Finite Group Databasebeta

← #36335629

Subgroups — #36335629

Order 21376.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128858C211yesmaximal, characteristic, complemented
2C22×#14581938C211yesmaximal, characteristic, complemented
3#24923928C211yesmaximal, characteristic, complemented
4#24923656C211yesmaximal, characteristic, complemented
5#24923513C211yesmaximal, characteristic, complemented
6#24923503C211yesmaximal, characteristic, complemented
7#24923484C211yesmaximal, characteristic, complemented
8#24923466C211yesmaximal, characteristic, complemented
9#24923420C211yesmaximal, characteristic, complemented
10#24923412C211yesmaximal, characteristic, complemented
11#24923407C211yesmaximal, characteristic, complemented
12#24923403C211yesmaximal, characteristic, complemented
13#24923369C211yesmaximal, characteristic, complemented
14#24923341C211yesmaximal, characteristic, complemented
15D42×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic, complemented
17C22×Dic334C2211yescharacteristic, complemented
18#19128858—24no—
19C2×#14581938—24no—
20C2×#14581938C2211yescharacteristic
21C2×#14581938C2211yescharacteristic
22C2×#14581938C2211yescharacteristic, complemented
23C2×#14581938C2211yescharacteristic, complemented
24C2×#14581938C2211yescharacteristic, complemented
25C2×#14581938C2211yescharacteristic, complemented
26C2×#14581935C2211yescharacteristic
27C2×#14581935C2211yescharacteristic, complemented
28C2×#14581935C2211yescharacteristic, complemented
29C2×#14581934C2211yescharacteristic, complemented
30C22×C167⋊C8C2211yescharacteristic, complemented
31#19128820C2211yescharacteristic, complemented
32#19128818—21no—
33#19128811C2211yescharacteristic, complemented
34#19128810—21no—
35#19128806C2211yescharacteristic, complemented
36#19128804—21no—
37C4×Dic334C2211yescharacteristic, complemented
38(C22⋊D167)⋊C4—21no—
39#19128765C2211yescharacteristic, complemented
40C4⋊(C167⋊C8)C2211yescharacteristic, complemented
41#19128755C2211yescharacteristic, complemented
42#19128755C2211yescharacteristic, complemented
43#19128753C2211yescharacteristic
44#19128753C2211yescharacteristic, complemented
45#19128753C2211yescharacteristic, complemented
46#19128752C2211yescharacteristic, complemented
47C22⋊(C167⋊C8)C2211yescharacteristic, complemented
48C22⋊(C167⋊C8)C2211yescharacteristic, complemented
49#19128743C2211yescharacteristic, complemented
50C22×D4×C167—21no—
51C22×D4×C167C2211yescharacteristic
52C22×D4×C167C2211yescharacteristic
53#127×C167C2211yescharacteristic
54#121×C167—21no—
55#121×C167C2211yescharacteristic
56#121×C167C2211yescharacteristic
57#120×C167—21no—
58#120×C167—21no—
59C4×D4×C167C2211yescharacteristic
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#14581938—22no—
74#14581938—22no—
75#14581938—22no—
76#14581938—22no—
77#14581938—41no—
78#14581938D414yescomplemented
79#14581935—22no—
80#14581935—22no—
81#14581935—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—
96C24×C167—21no—
97C24×C167—21no—
98C2×D4×C167—22no—
99C2×D4×C167—22no—
100C2×D4×C167—21no—
101C2×D4×C167—21no—
102C2×D4×C167—21no—
103C2×D4×C167—21no—
104C2×D4×C167—21no—
105C2×D4×C167—21no—
106C2×D4×C167—41no—
107C2×D4×C167—41no—
108C2×D4×C167—44no—
109C2×D4×C167D411yescharacteristic
110C2×D4×C167C2311yescharacteristic
111C2×D4×C167C2311yescharacteristic
112C2×D4×C167C2311yescharacteristic
113C22×C4×C167—21no—
114C22×C4×C167C2311yescharacteristic
115C22×C4×C167C2311yescharacteristic
116C4⋊C4×C167D411yescharacteristic
117C4⋊C4×C167C2311yescharacteristic
118C22⋊C4×C167—21no—
119C22⋊C4×C167—21no—
120C22⋊C4×C167—21no—
121C22⋊C4×C167D411yescharacteristic
122C22⋊C4×C167D411yescharacteristic
123C42×C167C2311yescharacteristic
124C22×D167—21no—
125Dic334—21no—
126Dic334—21no—
127Dic334—21no—
128Dic334—42no—
129Dic334C2×D412yescomplemented
130D668—21no—
131D668—21no—
132C22⋊D167—42no—
133C22⋊D167—42no—
134C2×Dic167—21no—
135C2×Dic167—21no—
136C2×Dic167—21no—
137C2×Dic167—42no—
138C4×D167—42no—
139C167⋊C8—21no—
140C167⋊C8—21no—
141C167⋊C8—21no—
142C167⋊C8C2×D412yescomplemented
143C23×C167—21no—
144C23×C167—21no—
145C23×C167—21no—
146C23×C167—42no—
147C23×C167—42no—
148C23×C167—42no—
149C23×C167—42no—
150C23×C167—81no—
151C23×C167—81no—
152C23×C167C2×D411yescharacteristic
153C23×C167C2×D411yescharacteristic
154D4×C167—22no—
155D4×C167—21no—
156D4×C167—21no—
157D4×C167—21no—
158D4×C167—21no—
159D4×C167—21no—
160D4×C167—42no—
161D4×C167—42no—
162D4×C167—42no—
163D4×C167—42no—
164D4×C167—42no—
165D4×C167—81no—
166D4×C167C2×D412yes—
167C2×C4×C167—21no—
168C2×C4×C167—21no—
169C2×C4×C167—21no—
170C2×C4×C167—21no—
171C2×C4×C167—21no—
172C2×C4×C167—42no—
173C2×C4×C167C2×D411yescharacteristic
174C2×C4×C167C2×D411yescharacteristic
175C2×C4×C167C2×D411yescharacteristic
176C2×C4×C167C2×D411yescharacteristic
177C2×C4×C167C2411yescharacteristic
178C22×C167—21no—
179C22×C167—21no—
180C22×C167—21no—
181C22×C167—22no—
182C22×C167—21no—
183C22×C167—21no—
184C22×C167—41no—
185C22×C167—41no—
186C22×C167—41no—
187C22×C167—41no—
188C22×C167—41no—
189C22×C167—41no—
190C22×C167—41no—
191C22×C167—41no—
192C22×C167—84no—
193C22×C167—84no—
194C22×C167AGL(1,ℤ/8)12yescomplemented
195C22×C167C22×D411yescharacteristic
196C2×D167—41no—
197C2×D167—41no—
198C2×D167—41no—
199C4×C167—21no—
200C4×C167—21no—
201C4×C167—21no—
202C4×C167—21no—
203C4×C167—41no—
204C4×C167C22×D411yescharacteristic
205C4×C167H(5,𝔽2)11yescharacteristic
206Dic167—22no—
207Dic167—41no—
208Dic167—41no—
209C2×C167—22no—
210C2×C167—41no—
211C2×C167—42no—
212C2×C167—81no—
213C2×C167—81no—
214C2×C167D4211yescharacteristic
215C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C167#57711yescharacteristic
217D167—81no—
218C167#326511yescharacteristic, complemented, Sylow
219#3265—1671nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1671no—
221C22×SD16—1671no—
222#548—1671no—
223D42D16711yescharacteristic, complemented
224#496—1671no—
225#468—1671no—
226#466—1671no—
227#463—1671no—
228#460—1671no—
229#450—1671no—
230#449—1671no—
231#448—1671no—
232#447—1671no—
233#440—1671no—
234#435—1671no—
235C2×#41—1671no—
236C22×Q8—1671no—
237C22×D4—21no—
238C22×D4C2×D16711yescharacteristic
239C22×D4C2×D16711yescharacteristic
240AGL(1,ℤ/8)—3344no—
241C2×SD16—3344no—
242C2×SD16—1671no—
243C2×SD16—1671no—
244C2×SD16—1671no—
245C2×SD16—1671no—
246C2×SD16—1671no—
247C2×SD16—1671no—
248C2×D8—1671no—
249C2×D8—1671no—
250C2×D8—1671no—
251C2×M16—1671no—
252C22×C8—1671no—
253#127C2×D16711yescharacteristic
254#124—3341no—
255#124—1671no—
256C22⋊Q8—3341no—
257C22⋊Q8—1671no—
258#121—21no—
259#121—3341no—
260#121—1671no—
261#121C2×D16711yescharacteristic
262#121C2×D16711yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—1671no—
266C4×D4—3341no—
267C4×D4C2×D16711yescharacteristic
268C4×D4C2×D16711yescharacteristic
269#107—1671no—
270C4⋊C8—1671no—
271#103—1671no—
272#103—1671no—
273#102—1671no—
274#102—1671no—
275#102—1671no—
276#102—1671no—
277C22⋊C8—1671no—
278C22⋊C8—1671no—
279#97—1671no—
280C24—21no—
281C24—21no—
282#41—3342no—
283#41—6681no—
284C2×Q8—3342no—
285C2×Q8—3341no—
286C2×Q8—6681no—
287C2×Q8—1671no—
288C2×Q8—1671no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—3341no—
301C2×D4—3341no—
302C2×D4—1671no—
303C2×D4C22⋊D16711yescharacteristic
304C2×D4C22×D16711yescharacteristic
305C2×D4C22×D16711yescharacteristic
306C2×D4C22×D16711yescharacteristic
307C22×C4—21no—
308C22×C4—3341no—
309C22×C4—3341no—
310C22×C4C22×D16711yescharacteristic
311C22×C4C22×D16711yescharacteristic
312SD16—3342no—
313SD16—3342no—
314SD16—3342no—
315SD16—3342no—
316SD16—6681no—
317SD16—1674no—
318D8—3342no—
319D8—3342no—
320D8—6681no—
321M16—3342no—
322C2×C8—3342no—
323C2×C8—1671no—
324C2×C8—1671no—
325C2×C8—1671no—
326C2×C8—1671no—
327C4⋊C4—3341no—
328C4⋊C4—3341no—
329C4⋊C4—1671no—
330C4⋊C4C22⋊D16711yescharacteristic
331C4⋊C4C22×D16711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—3341no—
336C22⋊C4—3341no—
337C22⋊C4—3341no—
338C22⋊C4—3341no—
339C22⋊C4C22⋊D16711yescharacteristic
340C22⋊C4C22⋊D16711yescharacteristic
341C42—3341no—
342C42C22×D16711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—3341no—
353C23C2×C22⋊D16711yescharacteristic
354C23C2×C22⋊D16711yescharacteristic
355Q8—3341no—
356Q8—3341no—
357Q8—3341no—
358Q8—6682no—
359Q8—1672no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—3341no—
373D4—3341no—
374D4—6682no—
375D4—6682no—
376D4D4×D16712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—3341no—
384C2×C4—3341no—
385C2×C4—3341no—
386C2×C4—6682no—
387C2×C4—6682no—
388C2×C4C2×C22⋊D16711yescharacteristic
389C2×C4C2×C22⋊D16711yescharacteristic
390C2×C4C2×C22⋊D16711yescharacteristic
391C2×C4C2×C22⋊D16711yescharacteristic
392C2×C4C23×D16711yescharacteristic
393C8—3341no—
394C8—3341no—
395C8—3341no—
396C8—1672no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—6681no—
414C22—6681no—
415C22—6681no—
416C22#1912885812yescomplemented
417C22C22×C22⋊D16711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—3342no—
424C4—6681no—
425C4—6681no—
426C4C2×D4×D16711yescharacteristic
427C4#1912887711yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—13361no—
434C2D4×C22⋊D16711yescharacteristic
435C2C2×#1912885811yescharacteristic
436C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingD42×C167C2
socleC22×C167C22×D4
hypercenterD42D167
hypocenterC167#3265
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#19128858C2semidirect
2. C22×#14581938C2semidirect
3. #24923928C2semidirect
4. #24923656C2semidirect
5. #24923513C2semidirect
6. #24923503C2semidirect
7. #24923484C2semidirect
8. #24923466C2semidirect
9. #24923420C2semidirect
10. #24923412C2semidirect
11. #24923407C2semidirect
12. #24923403C2semidirect
13. #24923369C2semidirect
14. #24923341C2semidirect
15. D42×C167C2semidirect
16. C2×#14581950C22semidirect
17. C22×Dic334C22semidirect
22. C2×#14581938C22semidirect
23. C2×#14581938C22semidirect
24. C2×#14581938C22semidirect
25. C2×#14581938C22semidirect
27. C2×#14581935C22semidirect
28. C2×#14581935C22semidirect
29. C2×#14581934C22semidirect
30. C22×C167⋊C8C22semidirect
31. #19128820C22semidirect
33. #19128811C22semidirect
35. #19128806C22semidirect
37. C4×Dic334C22semidirect
39. #19128765C22semidirect
40. C4⋊(C167⋊C8)C22semidirect
41. #19128755C22semidirect
42. #19128755C22semidirect
44. #19128753C22semidirect
45. #19128753C22semidirect
46. #19128752C22semidirect
47. C22⋊(C167⋊C8)C22semidirect
48. C22⋊(C167⋊C8)C22semidirect
49. #19128743C22semidirect
78. #14581938D4semidirect
129. Dic334C2×D4semidirect
142. C167⋊C8C2×D4semidirect
194. C22×C167AGL(1,ℤ/8)semidirect
218. C167#3265semidirect
223. D42D167semidirect
416. C22#19128858semidirect

Maximal subgroups

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

subgroupindex
C2×#191288582
C22×#145819382
#249239282
#249236562
#249235132
#249235032
#249234842
#249234662
#249234202
#249234122
#249234072
#249234032
#249233692
#249233412
D42×C1672
#3265167