4—21no—
89Dic167⋊C4—21no—
90Dic167⋊C4—21no—
91C22⋊Dic167—21no—
92C22⋊Dic167—21no—
93C22⋊Dic167—21no—
94(C2×D167)⋊C4—21no—
95(C2×D167)⋊C4—21no—
96(C2×D167)⋊C4—21no—
97C4×Dic167—21no—
98C4×Dic167—21no—
99C2×D4×C167D411yescharacteristic
100C22×C4×C167C2311yescharacteristic
101C22×C4×C167C2311yescharacteristic
102M16×C167—41no—
103C2×C8×C167—41no—
104C2×C8×C167C2311yescharacteristic
105C2×C8×C167C2311yescharacteristic
106C2×C8×C167C2311yescharacteristic
107C2×C8×C167C2311yescharacteristic
108C4⋊C4×C167D411yescharacteristic
109C22⋊C4×C167D411yescharacteristic
110C22⋊C4×C167D411yescharacteristic
111C42×C167C2311yescharacteristic
112C22×D167—21no—
113C22×D167—21no—
114C22×D167—21no—
115C22×D167—42no—
116C22×D167—42no—
117C22×D167—42no—
118C22×D167—42no—
119C22×D167—81no—
120C22×D167—81no—
121Dic334—21no—
122Dic334—21no—
123D668—21no—
124D668—21no—
125D668—21no—
126D668—21no—
127D668—21no—
128D668—42no—
129D668—42no—
130D668—42no—
131D668—81no—
132D668#4112yes—
133C22⋊D167—42no—
134C22⋊D167—42no—
135C2×Dic167—21no—
136C2×Dic167—21no—
137C2×Dic167—21no—
138C2×Dic167—21no—
139C2×Dic167—21no—
140C2×Dic167—42no—
141C4×D167—42no—
142C23×C167C2×D411yescharacteristic
143C23×C167C2×D411yescharacteristic
144D4×C167—22no—
145C2×C4×C167—21no—
146C2×C4×C167—21no—
147C2×C4×C167—21no—
148C2×C4×C167—21no—
149C2×C4×C167C2×D411yescharacteristic
150C2×C4×C167C2×D411yescharacteristic
151C2×C4×C167C2×D411yescharacteristic
152C2×C4×C167C2×D411yescharacteristic
153C2×C4×C167C2411yescharacteristic
154C8×C167—21no—
155C8×C167—21no—
156C8×C167—21no—
157C8×C167—21no—
158C22×C167—21no—
159C22×C167—21no—
160C22×C167—21no—
161C22×C167—21no—
162C22×C167—21no—
163C22×C167AGL(1,ℤ/8)12yes—
164C22×C167C22×D411yescharacteristic
165C2×D167—22no—
166C2×D167—41no—
167C2×D167—41no—
168C2×D167—41no—
169C2×D167—41no—
170C2×D167—41no—
171C2×D167—41no—
172C2×D167—41no—
173C2×D167—41no—
174C2×D167—84no—
175C2×D167—84no—
176C4×C167—21no—
177C4×C167—21no—
178C4×C167—21no—
179C4×C167—21no—
180C4×C167C2×#4111yescharacteristic
181C4×C167H(5,𝔽2)11yescharacteristic
182Dic167—41no—
183Dic167—41no—
184Dic167—41no—
185Dic167—41no—
186Dic167—41no—
187C2×C167—22no—
188C2×C167—41no—
189C2×C167#54611yescharacteristic
190C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C167#57711yescharacteristic
192D167—42no—
193D167—81no—
194D167—81no—
195C167#331011yescharacteristic, complemented, Sylow
196#3310—1671nomaximal, Sylow
197#552—1671no—
198D42—1671no—
199#490—1671no—
200#482—1671no—
201#480—1671no—
202#478—1671no—
203#468—1671no—
204#465—1671no—
205#460—1671no—
206#450—1671no—
207#450—1671no—
208#440—1671no—
209#435D16711yescharacteristic, complemented
210#418—1671no—
211C2×#102—1671no—
212C22×D4—3341no—
213C22×D4—1671no—
214C22×D4—1671no—
215C2×SD16—1671no—
216C2×SD16—1671no—
217C2×SD16—1671no—
218C2×M16C2×D16711yescharacteristic, complemented
219C22×C8C2×D16711yescharacteristic, complemented
220#127—1671no—
221#126—3341no—
222#125—3341no—
223#125—1671no—
224#123—3341no—
225C22⋊Q8—1671no—
226C22⋊Q8—1671no—
227#121—3341no—
228#121—1671no—
229#121—1671no—
230#120—3341no—
231#120—3341no—
232C4×Q8—1671no—
233C4×D4—1671no—
234C4×D4C2×D16711yescharacteristic
235#117—3341no—
236#117—1671no—
237C2×C4⋊C4—1671no—
238#107—1671no—
239#107—1671no—
240#106—1671no—
241C4⋊C8C2×D16711yescharacteristic, complemented
242#103—1671no—
243#102—1671no—
244#102—1671no—
245#102—1671no—
246#102—1671no—
247#102—1671no—
248#102—1671no—
249#102—1671no—
250#102—1671no—
251#102—1671no—
252C22⋊C8C2×D16711yescharacteristic, complemented
253C22⋊C8C2×D16711yescharacteristic, complemented
254#97C2×D16711yescharacteristic, complemented
255C24—3341no—
256C24—3341no—
257C2×Q8—1671no—
258C2×D4—3342no—
259C2×D4—3342no—
260C2×D4—3341no—
261C2×D4—3341no—
262C2×D4—3341no—
263C2×D4—3341no—
264C2×D4—3341no—
265C2×D4—3341no—
266C2×D4—6681no—
267C2×D4—6681no—
268C2×D4—6684no—
269C2×D4—1671no—
270C2×D4—1671no—
271C2×D4—1671no—
272C2×D4D66811yescharacteristic
273C22×C4—3341no—
274C22×C4—3341no—
275C22×C4C22×D16711yescharacteristic
276C22×C4C22×D16711yescharacteristic
277SD16—3342no—
278SD16—6681no—
279SD16—6681no—
280M16—41no—
281C2×C8—41no—
282C2×C8C22×D16711yescharacteristic
283C2×C8C22×D16711yescharacteristic
284C2×C8C22×D16711yescharacteristic
285C2×C8C22×D16711yescharacteristic
286C4⋊C4—3341no—
287C4⋊C4—3341no—
288C4⋊C4—3341no—
289C4⋊C4—3341no—
290C4⋊C4—3341no—
291C4⋊C4—3341no—
292C4⋊C4—1671no—
293C4⋊C4—1671no—
294C4⋊C4—1671no—
295C4⋊C4—1671no—
296C4⋊C4D66811yescharacteristic
297C22⋊C4—3341no—
298C22⋊C4—3341no—
299C22⋊C4—3341no—
300C22⋊C4—3341no—
301C22⋊C4—3341no—
302C22⋊C4—3341no—
303C22⋊C4D66811yescharacteristic
304C22⋊C4D66811yescharacteristic
305C42—3341no—
306C42—3341no—
307C42C22×D16711yescharacteristic
308C23—13361no—
309C23—13361no—
310C23—3341no—
311C23—3341no—
312C23—3341no—
313C23—6682no—
314C23—6682no—
315C23—6682no—
316C23—6682no—
317C23C2×D66811yescharacteristic
318C23C2×D66811yescharacteristic
319Q8—3341no—
320Q8—3341no—
321D4—22no—
322D4—13361no—
323D4—3341no—
324D4—3341no—
325D4—3341no—
326D4—3341no—
327D4—3341no—
328D4—6682no—
329D4—6682no—
330D4—6682no—
331D4—6682no—
332D4—6682no—
333D4—1672no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—3341no—
339C2×C4—3341no—
340C2×C4—3341no—
341C2×C4—3341no—
342C2×C4—3341no—
343C2×C4—6682no—
344C2×C4—6682no—
345C2×C4C2×D66811yescharacteristic
346C2×C4C2×D66811yescharacteristic
347C2×C4C2×D66811yescharacteristic
348C2×C4C2×D66811yescharacteristic
349C2×C4C23×D16711yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—13364no—
360C22—13364no—
361C22—3342no—
362C22—6681no—
363C22—6681no—
364C22—6681no—
365C22—6681no—
366C22—6681no—
367C22—6681no—
368C22—6681no—
369C22—6681no—
370C22#1912886012yes—
371C22C22×D66811yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—6681no—
377C4—6681no—
378C4—6681no—
379C4—6681no—
380C4—6681no—
381C4#41×D16711yescharacteristic
382C4#1912887711yescharacteristic
383C2—22no—
384C2—41no—
385C2—13361no—
386C2—13361no—
387C2—6682no—
388C2#2492391411yescharacteristic
389C2C2×#1912886011yescharacteristic
390C2#2492408111yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#435×C167C2
socleC22×C167C22×D4
prime radicalD4×D668C2
hypercenter#435D167
hypocenterC167#3310
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #24923959C2semidirect
3. #24923634C2semidirect
4. #24923594C2semidirect
5. #24923583C2semidirect
6. #24923570C2semidirect
7. #24923515C2semidirect
8. #24923498C2semidirect
9. #24923470C2semidirect
10. #24923422C2semidirect
11. #24923422C2semidirect
12. #24923371C2semidirect
13. #24923275C2semidirect
14. C2×#19128754C2semidirect
15. #435×C167C2semidirect
20. C2×#14581939C22semidirect
21. C2×#14581939C22semidirect
24. #19128824C22semidirect
27. #19128811C22semidirect
28. #19128811C22semidirect
34. C4×Dic334C22semidirect
36. #19128791C22semidirect
38. C2×C4⋊Dic167C22semidirect
39. #19128766C22semidirect
40. #19128766C22semidirect
41. #19128764C22semidirect
42. #19128757C22semidirect
46. #19128754C22semidirect
47. #19128754C22semidirect
48. #19128754C22semidirect
49. #19128754C22semidirect
50. #19128754C22semidirect
51. #19128754C22semidirect
52. C2×M16×C167C22semidirect
53. C22×C8×C167C22semidirect
55. C4⋊C8×C167C22semidirect
56. C22⋊C8×C167C22semidirect
57. C22⋊C8×C167C22semidirect
58. #97×C167C22semidirect
195. C167#3310semidirect
209. #435D167semidirect
218. C2×M16C2×D167semidirect
219. C22×C8C2×D167semidirect
241. C4⋊C8C2×D167semidirect
252. C22⋊C8C2×D167semidirect
253. C22⋊C8C2×D167semidirect
254. #97C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239592
D4×D6682
#249236342
#249235942
#249235832
#249235702
#249235152
#249234982
#249234702
#249234222
#249233712
#249232752
C2×#191287542
#435×C1672
#3310167