Finite Group Databasebeta

← #36363561

Subgroups — #36363561

Order 22144.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012038C211yesmaximal, characteristic, complemented
2C4×D4×D173C211yesmaximal, characteristic, complemented
3#26011806C211yesmaximal, characteristic, complemented
4#26011803C211yesmaximal, characteristic, complemented
5#26011770C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011685C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011626C211yesmaximal, characteristic, complemented
10#26011623C211yesmaximal, characteristic, complemented
11C4×D1384C211yesmaximal, characteristic, complemented
12C4×#14583811C211yesmaximal, characteristic, complemented
13C4×#14583811C211yesmaximal, characteristic, complemented
14#26011542C211yesmaximal, characteristic, complemented
15C4×D8×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×D4×D173C2211yescharacteristic
18C22×C4×D173—21no—
19C2×D1384C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22#19149678C2211yescharacteristic, complemented
23#19149675C2211yescharacteristic, complemented
24#19149665—21no—
25#19149654C2211yescharacteristic
26#19149652—21no—
27#19149650C2211yescharacteristic
28C4×D692C2211yescharacteristic
29C4×D692C2211yescharacteristic, complemented
30D4×Dic173C2211yescharacteristic
31#19149641C2211yescharacteristic
32C4×C22⋊D173—21no—
33C4×C22⋊D173—21no—
34#19149639—21no—
35C4⋊C4×D173C2211yescharacteristic
36C22⋊C4×D173—21no—
37C42×D173C2211yescharacteristic
38#19149613C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic, complemented
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51C4×#13063436C2211yescharacteristic, complemented
52C2×D8×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic, complemented
55#107×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic, complemented
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—44no—
61C2×Dic346—21no—
62C2×D692—21no—
63C2×D692C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69D4×D173—22no—
70D4×D173—22no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173—44no—
77C2×C4×D173D411yescharacteristic
78C2×C4×D173D411yescharacteristic
79C2×C4×D173C2311yescharacteristic
80D1384—22no—
81#14583811—22no—
82#14583811—22no—
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88#14583805—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173D411yescharacteristic
96C4×Dic173D411yescharacteristic
97C4×Dic173C2311yescharacteristic
98C2×D4×C173C2311yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173—21no—
101C22×C4×C173—21no—
102D8×C173—22no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C42×C173C2311yescharacteristic
110C22×D173—24no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173C2×D411yescharacteristic
116Dic346—42no—
117D692—21no—
118D692—21no—
119D692—21no—
120D692—42no—
121D692#4112yescomplemented
122C22⋊D173—24no—
123C22⋊D173—42no—
124C22⋊D173—42no—
125C22⋊D173—42no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—42no—
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—41no—
135C4×D173—42no—
136C4×D173—42no—
137C4×D173—42no—
138C4×D173—42no—
139C4×D173D814yescomplemented
140#13063436—21no—
141#13063436—21no—
142C23×C173—21no—
143C23×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—21no—
147D4×C173#4112yescomplemented
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—42no—
151C2×C4×C173—42no—
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C8×C173—21no—
156C8×C173—21no—
157C22×C173—22no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173C22×D411yescharacteristic
164C2×D173—21no—
165C2×D173—21no—
166C2×D173—22no—
167C2×D173—48no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173C2×D812yes—
174C4×C173—21no—
175C4×C173—41no—
176C4×C173—41no—
177C4×C173C2×D812yes—
178C4×C173C2×#4111yescharacteristic
179C4×C173C2×#4111yescharacteristic
180Dic173—21no—
181Dic173—21no—
182Dic173—41no—
183Dic173—41no—
184Dic173C2×D812yes—
185C2×C173—42no—
186C2×C173—81no—
187C2×C173#52511yescharacteristic
188C2×C173C22×D811yescharacteristic
189C2×C173#57511yescharacteristic
190D173—22no—
191D173—42no—
192D173—81no—
193C173#310011yescharacteristic, complemented, Sylow
194#3100—1731nomaximal, Sylow
195#532—1731no—
196C2×C4×D4—1731no—
197#480—1731no—
198#480—1731no—
199#474—1731no—
200#474—1731no—
201#459—1731no—
202#459—1731no—
203#447—1731no—
204#447—1731no—
205C4×D8—1731no—
206C4×D8—1731no—
207C4×D8—1731no—
208C4×D8D17311yescharacteristic, complemented
209#431—1731no—
210C2×#41—3461no—
211C22×D4—1731no—
212C23×C4—3461no—
213C2×D8—1731no—
214C2×D8—1731no—
215C2×D8—1731no—
216C2×D8C2×D17311yescharacteristic
217C4⋊Q8—1731no—
218#127—1731no—
219#124—3461no—
220#121—3461no—
221#121—1731no—
222#121—1731no—
223C4×D4—3461no—
224C4×D4—3461no—
225C4×D4—3461no—
226C4×D4—1731no—
227C4×D4—1731no—
228C4×D4—1731no—
229C4×D4—1731no—
230C4×D4C2×D17311yescharacteristic
231C4×D4C2×D17311yescharacteristic, complemented
232C2×C4⋊C4—1731no—
233C2×C22⋊C4—3461no—
234C2×C42—1731no—
235#107—1731no—
236#107—1731no—
237#107—1731no—
238#107C2×D17311yescharacteristic, complemented
239C4⋊C8—1731no—
240C4⋊C8—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102C2×D17311yescharacteristic
248#102C2×D17311yescharacteristic, complemented
249C22⋊C8—1731no—
250C22⋊C8—1731no—
251C4×C8—1731no—
252C4×C8C2×D17311yescharacteristic, complemented
253C24—3461no—
254#41—6924no—
255C2×Q8—3461no—
256C2×D4—3461no—
257C2×D4—3462no—
258C2×D4—3462no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—1731no—
264C2×D4—1731no—
265C2×D4C22×D17311yescharacteristic
266C2×D4C22×D17311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4—3461no—
272C22×C4—3461no—
273C22×C4—3461no—
274C22×C4—1731no—
275C22×C4—1731no—
276C22×C4—1731no—
277C22×C4—6924no—
278D8—22no—
279D8—3462no—
280D8—3462no—
281D8—3462no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8C22×D17311yescharacteristic
285C2×C8C22×D17311yescharacteristic
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—1731no—
289C4⋊C4—1731no—
290C4⋊C4C22×D17311yescharacteristic
291C4⋊C4C22×D17311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C42—1731no—
301C42—1731no—
302C42—1731no—
303C42C22×D17311yescharacteristic
304C23—21no—
305C23—21no—
306C23—3464no—
307C23—3461no—
308C23—3461no—
309C23—1731no—
310C23—6922no—
311C23—6922no—
312Q8—6922no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—3464no—
317D4—3461no—
318D4—3461no—
319D4—3461no—
320D4—1732no—
321D4—6922no—
322D4—6922no—
323D4—6922no—
324D4—6922no—
325D4#1458382612yescomplemented
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—1731no—
335C2×C4—1731no—
336C2×C4—1731no—
337C2×C4—1734no—
338C2×C4—6921no—
339C2×C4—6922no—
340C2×C4—6922no—
341C2×C4—6922no—
342C2×C4—6922no—
343C2×C4—6922no—
344C2×C4D4×D17311yescharacteristic
345C2×C4D4×D17311yescharacteristic
346C2×C4C23×D17311yescharacteristic
347C8—21no—
348C8—21no—
349C8—3461no—
350C8—3461no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—3461no—
358C22—3461no—
359C22—3462no—
360C22—1732no—
361C22—6928no—
362C22—6921no—
363C22—6921no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22C2×D4×D17311yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—3461no—
372C4—3461no—
373C4—1732no—
374C4—6921no—
375C4—6921no—
376C4D8×D17312yes—
377C4#41×D17311yescharacteristic
378C4C2×#1458382611yescharacteristic
379C2—42no—
380C2—81no—
381C2—3462no—
382C2—13841no—
383C2—6922no—
384C2#2601200011yescharacteristic
385C2C2×D8×D17311yescharacteristic
386C2#2601228811yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×D8×C173C2
socleC22×C173C22×D4
hypercenterC4×D8D173
hypocenterC173#3100
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #26012038C2semidirect
2. C4×D4×D173C2semidirect
3. #26011806C2semidirect
4. #26011803C2semidirect
5. #26011770C2semidirect
6. #26011769C2semidirect
7. #26011685C2semidirect
8. #26011684C2semidirect
9. #26011626C2semidirect
10. #26011623C2semidirect
11. C4×D1384C2semidirect
12. C4×#14583811C2semidirect
13. C4×#14583811C2semidirect
14. #26011542C2semidirect
15. C4×D8×C173C2semidirect
22. #19149678C22semidirect
23. #19149675C22semidirect
29. C4×D692C22semidirect
38. #19149613C22semidirect
39. #19149612C22semidirect
40. #19149612C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
44. #19149601C22semidirect
46. #19149600C22semidirect
48. #19149599C22semidirect
51. C4×#13063436C22semidirect
54. C4×D4×C173C22semidirect
55. #107×C173C22semidirect
57. #102×C173C22semidirect
58. C4×C8×C173C22semidirect
121. D692#41semidirect
139. C4×D173D8semidirect
147. D4×C173#41semidirect
193. C173#3100semidirect
208. C4×D8D173semidirect
231. C4×D4C2×D173semidirect
238. #107C2×D173semidirect
248. #102C2×D173semidirect
252. C4×C8C2×D173semidirect
325. D4#14583826semidirect

Maximal subgroups

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

subgroupindex
#260120382
C4×D4×D1732
#260118062
#260118032
#260117702
#260117692
#260116852
#260116842
#260116262
#260116232
C4×D13842
C4×#145838112
#260115422
C4×D8×C1732
#3100173