Finite Group Databasebeta

← #36365528

Subgroups — #36365528

Order 22144.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012141C211yesmaximal, characteristic, complemented
2#26012135C211yesmaximal, characteristic, complemented
3#26011849C211yesmaximal, characteristic, complemented
4#26011822C211yesmaximal, characteristic, complemented
5#26011808C211yesmaximal, characteristic, complemented
6#26011779C211yesmaximal, characteristic, complemented
7#26011741C211yesmaximal, characteristic, complemented
8#26011718C211yesmaximal, characteristic, complemented
9#26011704C211yesmaximal, characteristic, complemented
10#26011633C211yesmaximal, characteristic, complemented
11#26011603C211yesmaximal, characteristic, complemented
12#26011559C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14C2×#19149600C211yesmaximal, characteristic, complemented
15#447×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×#14583825C2211yescharacteristic
18C2×#14583814C2211yescharacteristic, complemented
19C2×#14583809C2211yescharacteristic, complemented
20C22×#13063436C2211yescharacteristic
21#19149678C2211yescharacteristic, complemented
22#19149666C2211yescharacteristic
23#19149661—21no—
24#19149660—21no—
25#19149660—21no—
26#19149658C2211yescharacteristic
27#19149656—21no—
28#19149651—21no—
29#19149650C2211yescharacteristic
30#19149650C2211yescharacteristic, complemented
31#19149648—21no—
32D4×Dic173C2211yescharacteristic
33D4×Dic173C2211yescharacteristic, complemented
34C4×C22⋊D173C2211yescharacteristic
35C2×C4⋊Dic173C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic, complemented
37C2×C22⋊Dic173—21no—
38#19149611C2211yescharacteristic, complemented
39#19149611C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×D8×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61#14583825—44no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173D411yescharacteristic
68C22×Dic173—21no—
69C22×Dic173—21no—
70C22×Dic173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74#14583814—41no—
75#14583809—41no—
76C2×#13063436—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173D411yescharacteristic
95#14583803D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97C24×C173—21no—
98C2×D4×C173—22no—
99C2×D4×C173—21no—
100C2×D4×C173—21no—
101C2×D4×C173—41no—
102C2×D4×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106D8×C173—22no—
107D8×C173—41no—
108C2×C8×C173C2311yescharacteristic
109C2×C8×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—42no—
116C22⋊D173—22no—
117C22⋊D173—42no—
118C22⋊D173—42no—
119C22⋊D173—81no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—42no—
125C2×Dic173—42no—
126C2×Dic173—42no—
127C2×Dic173—81no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—42no—
134#13063436—21no—
135#13063436—21no—
136C23×C173—21no—
137C23×C173—21no—
138C23×C173—21no—
139C23×C173—42no—
140C23×C173—42no—
141C23×C173—81no—
142C23×C173C2×D411yescharacteristic
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148D4×C173—42no—
149D4×C173—42no—
150D4×C173#4112yescomplemented
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2411yescharacteristic
156C8×C173—21no—
157C8×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—22no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—84no—
170C22×C173AGL(1,ℤ/8)12yes—
171C22×C173C22×D411yescharacteristic
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—21no—
175C4×C173—21no—
176C4×C173—41no—
177C4×C173C2×#4111yescharacteristic
178C4×C173H(5,𝔽2)11yescharacteristic
179Dic173—21no—
180Dic173—21no—
181Dic173—21no—
182Dic173—41no—
183Dic173—41no—
184Dic173—41no—
185Dic173—41no—
186C2×C173—22no—
187C2×C173—42no—
188C2×C173—81no—
189C2×C173—81no—
190C2×C173#54611yescharacteristic
191C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C173#57711yescharacteristic
193D173—41no—
194C173#330611yescharacteristic, complemented, Sylow
195#3306—1731nomaximal, Sylow
196#547—1731no—
197#546—1731no—
198#489—1731no—
199#483—1731no—
200#481—1731no—
201#476—1731no—
202#469—1731no—
203#465—1731no—
204#463—1731no—
205#449—1731no—
206#447D17311yescharacteristic, complemented
207#442—1731no—
208#435—1731no—
209#417—1731no—
210C2×#102—1731no—
211C2×#41—3461no—
212C2×#41—1731no—
213C22×D4C2×D17311yescharacteristic
214C2×SD16—1731no—
215C2×D8C2×D17311yescharacteristic
216C2×D8C2×D17311yescharacteristic
217C2×M16—1731no—
218C22×C8—1731no—
219C4⋊Q8—1731no—
220#124—1731no—
221#123—3461no—
222#123—3461no—
223#123—3461no—
224C22⋊Q8—3461no—
225C22⋊Q8—1731no—
226#121—3461no—
227#121—1731no—
228#121—1731no—
229#121C2×D17311yescharacteristic
230#120—3461no—
231C4×D4—1731no—
232C4×D4—1731no—
233C4×D4—1731no—
234C2×C4⋊C4—1731no—
235C2×C4⋊C4—1731no—
236C2×C22⋊C4—3461no—
237#106—1731no—
238#106—1731no—
239#106—1731no—
240C4⋊C8—1731no—
241#103—1731no—
242#103—1731no—
243#103—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102C2×D17311yescharacteristic
250#102C2×D17311yescharacteristic
251C22⋊C8—1731no—
252C22⋊C8C2×D17311yescharacteristic
253#97—1731no—
254C24—21no—
255#41—3462no—
256#41—6921no—
257#41—6924no—
258C2×Q8—3461no—
259C2×Q8—1731no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—1731no—
268C2×D4C22×D17311yescharacteristic
269C2×D4C22×D17311yescharacteristic
270C2×D4C22×D17311yescharacteristic
271C22×C4—3461no—
272C22×C4—3461no—
273C22×C4—3461no—
274C22×C4—3461no—
275C22×C4—3461no—
276C22×C4—1731no—
277C22×C4C22×D17311yescharacteristic
278SD16—6921no—
279D8—22no—
280D8—41no—
281M16—6921no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8—6921no—
285C2×C8C22×D17311yescharacteristic
286C2×C8C22×D17311yescharacteristic
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—1731no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4C22×D17311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—1731no—
305C22⋊C4—1731no—
306C42—1731no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—1731no—
314C23D4×D17311yescharacteristic
315Q8—3461no—
316Q8—3461no—
317Q8—6922no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—42no—
325D4—3462no—
326D4—13841no—
327D4—6922no—
328D4—6922no—
329D4#1458382512yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—13841no—
340C2×C4—1731no—
341C2×C4—1731no—
342C2×C4—1731no—
343C2×C4—6922no—
344C2×C4—6922no—
345C2×C4—6922no—
346C2×C4—6922no—
347C2×C4D4×D17311yescharacteristic
348C2×C4C23×D17311yescharacteristic
349C8—21no—
350C8—21no—
351C8—3461no—
352C8—3461no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—3461no—
366C22—3461no—
367C22—3461no—
368C22#1914970312yes—
369C22C2×D4×D17311yescharacteristic
370C4—21no—
371C4—41no—
372C4—3461no—
373C4—3461no—
374C4—3461no—
375C4—6921no—
376C4—6921no—
377C4—6921no—
378C4—6921no—
379C4C2×#1458382511yescharacteristic
380C4#1914972411yescharacteristic
381C2—22no—
382C2—42no—
383C2—81no—
384C2—81no—
385C2—6921no—
386C2#2601212811yescharacteristic
387C2C2×#1914970311yescharacteristic
388C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#447×C173C2
socleC22×C173C22×D4
hypercenter#447D173
hypocenterC173#3306
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #26012141C2semidirect
2. #26012135C2semidirect
3. #26011849C2semidirect
4. #26011822C2semidirect
5. #26011808C2semidirect
6. #26011779C2semidirect
7. #26011741C2semidirect
8. #26011718C2semidirect
9. #26011704C2semidirect
10. #26011633C2semidirect
11. #26011603C2semidirect
12. #26011559C2semidirect
13. #26011489C2semidirect
14. C2×#19149600C2semidirect
15. #447×C173C2semidirect
18. C2×#14583814C22semidirect
19. C2×#14583809C22semidirect
21. #19149678C22semidirect
30. #19149650C22semidirect
33. D4×Dic173C22semidirect
36. C4⋊C4×D173C22semidirect
38. #19149611C22semidirect
39. #19149611C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
49. #19149600C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
150. D4×C173#41semidirect
194. C173#3306semidirect
206. #447D173semidirect
329. D4#14583825semidirect

Maximal subgroups

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

subgroupindex
#260121412
#260121352
#260118492
#260118222
#260118082
#260117792
#260117412
#260117182
#260117042
#260116332
#260116032
#260115592
#260114892
C2×#191496002
#447×C1732
#3306173