Finite Group Databasebeta

← #36335709

Subgroups — #36335709

Order 21376.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128863C211yesmaximal, characteristic, complemented
2C22×#14581940C211yesmaximal, characteristic, complemented
3#24923928C211yesmaximal, characteristic
4#24923661C211yesmaximal, characteristic, complemented
5#24923525C211yesmaximal, characteristic, complemented
6#24923500C211yesmaximal, characteristic, complemented
7#24923491C211yesmaximal, characteristic, complemented
8#24923476C211yesmaximal, characteristic, complemented
9#24923429C211yesmaximal, characteristic, complemented
10#24923425C211yesmaximal, characteristic, complemented
11#24923424C211yesmaximal, characteristic, complemented
12#24923408C211yesmaximal, characteristic, complemented
13#24923374C211yesmaximal, characteristic, complemented
14#24923341C211yesmaximal, characteristic, complemented
15D4×Q8×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×Dic334C2211yescharacteristic
18#19128863—24no—
19C2×#14581940—24no—
20C2×#14581940C2211yescharacteristic
21C2×#14581940C2211yescharacteristic
22C2×#14581940C2211yescharacteristic
23C2×#14581940C2211yescharacteristic
24C2×#14581940C2211yescharacteristic
25C2×#14581940C2211yescharacteristic
26C2×#14581937C2211yescharacteristic
27C2×#14581937C2211yescharacteristic
28C2×#14581937C2211yescharacteristic
29C2×#14581934C2211yescharacteristic
30C22×C167⋊C8C2211yescharacteristic
31#19128820C2211yescharacteristic
32#19128818—21no—
33#19128811C2211yescharacteristic
34#19128810—21no—
35#19128806C2211yescharacteristic
36#19128804—21no—
37C4×Dic334C2211yescharacteristic
38(C22⋊D167)⋊C4—21no—
39#19128763C2211yescharacteristic
40C4⋊(C167⋊C8)C2211yescharacteristic
41#19128756C2211yescharacteristic
42#19128756C2211yescharacteristic
43#19128756C2211yescharacteristic
44#19128755C2211yescharacteristic
45#19128755C2211yescharacteristic
46#19128752C2211yescharacteristic
47C22⋊(C167⋊C8)C2211yescharacteristic
48C22⋊(C167⋊C8)C2211yescharacteristic
49#19128743C2211yescharacteristic
50C22×Q8×C167C2211yescharacteristic
51C22×Q8×C167C2211yescharacteristic
52C4⋊Q8×C167—21no—
53C4⋊Q8×C167C2211yescharacteristic
54C22⋊Q8×C167—21no—
55C22⋊Q8×C167—21no—
56C22⋊Q8×C167C2211yescharacteristic
57C22⋊Q8×C167C2211yescharacteristic
58C4×Q8×C167C2211yescharacteristic
59C4×D4×C167—21no—
60C4×D4×C167C2211yescharacteristic
61#14581950—22no—
62#14581950—41no—
63C2×Dic334—22no—
64C2×Dic334—21no—
65C2×Dic334—41no—
66C2×Dic334C2311yescharacteristic
67C2×Dic334C2311yescharacteristic
68C2×D668C2311yescharacteristic
69C2×C22⋊D167—21no—
70C2×C22⋊D167—21no—
71C22×Dic167—21no—
72C2×C4×D167—21no—
73#14581940—22no—
74#14581940—22no—
75#14581940—22no—
76#14581940—22no—
77#14581940—41no—
78#14581940D414yescomplemented
79#14581937—22no—
80#14581937—22no—
81#14581937—41no—
82#14581934—22no—
83C2×C167⋊C8—22no—
84C2×C167⋊C8C2311yescharacteristic
85C2×C167⋊C8C2311yescharacteristic
86C2×C167⋊C8C2311yescharacteristic
87C2×C167⋊C8C2311yescharacteristic
88C4⋊Dic167C2311yescharacteristic
89Dic167⋊C4—21no—
90Dic167⋊C4—21no—
91C22⋊Dic167—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95C4×Dic167—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—
124Dic334—21no—
125Dic334—21no—
126Dic334—21no—
127Dic334—42no—
128Dic334C2×D412yes—
129D668—21no—
130D668—21no—
131C22⋊D167—42no—
132C22⋊D167—42no—
133C2×Dic167—21no—
134C2×Dic167—21no—
135C2×Dic167—21no—
136C2×Dic167—42no—
137C4×D167—42no—
138C167⋊C8—21no—
139C167⋊C8—21no—
140C167⋊C8—21no—
141C167⋊C8C2×D412yes—
142C23×C167C2×D411yescharacteristic
143C23×C167C2×D411yescharacteristic
144Q8×C167—21no—
145Q8×C167—21no—
146Q8×C167—21no—
147Q8×C167—21no—
148Q8×C167—21no—
149Q8×C167—42no—
150Q8×C167—42no—
151Q8×C167C2×D412yes—
152D4×C167—22no—
153C2×C4×C167—21no—
154C2×C4×C167—21no—
155C2×C4×C167—21no—
156C2×C4×C167—21no—
157C2×C4×C167—21no—
158C2×C4×C167—21no—
159C2×C4×C167—21no—
160C2×C4×C167—21no—
161C2×C4×C167—42no—
162C2×C4×C167—42no—
163C2×C4×C167C2×D411yescharacteristic
164C2×C4×C167C2×D411yescharacteristic
165C2×C4×C167C2×D411yescharacteristic
166C2×C4×C167C2×D411yescharacteristic
167C2×C4×C167C2411yescharacteristic
168C22×C167—21no—
169C22×C167—21no—
170C22×C167—21no—
171C22×C167—21no—
172C22×C167—21no—
173C22×C167#13712yescomplemented
174C22×C167C22×D411yescharacteristic
175C2×D167—41no—
176C2×D167—41no—
177C2×D167—41no—
178C4×C167—21no—
179C4×C167—21no—
180C4×C167—22no—
181C4×C167—21no—
182C4×C167—21no—
183C4×C167—41no—
184C4×C167—41no—
185C4×C167—41no—
186C4×C167C22×D411yescharacteristic
187C4×C167H(5,𝔽2)11yescharacteristic
188Dic167—22no—
189Dic167—41no—
190Dic167—41no—
191C2×C167—22no—
192C2×C167—41no—
193C2×C167D4211yescharacteristic
194C2×C167C2×#13711yescharacteristic
195C2×C167#57811yescharacteristic
196D167—81no—
197C167#327011yescharacteristic, complemented, Sylow
198#3270—1671nomaximal, Sylow
199C2×#137—1671no—
200C22×Q16—1671no—
201D4×Q8D16711yescharacteristic, complemented
202#548—1671no—
203#497—1671no—
204#470—1671no—
205#465—1671no—
206#464—1671no—
207#462—1671no—
208#452—1671no—
209#451—1671no—
210#451—1671no—
211#448—1671no—
212#441—1671no—
213#435—1671no—
214C2×#41—1671no—
215C22×Q8—1671no—
216C22×Q8C2×D16711yescharacteristic
217C22×Q8C2×D16711yescharacteristic
218#137—3344no—
219C2×Q16—3344no—
220C2×Q16—1671no—
221C2×Q16—1671no—
222C2×Q16—1671no—
223C2×Q16—1671no—
224C2×Q16—1671no—
225C2×Q16—1671no—
226C2×SD16—1671no—
227C2×SD16—1671no—
228C2×SD16—1671no—
229C2×M16—1671no—
230C22×C8—1671no—
231C4⋊Q8—21no—
232C4⋊Q8C2×D16711yescharacteristic
233#124—3341no—
234#124—1671no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—3341no—
238C22⋊Q8—1671no—
239C22⋊Q8C2×D16711yescharacteristic
240C22⋊Q8C2×D16711yescharacteristic
241#121—3341no—
242#121—1671no—
243C4×Q8—1671no—
244C4×Q8C2×D16711yescharacteristic
245C4×D4—21no—
246C4×D4—3341no—
247C4×D4C2×D16711yescharacteristic
248#106—1671no—
249C4⋊C8—1671no—
250#103—1671no—
251#103—1671no—
252#103—1671no—
253#103—1671no—
254#103—1671no—
255#102—1671no—
256C22⋊C8—1671no—
257C22⋊C8—1671no—
258#97—1671no—
259#41—3342no—
260#41—6681no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—3342no—
268C2×Q8—3341no—
269C2×Q8—6681no—
270C2×Q8—1671no—
271C2×Q8—1671no—
272C2×Q8C22×D16711yescharacteristic
273C2×Q8C22×D16711yescharacteristic
274C2×Q8C22×D16711yescharacteristic
275C2×D4—3341no—
276C2×D4—3341no—
277C2×D4—1671no—
278C2×D4C22⋊D16711yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3341no—
282C22×C4—3341no—
283C22×C4C22×D16711yescharacteristic
284C22×C4C22×D16711yescharacteristic
285Q16—3342no—
286Q16—3342no—
287Q16—3342no—
288Q16—3342no—
289Q16—6681no—
290Q16—1674no—
291SD16—3342no—
292SD16—3342no—
293SD16—6681no—
294M16—3342no—
295C2×C8—3342no—
296C2×C8—1671no—
297C2×C8—1671no—
298C2×C8—1671no—
299C2×C8—1671no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—3341no—
306C4⋊C4—3341no—
307C4⋊C4—1671no—
308C4⋊C4C22⋊D16711yescharacteristic
309C4⋊C4C22×D16711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3341no—
313C22⋊C4—3341no—
314C22⋊C4—3341no—
315C22⋊C4—3341no—
316C22⋊C4C22⋊D16711yescharacteristic
317C22⋊C4C22⋊D16711yescharacteristic
318C42—21no—
319C42—3341no—
320C42C22×D16711yescharacteristic
321C23—3341no—
322C23C2×C22⋊D16711yescharacteristic
323C23C2×C22⋊D16711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—42no—
331Q8—3341no—
332Q8—3341no—
333Q8—3341no—
334Q8—6682no—
335Q8—1672no—
336Q8D4×D16712yes—
337D4—22no—
338D4—3341no—
339D4—3341no—
340D4—6682no—
341D4—6682no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—3341no—
353C2×C4—3341no—
354C2×C4—3341no—
355C2×C4—6682no—
356C2×C4—6682no—
357C2×C4C2×C22⋊D16711yescharacteristic
358C2×C4C2×C22⋊D16711yescharacteristic
359C2×C4C2×C22⋊D16711yescharacteristic
360C2×C4C2×C22⋊D16711yescharacteristic
361C2×C4C23×D16711yescharacteristic
362C8—3341no—
363C8—3341no—
364C8—3341no—
365C8—1672no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—6681no—
372C22—6681no—
373C22—6681no—
374C22#1912886312yescomplemented
375C22C22×C22⋊D16711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3342no—
385C4—6681no—
386C4—6681no—
387C4C2×D4×D16711yescharacteristic
388C4#1912887711yescharacteristic
389C2—22no—
390C2—41no—
391C2—13361no—
392C2D4×C22⋊D16711yescharacteristic
393C2C2×#1912886311yescharacteristic
394C2#2492408411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingD4×Q8×C167C2
socleC22×C167C22×D4
prime radical#24923928C2
hypercenterD4×Q8D167
hypocenterC167#3270
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×#14581940C2semidirect
4. #24923661C2semidirect
5. #24923525C2semidirect
6. #24923500C2semidirect
7. #24923491C2semidirect
8. #24923476C2semidirect
9. #24923429C2semidirect
10. #24923425C2semidirect
11. #24923424C2semidirect
12. #24923408C2semidirect
13. #24923374C2semidirect
14. #24923341C2semidirect
15. D4×Q8×C167C2semidirect
78. #14581940D4semidirect
173. C22×C167#137semidirect
197. C167#3270semidirect
201. D4×Q8D167semidirect
374. 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×#145819402
#249239282
#249236612
#249235252
#249235002
#249234912
#249234762
#249234292
#249234252
#249234242
#249234082
#249233742
#249233412
D4×Q8×C1672
#3270167