Finite Group Databasebeta

← #36336189

Subgroups — #36336189

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#24923959C211yesmaximal, characteristic
2#24923637C211yesmaximal, characteristic, complemented
3#24923607C211yesmaximal, characteristic, complemented
4#24923597C211yesmaximal, characteristic, complemented
5#24923576C211yesmaximal, characteristic, complemented
6#24923525C211yesmaximal, characteristic, complemented
7#24923508C211yesmaximal, characteristic, complemented
8#24923476C211yesmaximal, characteristic, complemented
9#24923425C211yesmaximal, characteristic, complemented
10#24923425C211yesmaximal, characteristic, complemented
11#24923374C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic
13#24923278C211yesmaximal, characteristic, complemented
14C2×#19128756C211yesmaximal, characteristic, complemented
15D4×Q8×C167C211yesmaximal, characteristic
16C2×#14581940C2211yescharacteristic
17C2×#14581940C2211yescharacteristic
18C2×#14581940C2211yescharacteristic
19C2×#14581934C2211yescharacteristic
20C22×C167⋊C8C2211yescharacteristic
21#19128825—21no—
22#19128824C2211yescharacteristic
23#19128822—21no—
24#19128813—21no—
25#19128811C2211yescharacteristic
26#19128811C2211yescharacteristic
27C4×Dic334C2211yescharacteristic
28#19128791C2211yescharacteristic
29#19128788—21no—
30C2×C4⋊Dic167C2211yescharacteristic
31#19128765C2211yescharacteristic
32#19128763C2211yescharacteristic
33#19128763C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35#19128756C2211yescharacteristic
36#19128756C2211yescharacteristic
37#19128756C2211yescharacteristic
38#19128756C2211yescharacteristic
39#19128756C2211yescharacteristic
40#19128756C2211yescharacteristic
41#19128756C2211yescharacteristic
42#19128756C2211yescharacteristic
43#19128756C2211yescharacteristic
44#19128755C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic
47#19128743C2211yescharacteristic
48C22×Q8×C167C2211yescharacteristic
49C22×Q8×C167C2211yescharacteristic
50C4⋊Q8×C167—21no—
51C4⋊Q8×C167C2211yescharacteristic
52C22⋊Q8×C167—21no—
53C22⋊Q8×C167—21no—
54C22⋊Q8×C167C2211yescharacteristic
55C22⋊Q8×C167C2211yescharacteristic
56C4×Q8×C167C2211yescharacteristic
57C4×D4×C167—21no—
58C4×D4×C167C2211yescharacteristic
59C2×Dic334C2311yescharacteristic
60C22×Dic167—21no—
61#14581940—22no—
62#14581940—41no—
63#14581940—41no—
64#14581934—41no—
65C2×C167⋊C8—41no—
66C2×C167⋊C8C2311yescharacteristic
67C2×C167⋊C8C2311yescharacteristic
68C2×C167⋊C8C2311yescharacteristic
69C2×C167⋊C8C2311yescharacteristic
70C4⋊Dic167—21no—
71C4⋊Dic167C2311yescharacteristic
72C4⋊Dic167C2311yescharacteristic
73C4⋊Dic167C2311yescharacteristic
74Dic167⋊C4—21no—
75Dic167⋊C4—21no—
76Dic167⋊C4—21no—
77Dic167⋊C4—21no—
78Dic167⋊C4—21no—
79C22⋊Dic167—21no—
80C22⋊Dic167—21no—
81C22⋊Dic167—21no—
82C4×Dic167—21no—
83C4×Dic167—21no—
84C2×Q8×C167—21no—
85C2×Q8×C167—21no—
86C2×Q8×C167—22no—
87C2×Q8×C167—22no—
88C2×Q8×C167—41no—
89C2×Q8×C167—41no—
90C2×Q8×C167C2311yescharacteristic
91C2×Q8×C167C2311yescharacteristic
92C2×Q8×C167C2311yescharacteristic
93C2×D4×C167D411yescharacteristic
94C22×C4×C167—21no—
95C22×C4×C167—21no—
96C22×C4×C167C2311yescharacteristic
97C22×C4×C167C2311yescharacteristic
98C4⋊C4×C167—21no—
99C4⋊C4×C167—21no—
100C4⋊C4×C167—21no—
101C4⋊C4×C167—21no—
102C4⋊C4×C167—21no—
103C4⋊C4×C167D411yescharacteristic
104C4⋊C4×C167C2311yescharacteristic
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167D411yescharacteristic
108C22⋊C4×C167D411yescharacteristic
109C42×C167—21no—
110C42×C167C2311yescharacteristic
111Dic334—21no—
112Dic334—21no—
113C2×Dic167—21no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—42no—
118C167⋊C8—21no—
119C167⋊C8—21no—
120C167⋊C8—21no—
121C167⋊C8—21no—
122C23×C167C2×D411yescharacteristic
123C23×C167C2×D411yescharacteristic
124Q8×C167—21no—
125Q8×C167—21no—
126Q8×C167—21no—
127Q8×C167—21no—
128Q8×C167—21no—
129Q8×C167—42no—
130Q8×C167—42no—
131Q8×C167#4112yes—
132D4×C167—22no—
133C2×C4×C167—21no—
134C2×C4×C167—21no—
135C2×C4×C167—21no—
136C2×C4×C167—21no—
137C2×C4×C167—21no—
138C2×C4×C167—21no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167—42no—
142C2×C4×C167—42no—
143C2×C4×C167C2×D411yescharacteristic
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2411yescharacteristic
148C22×C167—21no—
149C22×C167—21no—
150C22×C167—21no—
151C22×C167—21no—
152C22×C167—21no—
153C22×C167#13712yes—
154C22×C167C22×D411yescharacteristic
155C4×C167—21no—
156C4×C167—21no—
157C4×C167—22no—
158C4×C167—21no—
159C4×C167—21no—
160C4×C167—41no—
161C4×C167—41no—
162C4×C167—41no—
163C4×C167C2×#4111yescharacteristic
164C4×C167H(5,𝔽2)11yescharacteristic
165Dic167—41no—
166Dic167—41no—
167Dic167—41no—
168Dic167—41no—
169C2×C167—22no—
170C2×C167—41no—
171C2×C167#54611yescharacteristic
172C2×C167C2×#13711yescharacteristic
173C2×C167#57811yescharacteristic
174C167#331211yescharacteristic, complemented, Sylow
175#3312—1671nomaximal, Sylow
176#552—1671no—
177D4×Q8D16711yescharacteristic
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×#103—1671no—
191C22×Q8C2×D16711yescharacteristic
192C22×Q8C2×D16711yescharacteristic
193C2×Q16—1671no—
194C2×Q16—1671no—
195C2×Q16—1671no—
196C2×M16—1671no—
197C22×C8—1671no—
198C4⋊Q8—21no—
199C4⋊Q8C2×D16711yescharacteristic
200#126—3341no—
201#125—3341no—
202#125—1671no—
203#123—3341no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—1671no—
207C22⋊Q8—1671no—
208C22⋊Q8C2×D16711yescharacteristic
209C22⋊Q8C2×D16711yescharacteristic
210C4×Q8—1671no—
211C4×Q8C2×D16711yescharacteristic
212C4×D4—21no—
213C4×D4C2×D16711yescharacteristic
214#117—3341no—
215#117—1671no—
216C2×C4⋊C4—1671no—
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#103—1671no—
228#103—1671no—
229#103—1671no—
230#103—1671no—
231C22⋊C8—1671no—
232C22⋊C8—1671no—
233#97—1671no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—1671no—
241C2×Q8C22×D16711yescharacteristic
242C2×Q8C22×D16711yescharacteristic
243C2×Q8C22×D16711yescharacteristic
244C2×D4C22⋊D16711yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3341no—
248C22×C4C22×D16711yescharacteristic
249C22×C4C22×D16711yescharacteristic
250Q16—3342no—
251Q16—6681no—
252Q16—6681no—
253M16—6681no—
254C2×C8—6681no—
255C2×C8—1671no—
256C2×C8—1671no—
257C2×C8—1671no—
258C2×C8—1671no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
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—21no—
276C22⋊C4—21no—
277C22⋊C4—3341no—
278C22⋊C4—3341no—
279C22⋊C4—3341no—
280C22⋊C4C22⋊D16711yescharacteristic
281C22⋊C4C22⋊D16711yescharacteristic
282C42—21no—
283C42—3341no—
284C42—3341no—
285C42C22×D16711yescharacteristic
286C23C2×C22⋊D16711yescharacteristic
287C23C2×C22⋊D16711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—42no—
295Q8—3341no—
296Q8—3341no—
297Q8#1458194912yes—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—3341no—
310C2×C4—3341no—
311C2×C4—3341no—
312C2×C4—3341no—
313C2×C4—6682no—
314C2×C4C2×C22⋊D16711yescharacteristic
315C2×C4C2×C22⋊D16711yescharacteristic
316C2×C4C2×C22⋊D16711yescharacteristic
317C2×C4C2×C22⋊D16711yescharacteristic
318C2×C4C23×D16711yescharacteristic
319C8—3341no—
320C8—3341no—
321C8—3341no—
322C8—3341no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1912886312yes—
329C22C22×C22⋊D16711yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—6681no—
339C4—6681no—
340C4—6681no—
341C4—6681no—
342C4C2×#1458194911yescharacteristic
343C4#1912887711yescharacteristic
344C2—22no—
345C2—41no—
346C2#2492391311yescharacteristic
347C2C2×#1912886311yescharacteristic
348C2#2492408411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingD4×Q8×C167C2
socleC22×C167C22×D4
prime radicalC2×D4×C167D4
hypercenterD4×Q8D167
hypocenterC167#3312
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
2. #24923637C2semidirect
3. #24923607C2semidirect
4. #24923597C2semidirect
5. #24923576C2semidirect
6. #24923525C2semidirect
7. #24923508C2semidirect
8. #24923476C2semidirect
9. #24923425C2semidirect
10. #24923425C2semidirect
11. #24923374C2semidirect
13. #24923278C2semidirect
14. C2×#19128756C2semidirect
174. C167#3312semidirect

Maximal subgroups

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

subgroupindex
#249239592
#249236372
#249236072
#249235972
#249235762
#249235252
#249235082
#249234762
#249234252
#249233742
#249233412
#249232782
C2×#191287562
D4×Q8×C1672
#3312167