Finite Group Databasebeta

← #36335850

Subgroups — #36335850

Order 21376.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128854C211yesmaximal, characteristic, complemented
2C22×#14581939C211yesmaximal, characteristic, complemented
3#24923928C211yesmaximal, characteristic, complemented
4#24923915C211yesmaximal, characteristic, complemented
5#24923653C211yesmaximal, characteristic, complemented
6#24923498C211yesmaximal, characteristic, complemented
7#24923498C211yesmaximal, characteristic, complemented
8#24923492C211yesmaximal, characteristic, complemented
9#24923485C211yesmaximal, characteristic, complemented
10#24923433C211yesmaximal, characteristic, complemented
11#24923427C211yesmaximal, characteristic, complemented
12#24923416C211yesmaximal, characteristic, complemented
13#24923405C211yesmaximal, characteristic, complemented
14C4×#14581939C211yesmaximal, characteristic, complemented
15C8×D4×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C2×#14581950C2211yescharacteristic, complemented
18C22×Dic334C2211yescharacteristic
19C22×D668C2211yescharacteristic
20#19128854—24no—
21C2×Dic668C2211yescharacteristic, complemented
22C2×Dic668C2211yescharacteristic, complemented
23C2×#14581939—24no—
24C2×#14581939C2211yescharacteristic
25C2×#14581939C2211yescharacteristic
26C2×#14581939C2211yescharacteristic
27C2×#14581939C2211yescharacteristic
28C2×#14581939C2211yescharacteristic, complemented
29C2×D1336C2211yescharacteristic
30C2×D1336C2211yescharacteristic
31#19128820C2211yescharacteristic
32#19128820C2211yescharacteristic, complemented
33#19128818—21no—
34#19128817—21no—
35#19128811C2211yescharacteristic
36#19128811C2211yescharacteristic, complemented
37#19128810—21no—
38#19128806C2211yescharacteristic
39#19128806C2211yescharacteristic
40#19128805—21no—
41#19128804—21no—
42#19128800—21no—
43C4×Dic334C2211yescharacteristic, complemented
44C4×D668C2211yescharacteristic
45(C22⋊D167)⋊C4—21no—
46C22⋊C4×D167—21no—
47#19128764C2211yescharacteristic, complemented
48#19128757C2211yescharacteristic, complemented
49#19128757C2211yescharacteristic, complemented
50#19128757C2211yescharacteristic, complemented
51#19128754C2211yescharacteristic
52#19128754C2211yescharacteristic
53#19128754C2211yescharacteristic, complemented
54C22×C8×C167C2211yescharacteristic
55C22×C8×C167C2211yescharacteristic, complemented
56C4×D4×C167C2211yescharacteristic
57C4⋊C8×C167C2211yescharacteristic, complemented
58C22⋊C8×C167C2211yescharacteristic
59C22⋊C8×C167C2211yescharacteristic, complemented
60C4×C8×C167C2211yescharacteristic, complemented
61C23×D167—21no—
62#14581950—22no—
63#14581950—22no—
64#14581950—41no—
65#14581950—41no—
66C2×Dic334—22no—
67C2×Dic334—21no—
68C2×Dic334—41no—
69C2×Dic334C2311yescharacteristic
70C2×Dic334C2311yescharacteristic
71C2×Dic334C2311yescharacteristic
72C2×D668—22no—
73C2×D668—21no—
74C2×D668—41no—
75C2×D668C2311yescharacteristic
76C2×D668C2311yescharacteristic
77C2×D668C2311yescharacteristic
78C2×C22⋊D167—21no—
79C2×C22⋊D167—21no—
80C2×C22⋊D167—21no—
81C2×C22⋊D167—21no—
82C22×Dic167—21no—
83C2×C4×D167—21no—
84C2×C4×D167—21no—
85C2×C4×D167—21no—
86Dic668—22no—
87Dic668—41no—
88#14581939—22no—
89#14581939—22no—
90#14581939—22no—
91#14581939—22no—
92#14581939D414yescomplemented
93D1336—22no—
94D1336—41no—
95C4⋊Dic167C2311yescharacteristic
96C4⋊Dic167C2311yescharacteristic
97Dic167⋊C4—21no—
98Dic167⋊C4—21no—
99Dic167⋊C4—21no—
100C22⋊Dic167—21no—
101C22⋊Dic167—21no—
102(C2×D167)⋊C4—21no—
103(C2×D167)⋊C4—21no—
104(C2×D167)⋊C4—21no—
105(C2×D167)⋊C4—21no—
106(C2×D167)⋊C4—21no—
107(C2×D167)⋊C4—21no—
108(C2×D167)⋊C4—21no—
109C4×Dic167—21no—
110C2×D4×C167D411yescharacteristic
111C22×C4×C167C2311yescharacteristic
112C22×C4×C167C2311yescharacteristic
113C2×C8×C167—22no—
114C2×C8×C167—22no—
115C2×C8×C167C2311yescharacteristic
116C2×C8×C167C2311yescharacteristic
117C2×C8×C167C2311yescharacteristic
118C2×C8×C167C2311yescharacteristic
119C4⋊C4×C167D411yescharacteristic
120C22⋊C4×C167D411yescharacteristic
121C22⋊C4×C167D411yescharacteristic
122C42×C167C2311yescharacteristic
123C22×D167—21no—
124C22×D167—21no—
125C22×D167—21no—
126C22×D167—42no—
127C22×D167—42no—
128C22×D167—81no—
129Dic334—21no—
130Dic334—21no—
131Dic334—21no—
132Dic334—21no—
133Dic334—21no—
134Dic334—42no—
135Dic334C2×D412yes—
136D668—21no—
137D668—21no—
138D668—21no—
139D668—21no—
140D668—21no—
141D668—42no—
142D668C2×D412yes—
143C22⋊D167—42no—
144C22⋊D167—42no—
145C22⋊D167—42no—
146C22⋊D167—81no—
147C2×Dic167—21no—
148C2×Dic167—21no—
149C2×Dic167—21no—
150C2×Dic167—21no—
151C2×Dic167—21no—
152C2×Dic167—42no—
153C4×D167—42no—
154C4×D167—42no—
155C4×D167—81no—
156C23×C167C2×D411yescharacteristic
157C23×C167C2×D411yescharacteristic
158D4×C167—22no—
159C2×C4×C167—21no—
160C2×C4×C167—21no—
161C2×C4×C167—21no—
162C2×C4×C167—21no—
163C2×C4×C167C2×D411yescharacteristic
164C2×C4×C167C2×D411yescharacteristic
165C2×C4×C167C2×D411yescharacteristic
166C2×C4×C167C2×D411yescharacteristic
167C2×C4×C167C2411yescharacteristic
168C8×C167—21no—
169C8×C167—21no—
170C8×C167—21no—
171C8×C167C2×D412yes—
172C22×C167—21no—
173C22×C167—21no—
174C22×C167—21no—
175C22×C167—21no—
176C22×C167—21no—
177C22×C167#13512yescomplemented
178C22×C167C22×D411yescharacteristic
179C2×D167—22no—
180C2×D167—41no—
181C2×D167—41no—
182C2×D167—41no—
183C2×D167—41no—
184C2×D167—41no—
185C2×D167—41no—
186C2×D167—41no—
187C2×D167—41no—
188C2×D167—84no—
189C4×C167—21no—
190C4×C167—21no—
191C4×C167—21no—
192C4×C167—21no—
193C4×C167C22×D411yescharacteristic
194C4×C167H(5,𝔽2)11yescharacteristic
195Dic167—22no—
196Dic167—41no—
197Dic167—41no—
198Dic167—41no—
199Dic167—41no—
200C2×C167—22no—
201C2×C167—41no—
202C2×C167D4211yescharacteristic
203C2×C167C2×#13511yescharacteristic
204C2×C167#57711yescharacteristic
205D167—42no—
206D167—81no—
207D167—81no—
208C167#328211yescharacteristic, complemented, Sylow
209#3282—1671nomaximal, Sylow
210C2×#135—1671no—
211C22×SD16—1671no—
212#548—1671no—
213#546—1671no—
214#495—1671no—
215#465—1671no—
216#465—1671no—
217#464—1671no—
218#463—1671no—
219#452—1671no—
220#451—1671no—
221#449—1671no—
222#447—1671no—
223C4×SD16—1671no—
224C8×D4D16711yescharacteristic, complemented
225C2×#41—1671no—
226C2×#41—1671no—
227C22×Q8—1671no—
228C22×D4—1671no—
229#135—3344no—
230C2×Q16—1671no—
231C2×Q16—1671no—
232C2×SD16—3344no—
233C2×SD16—1671no—
234C2×SD16—1671no—
235C2×SD16—1671no—
236C2×SD16—1671no—
237C2×SD16—1671no—
238C2×D8—1671no—
239C2×D8—1671no—
240C22×C8C2×D16711yescharacteristic
241C22×C8C2×D16711yescharacteristic, complemented
242#124—3341no—
243#124—1671no—
244#124—1671no—
245#123—3341no—
246C22⋊Q8—3341no—
247C22⋊Q8—1671no—
248C22⋊Q8—1671no—
249#121—3341no—
250#121—3341no—
251#121—1671no—
252#121—1671no—
253#120—3341no—
254C4×Q8—1671no—
255C4×D4—3341no—
256C4×D4—1671no—
257C4×D4C2×D16711yescharacteristic
258C2×C22⋊C4—3341no—
259#106—1671no—
260C4⋊C8C2×D16711yescharacteristic, complemented
261#103—1671no—
262#103—1671no—
263#103—1671no—
264#102—1671no—
265#102—1671no—
266#102—1671no—
267C22⋊C8C2×D16711yescharacteristic
268C22⋊C8C2×D16711yescharacteristic, complemented
269C4×C8C2×D16711yescharacteristic, complemented
270C24—3341no—
271#41—3342no—
272#41—3342no—
273#41—6681no—
274#41—6681no—
275C2×Q8—3342no—
276C2×Q8—3341no—
277C2×Q8—6681no—
278C2×Q8—1671no—
279C2×Q8—1671no—
280C2×Q8—1671no—
281C2×D4—3342no—
282C2×D4—3341no—
283C2×D4—3341no—
284C2×D4—3341no—
285C2×D4—3341no—
286C2×D4—3341no—
287C2×D4—6681no—
288C2×D4—1671no—
289C2×D4—1671no—
290C2×D4—1671no—
291C2×D4D66811yescharacteristic
292C22×C4—3341no—
293C22×C4—3341no—
294C22×C4—3341no—
295C22×C4—3341no—
296C22×C4C22×D16711yescharacteristic
297C22×C4C22×D16711yescharacteristic
298Q16—3342no—
299Q16—6681no—
300SD16—3342no—
301SD16—3342no—
302SD16—3342no—
303SD16—3342no—
304SD16—1674no—
305D8—3342no—
306D8—6681no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8C22×D16711yescharacteristic
310C2×C8C22×D16711yescharacteristic
311C2×C8C22×D16711yescharacteristic
312C2×C8C22×D16711yescharacteristic
313C4⋊C4—3341no—
314C4⋊C4—3341no—
315C4⋊C4—3341no—
316C4⋊C4—1671no—
317C4⋊C4—1671no—
318C4⋊C4D66811yescharacteristic
319C22⋊C4—3341no—
320C22⋊C4—3341no—
321C22⋊C4—3341no—
322C22⋊C4—3341no—
323C22⋊C4—3341no—
324C22⋊C4—3341no—
325C22⋊C4—3341no—
326C22⋊C4—3341no—
327C22⋊C4—3341no—
328C22⋊C4D66811yescharacteristic
329C22⋊C4D66811yescharacteristic
330C42—3341no—
331C42C22×D16711yescharacteristic
332C23—13361no—
333C23—3341no—
334C23—3341no—
335C23—3341no—
336C23—6682no—
337C23—6682no—
338C23C2×D66811yescharacteristic
339C23C2×D66811yescharacteristic
340Q8—3341no—
341Q8—3341no—
342Q8—3341no—
343Q8—3341no—
344Q8—3341no—
345Q8—6682no—
346Q8—1672no—
347D4—22no—
348D4—13361no—
349D4—3341no—
350D4—3341no—
351D4—3341no—
352D4—3341no—
353D4—3341no—
354D4—6682no—
355D4—6682no—
356D4—6682no—
357D4—6682no—
358D4—1672no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—13361no—
364C2×C4—3341no—
365C2×C4—3341no—
366C2×C4—3341no—
367C2×C4—3341no—
368C2×C4—3341no—
369C2×C4—6682no—
370C2×C4—6682no—
371C2×C4—6682no—
372C2×C4C2×D66811yescharacteristic
373C2×C4C2×D66811yescharacteristic
374C2×C4C2×D66811yescharacteristic
375C2×C4C2×D66811yescharacteristic
376C2×C4C23×D16711yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D16712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—13364no—
387C22—3342no—
388C22—6681no—
389C22—6681no—
390C22—6681no—
391C22—6681no—
392C22—6681no—
393C22—6681no—
394C22—6681no—
395C22—6681no—
396C22#1912885412yescomplemented
397C22C22×D66811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—3342no—
403C4—6681no—
404C4—6681no—
405C4—6681no—
406C4—6681no—
407C4C2×D4×D16711yescharacteristic
408C4#1912887811yescharacteristic
409C2—22no—
410C2—41no—
411C2—13361no—
412C2—13361no—
413C2—6682no—
414C2D4×D66811yescharacteristic
415C2C2×#1912885411yescharacteristic
416C2#2492408111yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC8×D4×C167C2
socleC22×C167C22×D4
hypercenterC8×D4D167
hypocenterC167#3282
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19128854C2semidirect
2. C22×#14581939C2semidirect
3. #24923928C2semidirect
4. #24923915C2semidirect
5. #24923653C2semidirect
6. #24923498C2semidirect
7. #24923498C2semidirect
8. #24923492C2semidirect
9. #24923485C2semidirect
10. #24923433C2semidirect
11. #24923427C2semidirect
12. #24923416C2semidirect
13. #24923405C2semidirect
14. C4×#14581939C2semidirect
15. C8×D4×C167C2semidirect
17. C2×#14581950C22semidirect
21. C2×Dic668C22semidirect
22. C2×Dic668C22semidirect
28. C2×#14581939C22semidirect
32. #19128820C22semidirect
36. #19128811C22semidirect
43. C4×Dic334C22semidirect
47. #19128764C22semidirect
48. #19128757C22semidirect
49. #19128757C22semidirect
50. #19128757C22semidirect
53. #19128754C22semidirect
55. C22×C8×C167C22semidirect
57. C4⋊C8×C167C22semidirect
59. C22⋊C8×C167C22semidirect
60. C4×C8×C167C22semidirect
92. #14581939D4semidirect
177. C22×C167#135semidirect
208. C167#3282semidirect
224. C8×D4D167semidirect
241. C22×C8C2×D167semidirect
260. C4⋊C8C2×D167semidirect
268. C22⋊C8C2×D167semidirect
269. C4×C8C2×D167semidirect
396. C22#19128854semidirect

Maximal subgroups

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

subgroupindex
C2×#191288542
C22×#145819392
#249239282
#249239152
#249236532
#249234982
#249234922
#249234852
#249234332
#249234272
#249234162
#249234052
C4×#145819392
C8×D4×C1672
#3282167