Finite Group Databasebeta

← #36335745

Subgroups — #36335745

Order 21376.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128858C211yesmaximal, characteristic, complemented
2C2×#19128857C211yesmaximal, characteristic, complemented
3#24923915C211yesmaximal, characteristic, complemented
4#24923651C211yesmaximal, characteristic, complemented
5#24923517C211yesmaximal, characteristic, complemented
6#24923513C211yesmaximal, characteristic, complemented
7#24923484C211yesmaximal, characteristic, complemented
8#24923481C211yesmaximal, characteristic, complemented
9#24923431C211yesmaximal, characteristic, complemented
10#24923429C211yesmaximal, characteristic, complemented
11#24923403C211yesmaximal, characteristic, complemented
12#24923402C211yesmaximal, characteristic, complemented
13C4×#14581935C211yesmaximal, characteristic, complemented
14#24923347C211yesmaximal, characteristic, complemented
15#546×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×D668C2211yescharacteristic, complemented
18#19128858—24no—
19#19128857—24no—
20C2×#14581940C2211yescharacteristic, complemented
21C2×#14581938C2211yescharacteristic, complemented
22C2×#14581938C2211yescharacteristic, complemented
23C2×#14581937C2211yescharacteristic, complemented
24C2×#14581937C2211yescharacteristic, complemented
25C2×#14581935C2211yescharacteristic
26C2×#14581935C2211yescharacteristic
27C2×#14581935C2211yescharacteristic
28C2×#14581935C2211yescharacteristic
29C2×#14581934C2211yescharacteristic, complemented
30C2×#14581934C2211yescharacteristic, complemented
31#19128820C2211yescharacteristic, complemented
32#19128817—21no—
33#19128811C2211yescharacteristic, complemented
34#19128806C2211yescharacteristic
35#19128805—21no—
36#19128800—21no—
37C4×D668C2211yescharacteristic, complemented
38C22⋊C4×D167—21no—
39#19128765C2211yescharacteristic, complemented
40C4⋊(C167⋊C8)C2211yescharacteristic, complemented
41#19128756C2211yescharacteristic, complemented
42#19128755C2211yescharacteristic, complemented
43#19128753C2211yescharacteristic, complemented
44#19128753C2211yescharacteristic, complemented
45#19128752C2211yescharacteristic, complemented
46#19128752C2211yescharacteristic, complemented
47C22⋊(C167⋊C8)C2211yescharacteristic, complemented
48C22⋊(C167⋊C8)C2211yescharacteristic, complemented
49C4×C167⋊C8C2211yescharacteristic, complemented
50C2×#41×C167C2211yescharacteristic
51C22×D4×C167C2211yescharacteristic, complemented
52#124×C167C2211yescharacteristic, complemented
53#123×C167—21no—
54C22⋊Q8×C167C2211yescharacteristic, complemented
55#121×C167—21no—
56#121×C167C2211yescharacteristic
57#120×C167—21no—
58C4×D4×C167C2211yescharacteristic
59C4×D4×C167C2211yescharacteristic, complemented
60C2×C22⋊C4×C167—21no—
61C23×D167—21no—
62#14581950—22no—
63#14581950—41no—
64C2×Dic334C2311yescharacteristic
65C2×D668—22no—
66C2×D668—21no—
67C2×D668—41no—
68C2×D668C2311yescharacteristic
69C2×D668C2311yescharacteristic
70C2×C22⋊D167—21no—
71C2×C22⋊D167—21no—
72C2×C4×D167—21no—
73C2×C4×D167—21no—
74#14581940—41no—
75#14581938—22no—
76#14581938—22no—
77#14581937—22no—
78#14581937—22no—
79#14581935—22no—
80#14581935—22no—
81#14581935—41no—
82#14581935D414yescomplemented
83#14581934—22no—
84#14581934—22no—
85C2×C167⋊C8C2311yescharacteristic
86C2×C167⋊C8C2311yescharacteristic
87C2×C167⋊C8C2311yescharacteristic
88C2×C167⋊C8C2311yescharacteristic
89C4⋊Dic167C2311yescharacteristic
90Dic167⋊C4—21no—
91C22⋊Dic167—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95(C2×D167)⋊C4—21no—
96C24×C167—21no—
97#41×C167—22no—
98#41×C167—41no—
99C2×Q8×C167C2311yescharacteristic
100C2×D4×C167—22no—
101C2×D4×C167—21no—
102C2×D4×C167—21no—
103C2×D4×C167—21no—
104C2×D4×C167—41no—
105C2×D4×C167D411yescharacteristic
106C2×D4×C167C2311yescharacteristic
107C2×D4×C167C2311yescharacteristic
108C22×C4×C167—21no—
109C22×C4×C167—21no—
110C22×C4×C167C2311yescharacteristic
111C22×C4×C167C2311yescharacteristic
112C4⋊C4×C167—21no—
113C4⋊C4×C167D411yescharacteristic
114C4⋊C4×C167C2311yescharacteristic
115C22⋊C4×C167—21no—
116C22⋊C4×C167—21no—
117C22⋊C4×C167—21no—
118C22⋊C4×C167—21no—
119C22⋊C4×C167—21no—
120C22⋊C4×C167D411yescharacteristic
121C22⋊C4×C167D411yescharacteristic
122C42×C167C2311yescharacteristic
123C22×D167—21no—
124C22×D167—21no—
125C22×D167—42no—
126C22×D167—42no—
127C22×D167—81no—
128Dic334—21no—
129Dic334—21no—
130D668—21no—
131D668—21no—
132D668—21no—
133D668—42no—
134D668C2×D412yes—
135C22⋊D167—42no—
136C22⋊D167—81no—
137C2×Dic167—21no—
138C2×Dic167—21no—
139C4×D167—42no—
140C4×D167—81no—
141C167⋊C8—21no—
142C167⋊C8—21no—
143C167⋊C8—21no—
144C167⋊C8C2×D412yes—
145C23×C167—21no—
146C23×C167—21no—
147C23×C167—42no—
148C23×C167—42no—
149C23×C167—81no—
150C23×C167C2×D411yescharacteristic
151C23×C167C2×D411yescharacteristic
152Q8×C167—21no—
153Q8×C167—21no—
154D4×C167—21no—
155D4×C167—21no—
156D4×C167—21no—
157D4×C167—41no—
158D4×C167—42no—
159D4×C167—42no—
160D4×C167—81no—
161D4×C167C2×D412yes—
162C2×C4×C167—21no—
163C2×C4×C167—21no—
164C2×C4×C167—21no—
165C2×C4×C167—21no—
166C2×C4×C167—21no—
167C2×C4×C167—21no—
168C2×C4×C167—42no—
169C2×C4×C167—81no—
170C2×C4×C167C2×D411yescharacteristic
171C2×C4×C167C2×D411yescharacteristic
172C2×C4×C167C2×D411yescharacteristic
173C2×C4×C167C2×D411yescharacteristic
174C2×C4×C167C2411yescharacteristic
175C22×C167—21no—
176C22×C167—21no—
177C22×C167—22no—
178C22×C167—21no—
179C22×C167—21no—
180C22×C167—21no—
181C22×C167—21no—
182C22×C167—41no—
183C22×C167—41no—
184C22×C167—41no—
185C22×C167—41no—
186C22×C167—41no—
187C22×C167—84no—
188C22×C167C22×D411yescharacteristic
189C2×D167—22no—
190C2×D167—41no—
191C2×D167—41no—
192C2×D167—41no—
193C2×D167—41no—
194C2×D167—41no—
195C2×D167—84no—
196C4×C167—21no—
197C4×C167—21no—
198C4×C167—21no—
199C4×C167—21no—
200C4×C167—41no—
201C4×C167—41no—
202C4×C167C22×D411yescharacteristic
203C4×C167H(5,𝔽2)11yescharacteristic
204Dic167—41no—
205Dic167—41no—
206C2×C167—41no—
207C2×C167—41no—
208C2×C167—42no—
209C2×C167—81no—
210C2×C167D4211yescharacteristic
211C2×C167#57511yescharacteristic
212C2×C167#57611yescharacteristic
213D167—42no—
214D167—81no—
215C167#327311yescharacteristic, complemented, Sylow
216#3273—1671nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1671no—
218C2×AGL(1,ℤ/8)—1671no—
219#546—1671no—
220#546D16711yescharacteristic, complemented
221#495—1671no—
222#468—1671no—
223#468—1671no—
224#463—1671no—
225#463—1671no—
226#452—1671no—
227#452—1671no—
228#447—1671no—
229#447—1671no—
230C4×D8—1671no—
231#436—1671no—
232C2×#41—1671no—
233C2×#41C2×D16711yescharacteristic
234C22×D4—1671no—
235C22×D4C2×D16711yescharacteristic, complemented
236AGL(1,ℤ/8)—3344no—
237AGL(1,ℤ/8)—3344no—
238C2×Q16—1671no—
239C2×SD16—1671no—
240C2×SD16—1671no—
241C2×SD16—1671no—
242C2×SD16—1671no—
243C2×D8—1671no—
244C2×D8—1671no—
245C2×D8—1671no—
246C2×D8—1671no—
247C2×M16—1671no—
248C2×M16—1671no—
249#124—1671no—
250#124C2×D16711yescharacteristic, complemented
251#123—21no—
252#123—3341no—
253C22⋊Q8—1671no—
254C22⋊Q8C2×D16711yescharacteristic, complemented
255#121—21no—
256#121—3341no—
257#121—1671no—
258#121C2×D16711yescharacteristic
259#120—21no—
260#120—3341no—
261C4×D4—1671no—
262C4×D4C2×D16711yescharacteristic
263C4×D4C2×D16711yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—3341no—
266#107—1671no—
267C4⋊C8—1671no—
268#103—1671no—
269#103—1671no—
270#102—1671no—
271#102—1671no—
272#102—1671no—
273#102—1671no—
274C22⋊C8—1671no—
275C22⋊C8—1671no—
276C4×C8—1671no—
277C24—21no—
278C24—3341no—
279#41—22no—
280#41—41no—
281#41—3342no—
282#41—6681no—
283C2×Q8—1671no—
284C2×Q8C22×D16711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—3342no—
291C2×D4—3341no—
292C2×D4—3341no—
293C2×D4—3341no—
294C2×D4—6681no—
295C2×D4—1671no—
296C2×D4—1671no—
297C2×D4C22⋊D16711yescharacteristic
298C2×D4C22×D16711yescharacteristic
299C2×D4C22×D16711yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—3341no—
303C22×C4—3341no—
304C22×C4C22×D16711yescharacteristic
305C22×C4C22×D16711yescharacteristic
306Q16—6681no—
307SD16—3342no—
308SD16—3342no—
309SD16—3342no—
310SD16—3342no—
311D8—3342no—
312D8—3342no—
313D8—6681no—
314D8—1674no—
315M16—3342no—
316M16—3342no—
317C2×C8—1671no—
318C2×C8—1671no—
319C2×C8—1671no—
320C2×C8—1671no—
321C4⋊C4—21no—
322C4⋊C4—3341no—
323C4⋊C4—1671no—
324C4⋊C4C22⋊D16711yescharacteristic
325C4⋊C4C22×D16711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—3341no—
332C22⋊C4—3341no—
333C22⋊C4—3341no—
334C22⋊C4—3341no—
335C22⋊C4—3341no—
336C22⋊C4C22⋊D16711yescharacteristic
337C22⋊C4C22⋊D16711yescharacteristic
338C42C22×D16711yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—13361no—
345C23—3341no—
346C23—3341no—
347C23—6682no—
348C23—6682no—
349C23C2×C22⋊D16711yescharacteristic
350C23C2×C22⋊D16711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3341no—
354Q8—3341no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—13361no—
363D4—3341no—
364D4—3341no—
365D4—3341no—
366D4—6682no—
367D4—6682no—
368D4—1672no—
369D4D4×D16712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—13361no—
379C2×C4—3341no—
380C2×C4—3341no—
381C2×C4—6682no—
382C2×C4C2×C22⋊D16711yescharacteristic
383C2×C4C2×C22⋊D16711yescharacteristic
384C2×C4C2×C22⋊D16711yescharacteristic
385C2×C4C2×C22⋊D16711yescharacteristic
386C2×C4C23×D16711yescharacteristic
387C8—3341no—
388C8—3341no—
389C8—3341no—
390C8—1672no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—13364no—
405C22—3342no—
406C22—6681no—
407C22—6681no—
408C22—6681no—
409C22—6681no—
410C22—6681no—
411C22C22×C22⋊D16711yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—6681no—
419C4—6681no—
420C4C2×D4×D16711yescharacteristic
421C4#1912887711yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—13361no—
427C2—6682no—
428C2D4×C22⋊D16711yescharacteristic
429C2#2492406711yescharacteristic
430C2#2492407211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#546×C167C2
socleC22×C167C22×D4
hypercenter#546D167
hypocenterC167#3273
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#19128858C2semidirect
2. C2×#19128857C2semidirect
3. #24923915C2semidirect
4. #24923651C2semidirect
5. #24923517C2semidirect
6. #24923513C2semidirect
7. #24923484C2semidirect
8. #24923481C2semidirect
9. #24923431C2semidirect
10. #24923429C2semidirect
11. #24923403C2semidirect
12. #24923402C2semidirect
13. C4×#14581935C2semidirect
14. #24923347C2semidirect
15. #546×C167C2semidirect
17. C22×D668C22semidirect
20. C2×#14581940C22semidirect
21. C2×#14581938C22semidirect
22. C2×#14581938C22semidirect
23. C2×#14581937C22semidirect
24. C2×#14581937C22semidirect
29. C2×#14581934C22semidirect
30. C2×#14581934C22semidirect
31. #19128820C22semidirect
33. #19128811C22semidirect
37. C4×D668C22semidirect
39. #19128765C22semidirect
40. C4⋊(C167⋊C8)C22semidirect
41. #19128756C22semidirect
42. #19128755C22semidirect
43. #19128753C22semidirect
44. #19128753C22semidirect
45. #19128752C22semidirect
46. #19128752C22semidirect
47. C22⋊(C167⋊C8)C22semidirect
48. C22⋊(C167⋊C8)C22semidirect
49. C4×C167⋊C8C22semidirect
51. C22×D4×C167C22semidirect
52. #124×C167C22semidirect
54. C22⋊Q8×C167C22semidirect
59. C4×D4×C167C22semidirect
82. #14581935D4semidirect
215. C167#3273semidirect
220. #546D167semidirect
235. C22×D4C2×D167semidirect
250. #124C2×D167semidirect
254. C22⋊Q8C2×D167semidirect
263. C4×D4C2×D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288582
C2×#191288572
#249239152
#249236512
#249235172
#249235132
#249234842
#249234812
#249234312
#249234292
#249234032
#249234022
C4×#145819352
#249233472
#546×C1672
#3273167