Finite Group Databasebeta

← #36327278

Subgroups — #36327278

Order 21376.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128802C211yesmaximal, characteristic
2#24923171C211yesmaximal, characteristic, complemented
3#24923170C211yesmaximal, characteristic, complemented
4#24923169C211yesmaximal, characteristic, complemented
5#24923162C211yesmaximal, characteristic, complemented
6#24923161C211yesmaximal, characteristic, complemented
7#24923160C211yesmaximal, characteristic, complemented
8#24923159C211yesmaximal, characteristic, complemented
9#24923158C211yesmaximal, characteristic, complemented
10#24923147C211yesmaximal, characteristic, complemented
11#24923147C211yesmaximal, characteristic, complemented
12#24923145C211yesmaximal, characteristic, complemented
13#24923124C211yesmaximal, characteristic, complemented
14#24923124C211yesmaximal, characteristic, complemented
15#388×C167C211yesmaximal, characteristic, complemented
16C24×D167C2211yescharacteristic
17C22×D668C2211yescharacteristic
18C22×D668C2211yescharacteristic
19C22×C22⋊D167C2211yescharacteristic
20C22×C4×D167C2211yescharacteristic, complemented
21#19128802C418yescomplemented
22C2×C4⋊Dic167C2211yescharacteristic, complemented
23C2×Dic167⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic167C2211yescharacteristic
25C2×C22⋊Dic167C2211yescharacteristic
26C2×(C2×D167)⋊C4C2211yescharacteristic
27C2×(C2×D167)⋊C4C2211yescharacteristic
28C2×(C2×D167)⋊C4C2211yescharacteristic
29C2×(C2×D167)⋊C4C2211yescharacteristic
30C2×(C2×D167)⋊C4C2211yescharacteristic
31C2×(C2×D167)⋊C4C2211yescharacteristic
32C2×(C2×D167)⋊C4C2211yescharacteristic
33C2×(C2×D167)⋊C4C2211yescharacteristic
34C2×(C2×D167)⋊C4C2211yescharacteristic
35C2×(C2×D167)⋊C4C2211yescharacteristic
36C2×(C2×D167)⋊C4C2211yescharacteristic, complemented
37C2×(C2×D167)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D167)⋊C4C2211yescharacteristic, complemented
39C2×C4×Dic167C2211yescharacteristic, complemented
40C2×C4×Dic167C2211yescharacteristic, complemented
41C167⋊#95C2211yescharacteristic, complemented
42C167⋊#95C2211yescharacteristic, complemented
43C167⋊#95C2211yescharacteristic, complemented
44C167⋊#95C2211yescharacteristic, complemented
45C2×C4⋊C4×C167C2211yescharacteristic, complemented
46C2×C22⋊C4×C167C2211yescharacteristic
47C2×C22⋊C4×C167C2211yescharacteristic
48C2×C22⋊C4×C167C2211yescharacteristic
49C2×C42×C167C2211yescharacteristic, complemented
50#95×C167C2211yescharacteristic, complemented
51#95×C167C2211yescharacteristic, complemented
52C23×D167—41no—
53C23×D167—41no—
54C23×D167—41no—
55C23×D167—41no—
56C23×D167—41no—
57C23×D167—41no—
58C23×D167C2×C414yescomplemented
59C23×D167C2311yescharacteristic
60C23×D167C2311yescharacteristic
61C23×D167C2311yescharacteristic
62C2×D668—41no—
63C2×D668—41no—
64C2×D668—41no—
65C2×D668—41no—
66C2×D668C2×C414yes—
67C2×D668C2×C414yes—
68C2×C22⋊D167—41no—
69C2×C22⋊D167—41no—
70C2×C22⋊D167C2×C414yes—
71C22×Dic167C2311yescharacteristic
72C22×Dic167C2311yescharacteristic
73C22×Dic167C2311yescharacteristic
74C22×Dic167C2311yescharacteristic
75C22×Dic167C2311yescharacteristic
76C2×C4×D167—41no—
77C2×C4×D167—41no—
78C2×C4×D167—41no—
79C4⋊Dic167—41no—
80Dic167⋊C4—41no—
81C22⋊Dic167—41no—
82C22⋊Dic167—41no—
83(C2×D167)⋊C4—41no—
84(C2×D167)⋊C4—41no—
85(C2×D167)⋊C4—41no—
86(C2×D167)⋊C4—41no—
87(C2×D167)⋊C4—41no—
88(C2×D167)⋊C4—41no—
89(C2×D167)⋊C4—41no—
90(C2×D167)⋊C4—41no—
91(C2×D167)⋊C4—41no—
92(C2×D167)⋊C4—41no—
93(C2×D167)⋊C4—41no—
94(C2×D167)⋊C4C2×C414yescomplemented
95(C2×D167)⋊C4C2×C414yescomplemented
96C4×Dic167—41no—
97C4×Dic167—41no—
98C24×C167C2311yescharacteristic
99C22×C4×C167C2311yescharacteristic
100C22×C4×C167C2311yescharacteristic
101C22×C4×C167C2311yescharacteristic
102C22×C4×C167C2311yescharacteristic
103C22×C4×C167C2311yescharacteristic
104C22×C4×C167C2311yescharacteristic
105C4⋊C4×C167—41no—
106C22⋊C4×C167—41no—
107C22⋊C4×C167—41no—
108C22⋊C4×C167C2×C414yescomplemented
109C42×C167—41no—
110C22×D167—21no—
111C22×D167—21no—
112C22×D167—21no—
113C22×D167—21no—
114C22×D167—21no—
115C22×D167—21no—
116C22×D167—21no—
117C22×D167—21no—
118C22×D167—21no—
119C22×D167—21no—
120C22×D167—21no—
121C22×D167—21no—
122C22×D167—21no—
123C22×D167—21no—
124C22×D167—21no—
125C22×D167—21no—
126C22×D167—21no—
127C22×D167—21no—
128C22×D167—44no—
129C22×D167—44no—
130C22×D167—44no—
131C22×D167—44no—
132C22×D167—44no—
133C22×D167—44no—
134C22×D167—44no—
135C22×D167C22×C412yes—
136C22×D167C22×C412yes—
137C22×D167C22×C412yes—
138D668—44no—
139D668—44no—
140C22⋊D167—44no—
141C2×Dic167—21no—
142C2×Dic167—21no—
143C2×Dic167—21no—
144C2×Dic167—21no—
145C2×Dic167—21no—
146C2×Dic167—21no—
147C2×Dic167—21no—
148C2×Dic167—21no—
149C2×Dic167—21no—
150C2×Dic167—21no—
151C2×Dic167—21no—
152C2×Dic167—21no—
153C2×Dic167—21no—
154C2×Dic167—21no—
155C2×Dic167C22×C412yes—
156C4×D167—44no—
157C23×C167—21no—
158C23×C167—21no—
159C23×C167—21no—
160C23×C167—21no—
161C23×C167—21no—
162C23×C167—21no—
163C23×C167C22×C412yes—
164C23×C167C2411yescharacteristic
165C2×C4×C167—21no—
166C2×C4×C167—21no—
167C2×C4×C167—21no—
168C2×C4×C167—21no—
169C2×C4×C167—21no—
170C2×C4×C167—21no—
171C2×C4×C167—21no—
172C2×C4×C167—21no—
173C2×C4×C167—21no—
174C2×C4×C167—21no—
175C2×C4×C167—21no—
176C2×C4×C167—21no—
177C2×C4×C167—21no—
178C2×C4×C167—21no—
179C2×C4×C167—21no—
180C2×C4×C167—21no—
181C2×C4×C167C22×C412yes—
182C2×C4×C167C22×C412yes—
183C22×C167—22no—
184C22×C167—22no—
185C22×C167—22no—
186C22×C167—41no—
187C22×C167—41no—
188C22×C167—41no—
189C22×C167—41no—
190C22×C167C23×C411yescharacteristic
191C22×C167H(5,𝔽2)11yescharacteristic
192C22×C167H(5,𝔽2)11yescharacteristic
193C22×C167H(5,𝔽2)11yescharacteristic
194C22×C167H(5,𝔽2)11yescharacteristic
195C22×C167H(5,𝔽2)11yescharacteristic
196C22×C167H(5,𝔽2)11yescharacteristic
197C2×D167—22no—
198C2×D167—22no—
199C2×D167—22no—
200C2×D167—22no—
201C2×D167—22no—
202C2×D167—22no—
203C2×D167—22no—
204C2×D167—22no—
205C2×D167—22no—
206C2×D167—416no—
207C2×D167—41no—
208C2×D167—41no—
209C2×D167—41no—
210C2×D167—41no—
211C2×D167—41no—
212C2×D167—41no—
213C2×D167—41no—
214C2×D167—41no—
215C2×D167—41no—
216C2×D167—41no—
217C2×D167—41no—
218C2×D167—41no—
219C4×C167—22no—
220C4×C167—22no—
221C4×C167—41no—
222C4×C167—41no—
223C4×C167—41no—
224C4×C167—41no—
225Dic167—22no—
226Dic167—41no—
227Dic167—41no—
228Dic167—41no—
229Dic167—41no—
230C2×C167—42no—
231C2×C167#51811yescharacteristic
232C2×C167#51811yescharacteristic
233C2×C167#51811yescharacteristic
234C2×C167#56011yescharacteristic
235C2×C167#56011yescharacteristic
236C2×C167#56011yescharacteristic
237C2×C167H(3,𝔽4)11yescharacteristic
238D167—42no—
239D167—42no—
240D167—42no—
241C167#237511yescharacteristic, complemented, Sylow
242#2375—1671nomaximal, Sylow
243C2×#120—1671no—
244#390—1671no—
245#390—1671no—
246#390—1671no—
247#388—1671no—
248#388—1671no—
249#388—1671no—
250#388—1671no—
251#388—1671no—
252#388D16711yescharacteristic, complemented
253#386—1671no—
254#386—1671no—
255#386—1671no—
256#379—1671no—
257#379—1671no—
258C25—1671no—
259C22×D4—1671no—
260C22×D4—1671no—
261C22×D4—1671no—
262C23×C4—1671no—
263#120—1678no—
264C2×C4⋊C4—1671no—
265C2×C4⋊C4—1671no—
266C2×C4⋊C4C2×D16711yescharacteristic, complemented
267C2×C22⋊C4—1671no—
268C2×C22⋊C4—1671no—
269C2×C22⋊C4—1671no—
270C2×C22⋊C4—1671no—
271C2×C22⋊C4—1671no—
272C2×C22⋊C4—1671no—
273C2×C22⋊C4—1671no—
274C2×C22⋊C4—1671no—
275C2×C22⋊C4—1671no—
276C2×C22⋊C4—1671no—
277C2×C22⋊C4—1671no—
278C2×C22⋊C4—1671no—
279C2×C22⋊C4—1671no—
280C2×C22⋊C4—1671no—
281C2×C22⋊C4—1671no—
282C2×C22⋊C4C2×D16711yescharacteristic
283C2×C22⋊C4C2×D16711yescharacteristic
284C2×C22⋊C4C2×D16711yescharacteristic
285C2×C42—1671no—
286C2×C42—1671no—
287C2×C42C2×D16711yescharacteristic, complemented
288#95—1671no—
289#95—1671no—
290#95—1671no—
291#95—1671no—
292#95C2×D16711yescharacteristic, complemented
293#95C2×D16711yescharacteristic, complemented
294C24—6681no—
295C24—6681no—
296C24—6681no—
297C24—6681no—
298C24—6681no—
299C24—6681no—
300C24—1671no—
301C24—1671no—
302C24—1671no—
303C24—1674no—
304C24C22×D16711yescharacteristic
305C2×D4—6681no—
306C2×D4—6681no—
307C2×D4—6681no—
308C2×D4—6681no—
309C2×D4—6681no—
310C2×D4—6681no—
311C2×D4—1674no—
312C2×D4—1674no—
313C2×D4—1674no—
314C22×C4—6681no—
315C22×C4—6681no—
316C22×C4—6681no—
317C22×C4—1671no—
318C22×C4—1671no—
319C22×C4—1671no—
320C22×C4—1671no—
321C22×C4—1671no—
322C22×C4C22×D16711yescharacteristic
323C22×C4C22×D16711yescharacteristic
324C22×C4C22×D16711yescharacteristic
325C22×C4C22×D16711yescharacteristic
326C22×C4C22×D16711yescharacteristic
327C22×C4C22×D16711yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—6681no—
330C4⋊C4—6681no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—6681no—
334C22⋊C4—6681no—
335C22⋊C4—6681no—
336C22⋊C4—6681no—
337C22⋊C4—6681no—
338C22⋊C4—6681no—
339C22⋊C4—6681no—
340C22⋊C4—6681no—
341C22⋊C4—6681no—
342C22⋊C4—6681no—
343C22⋊C4—6681no—
344C22⋊C4—6681no—
345C22⋊C4—6681no—
346C22⋊C4—1674no—
347C22⋊C4—1674no—
348C22⋊C4C4×D16714yescomplemented
349C42—41no—
350C42—6681no—
351C42—6681no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—3341no—
359C23—3341no—
360C23—3341no—
361C23—3341no—
362C23—3341no—
363C23—3341no—
364C23—3341no—
365C23—3341no—
366C23—3341no—
367C23—3341no—
368C23—3341no—
369C23—3341no—
370C23—3341no—
371C23—3341no—
372C23—3341no—
373C23—3341no—
374C23—3341no—
375C23—3341no—
376C23—6684no—
377C23—6684no—
378C23—6684no—
379C23—6684no—
380C23—6684no—
381C23—6684no—
382C23—6684no—
383C23—1672no—
384C23—1672no—
385C23—1672no—
386C23C2×C4×D16712yes—
387C23C23×D16711yescharacteristic
388D4—6684no—
389D4—6684no—
390D4—6684no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—3341no—
408C2×C4—3341no—
409C2×C4—3341no—
410C2×C4—3341no—
411C2×C4—3341no—
412C2×C4—3341no—
413C2×C4—3341no—
414C2×C4—3341no—
415C2×C4—3341no—
416C2×C4—3341no—
417C2×C4—3341no—
418C2×C4—3341no—
419C2×C4—3341no—
420C2×C4—3341no—
421C2×C4—6684no—
422C2×C4—1672no—
423C2×C4C2×C4×D16712yes—
424C2×C4C2×C4×D16712yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—3342no—
433C22—3342no—
434C22—3342no—
435C22—3342no—
436C22—3342no—
437C22—3342no—
438C22—3342no—
439C22—3342no—
440C22—3342no—
441C22—66816no—
442C22—6681no—
443C22—6681no—
444C22—6681no—
445C22—6681no—
446C22—6681no—
447C22—6681no—
448C22—6681no—
449C22—6681no—
450C22—6681no—
451C22—6681no—
452C22—6681no—
453C22—6681no—
454C22C22×C4×D16711yescharacteristic
455C22#1912887711yescharacteristic
456C22#1912887711yescharacteristic
457C22#1912887811yescharacteristic
458C22#1912887811yescharacteristic
459C22#1912887811yescharacteristic
460C22#1912887811yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—3342no—
468C4—6681no—
469C4—6681no—
470C4—6681no—
471C4—6681no—
472C2—42no—
473C2—6682no—
474C2—6682no—
475C2—6682no—
476C2#2492374111yescharacteristic
477C2#2492374111yescharacteristic
478C2#2492374311yescharacteristic
479C2#2492400711yescharacteristic
480C2#2492400911yescharacteristic
481C2#2492401011yescharacteristic
482C2#2492401111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D167
commutatorC22×C167C23×C4
Fitting#388×C167C2
socle, cosocleC23×C167C24
prime radicalC2×#19128802C2
hypercenter#388D167
hypocenterC167#2375
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #24923171C2semidirect
3. #24923170C2semidirect
4. #24923169C2semidirect
5. #24923162C2semidirect
6. #24923161C2semidirect
7. #24923160C2semidirect
8. #24923159C2semidirect
9. #24923158C2semidirect
10. #24923147C2semidirect
11. #24923147C2semidirect
12. #24923145C2semidirect
13. #24923124C2semidirect
14. #24923124C2semidirect
15. #388×C167C2semidirect
20. C22×C4×D167C22semidirect
21. #19128802C4semidirect
22. C2×C4⋊Dic167C22semidirect
23. C2×Dic167⋊C4C22semidirect
36. C2×(C2×D167)⋊C4C22semidirect
37. C2×(C2×D167)⋊C4C22semidirect
38. C2×(C2×D167)⋊C4C22semidirect
39. C2×C4×Dic167C22semidirect
40. C2×C4×Dic167C22semidirect
41. C167⋊#95C22semidirect
42. C167⋊#95C22semidirect
43. C167⋊#95C22semidirect
44. C167⋊#95C22semidirect
45. C2×C4⋊C4×C167C22semidirect
49. C2×C42×C167C22semidirect
50. #95×C167C22semidirect
51. #95×C167C22semidirect
58. C23×D167C2×C4semidirect
94. (C2×D167)⋊C4C2×C4semidirect
95. (C2×D167)⋊C4C2×C4semidirect
108. C22⋊C4×C167C2×C4semidirect
241. C167#2375semidirect
252. #388D167semidirect
266. C2×C4⋊C4C2×D167semidirect
287. C2×C42C2×D167semidirect
292. #95C2×D167semidirect
293. #95C2×D167semidirect
348. C22⋊C4C4×D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288022
#249231712
#249231702
#249231692
#249231622
#249231612
#249231602
#249231592
#249231582
#249231472
#249231452
#249231242
#388×C1672
#2375167