Finite Group Databasebeta

← #36365371

Subgroups — #36365371

Order 22144.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012141C211yesmaximal, characteristic, complemented
2#26012125C211yesmaximal, characteristic, complemented
3#26011836C211yesmaximal, characteristic, complemented
4#26011806C211yesmaximal, characteristic, complemented
5#26011802C211yesmaximal, characteristic, complemented
6#26011770C211yesmaximal, characteristic, complemented
7#26011726C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011624C211yesmaximal, characteristic, complemented
10#26011623C211yesmaximal, characteristic, complemented
11#26011571C211yesmaximal, characteristic, complemented
12C8×C22⋊D173C211yesmaximal, characteristic, complemented
13C2×#19149601C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#466×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C22×D692C2211yescharacteristic
18C2×D4×D173—21no—
19C2×D4×D173C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583811C2211yescharacteristic, complemented
22C2×C8×D173C2211yescharacteristic, complemented
23#19149678C2211yescharacteristic, complemented
24#19149675C2211yescharacteristic
25#19149661—21no—
26#19149660—21no—
27#19149656—21no—
28#19149654C2211yescharacteristic
29#19149652—21no—
30#19149650C2211yescharacteristic, complemented
31#19149649—21no—
32#19149647—21no—
33D4×Dic173C2211yescharacteristic, complemented
34#19149641C2211yescharacteristic
35C4×C22⋊D173C2211yescharacteristic
36C2×C4⋊Dic173C2211yescharacteristic, complemented
37C4⋊C4×D173C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149608C2211yescharacteristic, complemented
41#19149601C2211yescharacteristic
42#19149601C2211yescharacteristic
43#19149601C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×D8×C173C2211yescharacteristic, complemented
53C22×C8×C173C2211yescharacteristic, complemented
54#121×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic, complemented
56#107×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173—21no—
61#14583825—44no—
62C2×Dic346—21no—
63C2×D692—22no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—41no—
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173D411yescharacteristic
74D4×D173—22no—
75D4×D173—41no—
76D4×D173—44no—
77C22×Dic173—21no—
78C22×Dic173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173C2311yescharacteristic
82#14583811—22no—
83#14583811—41no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C8×D173—41no—
87C4⋊Dic173—21no—
88C4⋊Dic173C2311yescharacteristic
89C4⋊Dic173C2311yescharacteristic
90#14583805—21no—
91#14583805—21no—
92#14583805D411yescharacteristic
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173D411yescharacteristic
96#14583803—21no—
97#14583803—21no—
98#14583803D411yescharacteristic
99C4×Dic173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173—21no—
102C2×D4×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105D8×C173—41no—
106C2×C8×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—42no—
116C22×D173—42no—
117C22×D173—42no—
118C22×D173—42no—
119C22×D173—81no—
120C22×D173—81no—
121C22×D173C2×D411yescharacteristic
122Dic346—42no—
123D692—21no—
124D692—21no—
125D692—21no—
126D692—42no—
127D692—42no—
128D692#4112yescomplemented
129C22⋊D173—42no—
130C22⋊D173—42no—
131C22⋊D173—42no—
132C22⋊D173—81no—
133C22⋊D173D814yescomplemented
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—42no—
137C2×Dic173—42no—
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C2×Dic173C2×D411yescharacteristic
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—42no—
144C4×D173—42no—
145#13063436—21no—
146#13063436—21no—
147C23×C173—21no—
148C23×C173—21no—
149C23×C173C2×D411yescharacteristic
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153D4×C173—21no—
154D4×C173—42no—
155D4×C173—42no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2411yescharacteristic
162C8×C173—21no—
163C8×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173C2×D812yes—
173C22×C173C22×D411yescharacteristic
174C2×D173—21no—
175C2×D173—21no—
176C2×D173—22no—
177C2×D173—41no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—84no—
183C2×D173—84no—
184C2×D173C2×D812yes—
185C4×C173—21no—
186C4×C173—41no—
187C4×C173—41no—
188C4×C173C2×#4111yescharacteristic
189C4×C173H(5,𝔽2)11yescharacteristic
190Dic173—21no—
191Dic173—21no—
192Dic173—41no—
193Dic173—41no—
194Dic173C2×D812yes—
195C2×C173—22no—
196C2×C173—81no—
197C2×C173—81no—
198C2×C173#54611yescharacteristic
199C2×C173C22×D811yescharacteristic
200C2×C173#57711yescharacteristic
201D173—22no—
202D173—42no—
203D173—81no—
204C173#329411yescharacteristic, complemented, Sylow
205#3294—1731nomaximal, Sylow
206#547—1731no—
207D42—1731no—
208#486—1731no—
209#480—1731no—
210#480—1731no—
211#474—1731no—
212#466—1731no—
213#466D17311yescharacteristic, complemented
214#459—1731no—
215#447—1731no—
216#447—1731no—
217C4×D8—1731no—
218C8×D4—1731no—
219C2×#102—1731no—
220C2×#102—1731no—
221C2×#41—3461no—
222C22×D4—3461no—
223C22×D4—1731no—
224C22×D4—1731no—
225C2×D8—1731no—
226C2×D8—1731no—
227C2×D8C2×D17311yescharacteristic, complemented
228C22×C8—1731no—
229C22×C8C2×D17311yescharacteristic, complemented
230C4⋊Q8—1731no—
231#127—1731no—
232#123—3461no—
233#123—3461no—
234C22⋊Q8—3461no—
235#121—3461no—
236#121—1731no—
237#121—1731no—
238#121C2×D17311yescharacteristic
239#121C2×D17311yescharacteristic, complemented
240#120—3461no—
241#120—3461no—
242C4×D4—1731no—
243C4×D4—1731no—
244C4×D4—1731no—
245C2×C4⋊C4—1731no—
246C2×C4⋊C4—1731no—
247#107—1731no—
248#107—1731no—
249#107C2×D17311yescharacteristic, complemented
250C4⋊C8—1731no—
251#102—1731no—
252#102—1731no—
253#102—1731no—
254#102—1731no—
255#102—1731no—
256#102—1731no—
257#102—1731no—
258#102—1731no—
259#102C2×D17311yescharacteristic, complemented
260#102C2×D17311yescharacteristic, complemented
261C22⋊C8—1731no—
262C22⋊C8—1731no—
263C4×C8—1731no—
264C24—3461no—
265C24—3461no—
266#41—6924no—
267C2×Q8—3461no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—3462no—
271C2×D4—3461no—
272C2×D4—3462no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—1731no—
279C2×D4—1731no—
280C2×D4—1731no—
281C2×D4—6921no—
282C2×D4—6921no—
283C2×D4—6924no—
284C2×D4C22×D17311yescharacteristic
285C2×D4C22×D17311yescharacteristic
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—1731no—
291C22×C4C22×D17311yescharacteristic
292D8—41no—
293D8—3462no—
294D8—6921no—
295C2×C8—41no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8—6921no—
299C2×C8C22×D17311yescharacteristic
300C2×C8C22×D17311yescharacteristic
301C4⋊C4—3461no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—1731no—
305C4⋊C4—1731no—
306C4⋊C4—1731no—
307C4⋊C4C22×D17311yescharacteristic
308C4⋊C4C22×D17311yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—3461no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—1731no—
316C22⋊C4—1731no—
317C42—1731no—
318C23—21no—
319C23—21no—
320C23—3461no—
321C23—3461no—
322C23—13841no—
323C23—13841no—
324C23—1731no—
325C23—6922no—
326C23—6922no—
327C23—6922no—
328C23—6922no—
329C23D4×D17311yescharacteristic
330Q8—6922no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—42no—
336D4—42no—
337D4—3461no—
338D4—3461no—
339D4—3461no—
340D4—13841no—
341D4—1734no—
342D4—1732no—
343D4—6922no—
344D4—6922no—
345D4—6922no—
346D4—6922no—
347D4—6922no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—1731no—
357C2×C4—1731no—
358C2×C4—1731no—
359C2×C4—6922no—
360C2×C4—6922no—
361C2×C4—6922no—
362C2×C4—6922no—
363C2×C4D4×D17311yescharacteristic
364C2×C4C23×D17311yescharacteristic
365C8—21no—
366C8—21no—
367C8—3461no—
368C8—3461no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3461no—
378C22—3461no—
379C22—3462no—
380C22—13844no—
381C22—13844no—
382C22—1732no—
383C22—6921no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22—6921no—
388C22D8×D17312yes—
389C22C2×D4×D17311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—3461no—
394C4—3461no—
395C4—1732no—
396C4—6921no—
397C4—6921no—
398C4#41×D17311yescharacteristic
399C4#1914972411yescharacteristic
400C2—22no—
401C2—81no—
402C2—81no—
403C2—3462no—
404C2—13841no—
405C2—6922no—
406C2#2601213011yescharacteristic
407C2C2×D8×D17311yescharacteristic
408C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#466×C173C2
socleC22×C173C22×D4
hypercenter#466D173
hypocenterC173#3294
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #26012141C2semidirect
2. #26012125C2semidirect
3. #26011836C2semidirect
4. #26011806C2semidirect
5. #26011802C2semidirect
6. #26011770C2semidirect
7. #26011726C2semidirect
8. #26011684C2semidirect
9. #26011624C2semidirect
10. #26011623C2semidirect
11. #26011571C2semidirect
12. C8×C22⋊D173C2semidirect
13. C2×#19149601C2semidirect
14. #102×D173C2semidirect
15. #466×C173C2semidirect
21. C2×#14583811C22semidirect
22. C2×C8×D173C22semidirect
23. #19149678C22semidirect
30. #19149650C22semidirect
33. D4×Dic173C22semidirect
36. C2×C4⋊Dic173C22semidirect
37. C4⋊C4×D173C22semidirect
38. #19149612C22semidirect
39. #19149612C22semidirect
40. #19149608C22semidirect
43. #19149601C22semidirect
44. #19149601C22semidirect
45. #19149600C22semidirect
46. #19149600C22semidirect
48. #19149599C22semidirect
49. C22⋊#13063436C22semidirect
50. #19149591C22semidirect
51. C8×Dic173C22semidirect
52. C2×D8×C173C22semidirect
53. C22×C8×C173C22semidirect
55. #121×C173C22semidirect
56. #107×C173C22semidirect
57. #102×C173C22semidirect
58. #102×C173C22semidirect
128. D692#41semidirect
133. C22⋊D173D8semidirect
204. C173#3294semidirect
213. #466D173semidirect
227. C2×D8C2×D173semidirect
229. C22×C8C2×D173semidirect
239. #121C2×D173semidirect
249. #107C2×D173semidirect
259. #102C2×D173semidirect
260. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121412
#260121252
#260118362
#260118062
#260118022
#260117702
#260117262
#260116842
#260116242
#260116232
#260115712
C8×C22⋊D1732
C2×#191496012
#102×D1732
#466×C1732
#3294173