Finite Group Databasebeta

← #36365586

Subgroups — #36365586

Order 22144.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012181C211yesmaximal, characteristic, complemented
2D4×D692C211yesmaximal, characteristic
3#26011856C211yesmaximal, characteristic, complemented
4#26011816C211yesmaximal, characteristic, complemented
5#26011805C211yesmaximal, characteristic, complemented
6#26011792C211yesmaximal, characteristic, complemented
7#26011737C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011692C211yesmaximal, characteristic, complemented
10#26011644C211yesmaximal, characteristic, complemented
11#26011644C211yesmaximal, characteristic, complemented
12#26011593C211yesmaximal, characteristic, complemented
13#26011497C211yesmaximal, characteristic, complemented
14C2×#19149601C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×D4×D173—21no—
19C2×#14583815C2211yescharacteristic
20C2×#14583815C2211yescharacteristic, complemented
21C2×#14583815C2211yescharacteristic, complemented
22#19149676C2211yescharacteristic
23#19149672—21no—
24#19149671C2211yescharacteristic, complemented
25#19149669—21no—
26#19149660—21no—
27#19149658C2211yescharacteristic, complemented
28#19149658C2211yescharacteristic, complemented
29#19149654—21no—
30#19149653C2211yescharacteristic
31#19149653C2211yescharacteristic
32#19149649—21no—
33#19149649—21no—
34C4×Dic346C2211yescharacteristic, complemented
35C4×D692C2211yescharacteristic
36#19149638C2211yescharacteristic, complemented
37#19149635—21no—
38C2×C4⋊Dic173C2211yescharacteristic, complemented
39#19149613C2211yescharacteristic, complemented
40#19149613C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic, complemented
47#19149601C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52C2×M16×C173C2211yescharacteristic, complemented
53C22×C8×C173C2211yescharacteristic, complemented
54C4×D4×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic, complemented
56C22⋊C8×C173C2211yescharacteristic, complemented
57C22⋊C8×C173C2211yescharacteristic, complemented
58#97×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173—21no—
61C2×Dic346C2311yescharacteristic
62C2×D692—22no—
63C2×D692—22no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—41no—
69C2×D692—41no—
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75D4×D173—44no—
76C22×Dic173—21no—
77C2×C4×D173—21no—
78#14583815—22no—
79#14583815—41no—
80#14583815—41no—
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90#14583805—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99C2×D4×C173D411yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102M16×C173—41no—
103C2×C8×C173—41no—
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C22⋊C4×C173D411yescharacteristic
111C42×C173C2311yescharacteristic
112C22×D173—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—
121Dic346—21no—
122Dic346—21no—
123D692—21no—
124D692—21no—
125D692—21no—
126D692—21no—
127D692—21no—
128D692—42no—
129D692—42no—
130D692—42no—
131D692—81no—
132D692#4112yes—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—42no—
141C4×D173—42no—
142C23×C173C2×D411yescharacteristic
143C23×C173C2×D411yescharacteristic
144D4×C173—22no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2411yescharacteristic
154C8×C173—21no—
155C8×C173—21no—
156C8×C173—21no—
157C8×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173AGL(1,ℤ/8)12yes—
164C22×C173C22×D411yescharacteristic
165C2×D173—22no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—84no—
175C2×D173—84no—
176C4×C173—21no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173C2×#4111yescharacteristic
181C4×C173H(5,𝔽2)11yescharacteristic
182Dic173—41no—
183Dic173—41no—
184Dic173—41no—
185Dic173—41no—
186Dic173—41no—
187C2×C173—22no—
188C2×C173—41no—
189C2×C173#54611yescharacteristic
190C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C173#57711yescharacteristic
192D173—42no—
193D173—81no—
194D173—81no—
195C173#331011yescharacteristic, complemented, Sylow
196#3310—1731nomaximal, Sylow
197#552—1731no—
198D42—1731no—
199#490—1731no—
200#482—1731no—
201#480—1731no—
202#478—1731no—
203#468—1731no—
204#465—1731no—
205#460—1731no—
206#450—1731no—
207#450—1731no—
208#440—1731no—
209#435D17311yescharacteristic, complemented
210#418—1731no—
211C2×#102—1731no—
212C22×D4—3461no—
213C22×D4—1731no—
214C22×D4—1731no—
215C2×SD16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×M16C2×D17311yescharacteristic, complemented
219C22×C8C2×D17311yescharacteristic, complemented
220#127—1731no—
221#126—3461no—
222#125—3461no—
223#125—1731no—
224#123—3461no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227#121—3461no—
228#121—1731no—
229#121—1731no—
230#120—3461no—
231#120—3461no—
232C4×Q8—1731no—
233C4×D4—1731no—
234C4×D4C2×D17311yescharacteristic
235#117—3461no—
236#117—1731no—
237C2×C4⋊C4—1731no—
238#107—1731no—
239#107—1731no—
240#106—1731no—
241C4⋊C8C2×D17311yescharacteristic, complemented
242#103—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102—1731no—
252C22⋊C8C2×D17311yescharacteristic, complemented
253C22⋊C8C2×D17311yescharacteristic, complemented
254#97C2×D17311yescharacteristic, complemented
255C24—3461no—
256C24—3461no—
257C2×Q8—1731no—
258C2×D4—3462no—
259C2×D4—3462no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—1731no—
267C2×D4—1731no—
268C2×D4—1731no—
269C2×D4—6921no—
270C2×D4—6921no—
271C2×D4—6924no—
272C2×D4D69211yescharacteristic
273C22×C4—3461no—
274C22×C4—3461no—
275C22×C4C22×D17311yescharacteristic
276C22×C4C22×D17311yescharacteristic
277SD16—3462no—
278SD16—6921no—
279SD16—6921no—
280M16—41no—
281C2×C8—41no—
282C2×C8C22×D17311yescharacteristic
283C2×C8C22×D17311yescharacteristic
284C2×C8C22×D17311yescharacteristic
285C2×C8C22×D17311yescharacteristic
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4—1731no—
296C4⋊C4D69211yescharacteristic
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4D69211yescharacteristic
304C22⋊C4D69211yescharacteristic
305C42—3461no—
306C42—3461no—
307C42C22×D17311yescharacteristic
308C23—3461no—
309C23—3461no—
310C23—3461no—
311C23—13841no—
312C23—13841no—
313C23—6922no—
314C23—6922no—
315C23—6922no—
316C23—6922no—
317C23C2×D69211yescharacteristic
318C23C2×D69211yescharacteristic
319Q8—3461no—
320Q8—3461no—
321D4—22no—
322D4—3461no—
323D4—3461no—
324D4—3461no—
325D4—3461no—
326D4—3461no—
327D4—13841no—
328D4—1732no—
329D4—6922no—
330D4—6922no—
331D4—6922no—
332D4—6922no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—3461no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—6922no—
344C2×C4—6922no—
345C2×C4C2×D69211yescharacteristic
346C2×C4C2×D69211yescharacteristic
347C2×C4C2×D69211yescharacteristic
348C2×C4C2×D69211yescharacteristic
349C2×C4C23×D17311yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—3462no—
360C22—13844no—
361C22—13844no—
362C22—6921no—
363C22—6921no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22#1914970712yes—
371C22C22×D69211yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—6921no—
377C4—6921no—
378C4—6921no—
379C4—6921no—
380C4—6921no—
381C4#41×D17311yescharacteristic
382C4#1914972411yescharacteristic
383C2—22no—
384C2—41no—
385C2—13841no—
386C2—13841no—
387C2—6922no—
388C2#2601213611yescharacteristic
389C2C2×#1914970711yescharacteristic
390C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#435×C173C2
socleC22×C173C22×D4
prime radicalD4×D692C2
hypercenter#435D173
hypocenterC173#3310
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #26012181C2semidirect
3. #26011856C2semidirect
4. #26011816C2semidirect
5. #26011805C2semidirect
6. #26011792C2semidirect
7. #26011737C2semidirect
8. #26011720C2semidirect
9. #26011692C2semidirect
10. #26011644C2semidirect
11. #26011644C2semidirect
12. #26011593C2semidirect
13. #26011497C2semidirect
14. C2×#19149601C2semidirect
15. #435×C173C2semidirect
20. C2×#14583815C22semidirect
21. C2×#14583815C22semidirect
24. #19149671C22semidirect
27. #19149658C22semidirect
28. #19149658C22semidirect
34. C4×Dic346C22semidirect
36. #19149638C22semidirect
38. C2×C4⋊Dic173C22semidirect
39. #19149613C22semidirect
40. #19149613C22semidirect
41. #19149611C22semidirect
42. #19149604C22semidirect
46. #19149601C22semidirect
47. #19149601C22semidirect
48. #19149601C22semidirect
49. #19149601C22semidirect
50. #19149601C22semidirect
51. #19149601C22semidirect
52. C2×M16×C173C22semidirect
53. C22×C8×C173C22semidirect
55. C4⋊C8×C173C22semidirect
56. C22⋊C8×C173C22semidirect
57. C22⋊C8×C173C22semidirect
58. #97×C173C22semidirect
195. C173#3310semidirect
209. #435D173semidirect
218. C2×M16C2×D173semidirect
219. C22×C8C2×D173semidirect
241. C4⋊C8C2×D173semidirect
252. C22⋊C8C2×D173semidirect
253. C22⋊C8C2×D173semidirect
254. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121812
D4×D6922
#260118562
#260118162
#260118052
#260117922
#260117372
#260117202
#260116922
#260116442
#260115932
#260114972
C2×#191496012
#435×C1732
#3310173