Finite Group Databasebeta

← #36336185

Subgroups — #36336185

Order 21376.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923954C211yesmaximal, characteristic
2Q8×C22⋊D167C211yesmaximal, characteristic
3#24923636C211yesmaximal, characteristic, complemented
4#24923609C211yesmaximal, characteristic, complemented
5#24923599C211yesmaximal, characteristic, complemented
6#24923578C211yesmaximal, characteristic, complemented
7#24923526C211yesmaximal, characteristic, complemented
8#24923510C211yesmaximal, characteristic, complemented
9#24923478C211yesmaximal, characteristic, complemented
10#24923426C211yesmaximal, characteristic, complemented
11#24923425C211yesmaximal, characteristic, complemented
12#24923373C211yesmaximal, characteristic, complemented
13#24923340C211yesmaximal, characteristic
14#24923276C211yesmaximal, characteristic, complemented
15C2×#103×C167C211yesmaximal, characteristic, complemented
16C2×Q8×D167C2211yescharacteristic
17C2×Dic668C2211yescharacteristic
18C2×#14581940C2211yescharacteristic
19C2×#14581940C2211yescharacteristic
20C2×#14581933C2211yescharacteristic
21#19128830—21no—
22#19128830C2211yescharacteristic
23#19128827—21no—
24#19128823C2211yescharacteristic
25#19128822—21no—
26#19128815—21no—
27#19128812C2211yescharacteristic
28#19128811—21no—
29#19128811C2211yescharacteristic
30#19128811C2211yescharacteristic
31#19128808—21no—
32#19128808C2211yescharacteristic
33Q8×Dic167C2211yescharacteristic
34#19128797C2211yescharacteristic
35C4×C22⋊D167—21no—
36C4×C22⋊D167C2211yescharacteristic
37#19128791—21no—
38#19128790C2211yescharacteristic
39#19128766C2211yescharacteristic
40#19128763C2211yescharacteristic
41#19128763C2211yescharacteristic
42Dic167⋊C8C2211yescharacteristic
43#19128757C2211yescharacteristic
44#19128757C2211yescharacteristic
45#19128756C2211yescharacteristic
46#19128756C2211yescharacteristic
47#19128755C2211yescharacteristic
48#19128755C2211yescharacteristic
49C22⋊(C167⋊C8)C2211yescharacteristic
50(C2×D167)⋊C8C2211yescharacteristic
51#19128742C2211yescharacteristic
52C22×Q8×C167C2211yescharacteristic
53C22×C8×C167C2211yescharacteristic
54C2×C4⋊C4×C167C2211yescharacteristic
55#103×C167C2211yescharacteristic
56#103×C167C2211yescharacteristic
57#103×C167C2211yescharacteristic
58#103×C167C2211yescharacteristic
59C2×Dic334—21no—
60C2×Dic334C2311yescharacteristic
61C2×Dic334C2311yescharacteristic
62Q8×D167—22no—
63Q8×D167—41no—
64C2×C22⋊D167D411yescharacteristic
65C2×C4×D167—21no—
66C2×C4×D167C2311yescharacteristic
67Dic668—41no—
68#14581940—22no—
69#14581940—41no—
70#14581933—41no—
71C2×C167⋊C8C2311yescharacteristic
72C2×C167⋊C8C2311yescharacteristic
73C4⋊Dic167—21no—
74C4⋊Dic167—21no—
75C4⋊Dic167—21no—
76C4⋊Dic167C2311yescharacteristic
77C4⋊Dic167C2311yescharacteristic
78Dic167⋊C4—21no—
79Dic167⋊C4—21no—
80Dic167⋊C4—21no—
81Dic167⋊C4—21no—
82Dic167⋊C4—21no—
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4D411yescharacteristic
86C22⋊Dic167—21no—
87C22⋊Dic167—21no—
88C22⋊Dic167D411yescharacteristic
89(C2×D167)⋊C4—21no—
90(C2×D167)⋊C4—21no—
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4D411yescharacteristic
93C4×Dic167—21no—
94C4×Dic167—21no—
95C4×Dic167—21no—
96C4×Dic167C2311yescharacteristic
97C2×Q8×C167—21no—
98C2×Q8×C167—22no—
99C2×Q8×C167—41no—
100C2×Q8×C167C2311yescharacteristic
101C2×Q8×C167C2311yescharacteristic
102C22×C4×C167—21no—
103C22×C4×C167—21no—
104C22×C4×C167C2311yescharacteristic
105C2×C8×C167—41no—
106C2×C8×C167C2311yescharacteristic
107C2×C8×C167C2311yescharacteristic
108C4⋊C4×C167—21no—
109C4⋊C4×C167C2311yescharacteristic
110C4⋊C4×C167C2311yescharacteristic
111C22×D167C2×D411yescharacteristic
112Dic334—21no—
113Dic334—21no—
114Dic334—21no—
115Dic334—21no—
116Dic334—42no—
117C22⋊D167—22no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C2×Dic167—21no—
121C2×Dic167—21no—
122C2×Dic167C2×D411yescharacteristic
123C2×Dic167C2×D411yescharacteristic
124C2×Dic167C2×D411yescharacteristic
125C4×D167—21no—
126C4×D167—21no—
127C4×D167—42no—
128C167⋊C8—21no—
129C167⋊C8—21no—
130C23×C167C2×D411yescharacteristic
131Q8×C167—21no—
132Q8×C167—21no—
133Q8×C167—21no—
134Q8×C167—42no—
135Q8×C167#4112yes—
136C2×C4×C167—21no—
137C2×C4×C167—21no—
138C2×C4×C167—21no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167—21no—
142C2×C4×C167—42no—
143C2×C4×C167—42no—
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2411yescharacteristic
146C8×C167—21no—
147C8×C167—21no—
148C22×C167—21no—
149C22×C167—21no—
150C22×C167#13712yes—
151C22×C167C22×D411yescharacteristic
152C2×D167—21no—
153C2×D167—21no—
154C2×D167—21no—
155C4×C167—21no—
156C4×C167—22no—
157C4×C167—41no—
158C4×C167—41no—
159C4×C167—41no—
160C4×C167C2×#4111yescharacteristic
161C4×C167H(5,𝔽2)11yescharacteristic
162Dic167—21no—
163Dic167—21no—
164Dic167—21no—
165Dic167—41no—
166Dic167—41no—
167Dic167—41no—
168Dic167—41no—
169C2×C167—22no—
170C2×C167#54611yescharacteristic
171C2×C167C2×#13711yescharacteristic
172C2×C167#57811yescharacteristic
173D167—41no—
174C167#331211yescharacteristic, complemented, Sylow
175#3312—1671nomaximal, Sylow
176#552—1671no—
177D4×Q8—1671no—
178#491—1671no—
179#485—1671no—
180#483—1671no—
181#479—1671no—
182#470—1671no—
183#467—1671no—
184#462—1671no—
185#451—1671no—
186#451—1671no—
187#441—1671no—
188#435—1671no—
189#419—1671no—
190C2×#103D16711yescharacteristic, complemented
191C22×Q8—1671no—
192C22×Q8C2×D16711yescharacteristic
193C2×Q16—1671no—
194C2×Q16—1671no—
195C2×Q16—1671no—
196C2×M16—1671no—
197C22×C8C2×D16711yescharacteristic
198C4⋊Q8—3341no—
199C4⋊Q8—1671no—
200#126—3341no—
201#125—3341no—
202#125—1671no—
203#123—3341no—
204C22⋊Q8—3341no—
205C22⋊Q8—3341no—
206C22⋊Q8—1671no—
207C22⋊Q8—1671no—
208C22⋊Q8—1671no—
209C22⋊Q8—1671no—
210C4×Q8—1671no—
211C4×Q8—1671no—
212C4×D4—3341no—
213C4×D4—1671no—
214#117—3341no—
215#117—1671no—
216C2×C4⋊C4C2×D16711yescharacteristic
217#107—1671no—
218#106—1671no—
219#106—1671no—
220C4⋊C8—1671no—
221#103—1671no—
222#103—1671no—
223#103—1671no—
224#103—1671no—
225#103—1671no—
226#103—1671no—
227#103C2×D16711yescharacteristic
228#103C2×D16711yescharacteristic
229#103C2×D16711yescharacteristic
230#103C2×D16711yescharacteristic
231C22⋊C8—1671no—
232C22⋊C8—1671no—
233#97—1671no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—3341no—
238C2×Q8—3342no—
239C2×Q8—6681no—
240C2×Q8—1671no—
241C2×Q8—1671no—
242C2×Q8C22×D16711yescharacteristic
243C2×Q8C22×D16711yescharacteristic
244C2×D4—1671no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3341no—
248C22×C4—1671no—
249C22×C4C22×D16711yescharacteristic
250Q16—3342no—
251Q16—6681no—
252Q16—6681no—
253M16—6681no—
254C2×C8—41no—
255C2×C8—1671no—
256C2×C8—1671no—
257C2×C8C22×D16711yescharacteristic
258C2×C8C22×D16711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—3341no—
261C4⋊C4—3341no—
262C4⋊C4—3341no—
263C4⋊C4—3341no—
264C4⋊C4—3341no—
265C4⋊C4—3341no—
266C4⋊C4—3341no—
267C4⋊C4—3341no—
268C4⋊C4—3341no—
269C4⋊C4—3341no—
270C4⋊C4—1671no—
271C4⋊C4—1671no—
272C4⋊C4—1671no—
273C4⋊C4C22×D16711yescharacteristic
274C4⋊C4C22×D16711yescharacteristic
275C22⋊C4—3341no—
276C22⋊C4—3341no—
277C22⋊C4—3341no—
278C22⋊C4—3341no—
279C22⋊C4—3341no—
280C22⋊C4—1671no—
281C22⋊C4—1671no—
282C42—3341no—
283C42—3341no—
284C42—3341no—
285C42—1671no—
286C23—1671no—
287C23D4×D16711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—3341no—
293Q8—3341no—
294Q8—3341no—
295Q8—3341no—
296Q8—6682no—
297Q8#1458195012yes—
298D4—3342no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—3341no—
308C2×C4—3341no—
309C2×C4—3341no—
310C2×C4—3341no—
311C2×C4—3341no—
312C2×C4—3341no—
313C2×C4—6682no—
314C2×C4—1671no—
315C2×C4—1671no—
316C2×C4—1671no—
317C2×C4D4×D16711yescharacteristic
318C2×C4C23×D16711yescharacteristic
319C8—21no—
320C8—21no—
321C8—3341no—
322C8—3341no—
323C22—21no—
324C22—21no—
325C22—3341no—
326C22—3341no—
327C22—3341no—
328C22#1912886112yes—
329C22C2×D4×D16711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—3341no—
336C4—3341no—
337C4—3341no—
338C4—6681no—
339C4—6681no—
340C4—6681no—
341C4—6681no—
342C4C2×#1458195011yescharacteristic
343C4#1912887711yescharacteristic
344C2—22no—
345C2—6681no—
346C2#2492390911yescharacteristic
347C2C2×#1912886111yescharacteristic
348C2#2492408211yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC2×#103×C167C2
socleC22×C167C22×D4
prime radicalC2×C22⋊D167D4
hypercenterC2×#103D167
hypocenterC167#3312
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #24923636C2semidirect
4. #24923609C2semidirect
5. #24923599C2semidirect
6. #24923578C2semidirect
7. #24923526C2semidirect
8. #24923510C2semidirect
9. #24923478C2semidirect
10. #24923426C2semidirect
11. #24923425C2semidirect
12. #24923373C2semidirect
14. #24923276C2semidirect
15. C2×#103×C167C2semidirect
174. C167#3312semidirect
190. C2×#103D167semidirect

Maximal subgroups

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

subgroupindex
#249239542
Q8×C22⋊D1672
#249236362
#249236092
#249235992
#249235782
#249235262
#249235102
#249234782
#249234262
#249234252
#249233732
#249233402
#249232762
C2×#103×C1672
#3312167