Finite Group Databasebeta

← #36365415

Subgroups — #36365415

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012187C211yesmaximal, characteristic, complemented
2#26012137C211yesmaximal, characteristic, complemented
3#26011846C211yesmaximal, characteristic, complemented
4#26011827C211yesmaximal, characteristic, complemented
5#26011816C211yesmaximal, characteristic, complemented
6#26011792C211yesmaximal, characteristic, complemented
7#26011727C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011707C211yesmaximal, characteristic, complemented
10#26011638C211yesmaximal, characteristic, complemented
11#26011627C211yesmaximal, characteristic, complemented
12C4×D1384C211yesmaximal, characteristic, complemented
13#26011488C211yesmaximal, characteristic, complemented
14C2×#19149601C211yesmaximal, characteristic, complemented
15C8×D4×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583815C2211yescharacteristic, complemented
19C2×D1384C2211yescharacteristic
20C2×D1384C2211yescharacteristic
21#19149674—21no—
22#19149671C2211yescharacteristic, complemented
23#19149667C2211yescharacteristic
24#19149664—21no—
25#19149661—21no—
26#19149658C2211yescharacteristic
27#19149653C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149653C2211yescharacteristic, complemented
30#19149652—21no—
31#19149651—21no—
32#19149647—21no—
33C4×D692C2211yescharacteristic
34C4×D692C2211yescharacteristic, complemented
35#19149639—21no—
36#19149638C2211yescharacteristic, complemented
37C2×C4⋊Dic173C2211yescharacteristic
38C22⋊C4×D173—21no—
39#19149613C2211yescharacteristic, complemented
40#19149613C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic
51#19149601C2211yescharacteristic, complemented
52C22×C8×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic, complemented
54C4×D4×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic, complemented
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic, complemented
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—22no—
61#14583826—41no—
62C2×Dic346C2311yescharacteristic
63C2×D692—22no—
64C2×D692—21no—
65C2×D692—41no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C22×Dic173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77#14583815—41no—
78D1384—22no—
79D1384—41no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C4×Dic173—21no—
98C2×D4×C173D411yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C2×C8×C173—41no—
102C2×C8×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—42no—
115C22×D173—42no—
116C22×D173—81no—
117Dic346—21no—
118Dic346—21no—
119D692—21no—
120D692—21no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—42no—
125D692#4112yes—
126C22⋊D173—42no—
127C22⋊D173—42no—
128C22⋊D173—81no—
129C22⋊D173—81no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—81no—
136C4×D173—42no—
137C4×D173—81no—
138C4×D173—81no—
139C23×C173C2×D411yescharacteristic
140C23×C173C2×D411yescharacteristic
141D4×C173—22no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2411yescharacteristic
151C8×C173—21no—
152C8×C173—21no—
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173#13512yes—
161C22×C173C22×D411yescharacteristic
162C2×D173—22no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—84no—
172C4×C173—21no—
173C4×C173—21no—
174C4×C173—21no—
175C4×C173—21no—
176C4×C173C2×#4111yescharacteristic
177C4×C173H(5,𝔽2)11yescharacteristic
178Dic173—41no—
179Dic173—41no—
180Dic173—41no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—22no—
184C2×C173—41no—
185C2×C173#54611yescharacteristic
186C2×C173C2×#13511yescharacteristic
187C2×C173#57611yescharacteristic
188D173—42no—
189D173—81no—
190D173—81no—
191C173#329711yescharacteristic, complemented, Sylow
192#3297—1731nomaximal, Sylow
193#553—1731no—
194#546—1731no—
195#488—1731no—
196#484—1731no—
197#482—1731no—
198#478—1731no—
199#466—1731no—
200#465—1731no—
201#463—1731no—
202#449—1731no—
203#447—1731no—
204C4×D8—1731no—
205C8×D4D17311yescharacteristic, complemented
206#416—1731no—
207C2×#102—1731no—
208C2×#41—1731no—
209C22×D4—1731no—
210C2×SD16—1731no—
211C2×D8—1731no—
212C2×D8—1731no—
213C22×C8C2×D17311yescharacteristic
214C22×C8C2×D17311yescharacteristic, complemented
215#126—3461no—
216#125—1731no—
217#124—1731no—
218#123—3461no—
219#123—3461no—
220C22⋊Q8—1731no—
221#121—3461no—
222#121—3461no—
223#121—1731no—
224#121—1731no—
225#121—1731no—
226#120—3461no—
227C4×D4—3461no—
228C4×D4—1731no—
229C4×D4—1731no—
230C4×D4C2×D17311yescharacteristic
231#117—1731no—
232C2×C4⋊C4—1731no—
233C2×C22⋊C4—3461no—
234#107—1731no—
235#107—1731no—
236#106—1731no—
237C4⋊C8C2×D17311yescharacteristic, complemented
238#103—1731no—
239#103—1731no—
240#103—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248C22⋊C8C2×D17311yescharacteristic
249C22⋊C8C2×D17311yescharacteristic, complemented
250C4×C8C2×D17311yescharacteristic, complemented
251C24—3461no—
252#41—3462no—
253#41—6921no—
254C2×Q8—1731no—
255C2×D4—3462no—
256C2×D4—3461no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—1731no—
262C2×D4—1731no—
263C2×D4—1731no—
264C2×D4—6921no—
265C2×D4D69211yescharacteristic
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4C22×D17311yescharacteristic
271C22×C4C22×D17311yescharacteristic
272SD16—6921no—
273D8—3462no—
274D8—6921no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D17311yescharacteristic
278C2×C8C22×D17311yescharacteristic
279C2×C8C22×D17311yescharacteristic
280C2×C8C22×D17311yescharacteristic
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4—1731no—
288C4⋊C4—1731no—
289C4⋊C4D69211yescharacteristic
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4D69211yescharacteristic
300C22⋊C4D69211yescharacteristic
301C42—3461no—
302C42C22×D17311yescharacteristic
303C23—3461no—
304C23—3461no—
305C23—3461no—
306C23—13841no—
307C23—6922no—
308C23—6922no—
309C23C2×D69211yescharacteristic
310C23C2×D69211yescharacteristic
311Q8—3461no—
312Q8—3461no—
313D4—22no—
314D4—3461no—
315D4—3461no—
316D4—3461no—
317D4—3461no—
318D4—3461no—
319D4—13841no—
320D4—13841no—
321D4—1732no—
322D4—6922no—
323D4—6922no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—13841no—
335C2×C4—13841no—
336C2×C4—13841no—
337C2×C4—6922no—
338C2×C4C2×D69211yescharacteristic
339C2×C4C2×D69211yescharacteristic
340C2×C4C2×D69211yescharacteristic
341C2×C4C2×D69211yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—3462no—
353C22—13844no—
354C22—6921no—
355C22—6921no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22—6921no—
362C22#1914970112yes—
363C22C22×D69211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—6921no—
369C4—6921no—
370C4—6921no—
371C4—6921no—
372C4—6921no—
373C4#41×D17311yescharacteristic
374C4#1914972411yescharacteristic
375C2—22no—
376C2—41no—
377C2—13841no—
378C2—13841no—
379C2—6922no—
380C2#2601213611yescharacteristic
381C2C2×#1914970111yescharacteristic
382C2#2601229611yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC8×D4×C173C2
socleC22×C173C22×D4
hypercenterC8×D4D173
hypocenterC173#3297
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #26012187C2semidirect
2. #26012137C2semidirect
3. #26011846C2semidirect
4. #26011827C2semidirect
5. #26011816C2semidirect
6. #26011792C2semidirect
7. #26011727C2semidirect
8. #26011720C2semidirect
9. #26011707C2semidirect
10. #26011638C2semidirect
11. #26011627C2semidirect
12. C4×D1384C2semidirect
13. #26011488C2semidirect
14. C2×#19149601C2semidirect
15. C8×D4×C173C2semidirect
18. C2×#14583815C22semidirect
22. #19149671C22semidirect
29. #19149653C22semidirect
34. C4×D692C22semidirect
36. #19149638C22semidirect
39. #19149613C22semidirect
40. #19149613C22semidirect
41. #19149611C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
51. #19149601C22semidirect
53. C22×C8×C173C22semidirect
55. C4⋊C8×C173C22semidirect
57. C22⋊C8×C173C22semidirect
58. C4×C8×C173C22semidirect
191. C173#3297semidirect
205. C8×D4D173semidirect
214. C22×C8C2×D173semidirect
237. C4⋊C8C2×D173semidirect
249. C22⋊C8C2×D173semidirect
250. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121872
#260121372
#260118462
#260118272
#260118162
#260117922
#260117272
#260117202
#260117072
#260116382
#260116272
C4×D13842
#260114882
C2×#191496012
C8×D4×C1732
#3297173