Finite Group Databasebeta

← #36365372

Subgroups — #36365372

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#26012142C211yesmaximal, characteristic, complemented
2D4×C22⋊D173C211yesmaximal, characteristic, complemented
3#26011836C211yesmaximal, characteristic, complemented
4#26011804C211yesmaximal, characteristic, complemented
5#26011803C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011727C211yesmaximal, characteristic, complemented
8#26011726C211yesmaximal, characteristic, complemented
9#26011685C211yesmaximal, characteristic, complemented
10#26011626C211yesmaximal, characteristic, complemented
11#26011625C211yesmaximal, characteristic, complemented
12#26011571C211yesmaximal, characteristic, complemented
13C8×C22⋊D173C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15C2×#102×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C22×C22⋊D173—21no—
18C2×D4×D173C2211yescharacteristic
19C2×D1384C2211yescharacteristic, complemented
20C2×#14583811C2211yescharacteristic
21C2×#14583811C2211yescharacteristic, complemented
22C2×C8×D173C2211yescharacteristic, complemented
23#19149678C2211yescharacteristic, complemented
24#19149675C2211yescharacteristic
25#19149664—21no—
26#19149662—21no—
27#19149657—21no—
28#19149654C2211yescharacteristic, complemented
29#19149653C2211yescharacteristic
30#19149653C2211yescharacteristic, complemented
31#19149651—21no—
32#19149650C2211yescharacteristic
33#19149648—21no—
34#19149647—21no—
35D4×Dic173C2211yescharacteristic
36#19149641C2211yescharacteristic, complemented
37C4×C22⋊D173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic, complemented
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149612C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic, complemented
54C2×C4⋊C4×C173C2211yescharacteristic, complemented
55#102×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic, complemented
58#102×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—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—44no—
72C2×C22⋊D173D411yescharacteristic
73D4×D173—22no—
74D4×D173—41no—
75C22×Dic173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79D1384—41no—
80#14583811—22no—
81#14583811—41no—
82C2×#13063436C2311yescharacteristic
83C2×#13063436C2311yescharacteristic
84C8×D173—41no—
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173D411yescharacteristic
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803D411yescharacteristic
98C4×Dic173C2311yescharacteristic
99C24×C173—21no—
100C2×D4×C173—22no—
101C2×D4×C173—21no—
102C2×D4×C173—41no—
103C2×D4×C173C2311yescharacteristic
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173—21no—
106C22×C4×C173C2311yescharacteristic
107C2×C8×C173—41no—
108C2×C8×C173C2311yescharacteristic
109C2×C8×C173C2311yescharacteristic
110C4⋊C4×C173—21no—
111C4⋊C4×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—42no—
116C22×D173—42no—
117C22×D173—81no—
118C22×D173C2×D411yescharacteristic
119Dic346—42no—
120D692—21no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—42no—
125C22⋊D173—42no—
126C22⋊D173—42no—
127C22⋊D173—42no—
128C22⋊D173—42no—
129C22⋊D173—42no—
130C22⋊D173—81no—
131C22⋊D173D814yescomplemented
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—42no—
135C2×Dic173C2×D411yescharacteristic
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C4×D173—21no—
139C4×D173—21no—
140C4×D173—42no—
141C4×D173—42no—
142#13063436—21no—
143#13063436—21no—
144C23×C173—21no—
145C23×C173—21no—
146C23×C173—42no—
147C23×C173—42no—
148C23×C173—81no—
149C23×C173C2×D411yescharacteristic
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153D4×C173—42no—
154D4×C173#4112yescomplemented
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—42no—
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2411yescharacteristic
162C8×C173—21no—
163C8×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—22no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—84no—
173C22×C173C2×D812yes—
174C22×C173C22×D411yescharacteristic
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—41no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
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—42no—
197C2×C173—81no—
198C2×C173#54611yescharacteristic
199C2×C173C22×D811yescharacteristic
200C2×C173#57711yescharacteristic
201D173—22no—
202D173—81no—
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#466—1731no—
214#459—1731no—
215#447—1731no—
216#447—1731no—
217C4×D8—1731no—
218C8×D4—1731no—
219C2×#102—1731no—
220C2×#102D17311yescharacteristic, complemented
221C2×#41—3461no—
222C22×D4—3461no—
223C22×D4—1731no—
224C22×D4C2×D17311yescharacteristic
225C2×D8—1731no—
226C2×D8—1731no—
227C2×D8—1731no—
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#121—1731no—
239#121—1731no—
240#120—3461no—
241#120—3461no—
242C4×D4—1731no—
243C4×D4—1731no—
244C4×D4—1731no—
245C2×C4⋊C4—1731no—
246C2×C4⋊C4C2×D17311yescharacteristic, complemented
247#107—1731no—
248#107—1731no—
249#107—1731no—
250C4⋊C8—1731no—
251#102—1731no—
252#102—1731no—
253#102—1731no—
254#102—1731no—
255#102—1731no—
256#102—1731no—
257#102C2×D17311yescharacteristic
258#102C2×D17311yescharacteristic
259#102C2×D17311yescharacteristic, complemented
260#102C2×D17311yescharacteristic, complemented
261C22⋊C8—1731no—
262C22⋊C8—1731no—
263C4×C8—1731no—
264C24—21no—
265C24—3461no—
266#41—6924no—
267C2×Q8—3461no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—3461no—
272C2×D4—3462no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—3461no—
279C2×D4—1731no—
280C2×D4—1731no—
281C2×D4—1731no—
282C2×D4—6921no—
283C2×D4—6924no—
284C2×D4C22×D17311yescharacteristic
285C2×D4C22×D17311yescharacteristic
286C22×C4—21no—
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—1731no—
291C22×C4C22×D17311yescharacteristic
292D8—3462no—
293D8—6921no—
294D8—6921no—
295C2×C8—41no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8—6921no—
299C2×C8C22×D17311yescharacteristic
300C2×C8C22×D17311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—1731no—
305C4⋊C4—1731no—
306C4⋊C4—1731no—
307C4⋊C4C22×D17311yescharacteristic
308C4⋊C4C22×D17311yescharacteristic
309C22⋊C4—3461no—
310C22⋊C4—3461no—
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—42no—
321C23—42no—
322C23—81no—
323C23—3461no—
324C23—3461no—
325C23—13841no—
326C23—1731no—
327C23—6922no—
328C23—6922no—
329C23D4×D17311yescharacteristic
330Q8—6922no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—3461no—
336D4—3461no—
337D4—3461no—
338D4—3461no—
339D4—13841no—
340D4—1734no—
341D4—6922no—
342D4—6922no—
343D4—6922no—
344D4—6922no—
345D4—6922no—
346D4—6922no—
347D4#1458382612yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—1731no—
358C2×C4—1731no—
359C2×C4—1731no—
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—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—84no—
378C22—3461no—
379C22—3461no—
380C22—13844no—
381C22—1732no—
382C22—6921no—
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—
398C4C2×#1458382611yescharacteristic
399C4#1914972411yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—3462no—
404C2—13841no—
405C2—13841no—
406C2#2601213111yescharacteristic
407C2C2×D8×D17311yescharacteristic
408C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#102×C173C2
socleC22×C173C22×D4
hypercenterC2×#102D173
hypocenterC173#3294
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #26012142C2semidirect
2. D4×C22⋊D173C2semidirect
3. #26011836C2semidirect
4. #26011804C2semidirect
5. #26011803C2semidirect
6. #26011769C2semidirect
7. #26011727C2semidirect
8. #26011726C2semidirect
9. #26011685C2semidirect
10. #26011626C2semidirect
11. #26011625C2semidirect
12. #26011571C2semidirect
13. C8×C22⋊D173C2semidirect
14. #102×D173C2semidirect
15. C2×#102×C173C2semidirect
19. C2×D1384C22semidirect
21. C2×#14583811C22semidirect
22. C2×C8×D173C22semidirect
23. #19149678C22semidirect
28. #19149654C22semidirect
30. #19149653C22semidirect
36. #19149641C22semidirect
38. C4⋊C4×D173C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149612C22semidirect
42. #19149608C22semidirect
43. #19149601C22semidirect
44. #19149601C22semidirect
46. #19149600C22semidirect
47. #19149599C22semidirect
48. #19149599C22semidirect
49. C22⋊#13063436C22semidirect
50. #19149591C22semidirect
51. C8×Dic173C22semidirect
53. C22×C8×C173C22semidirect
54. C2×C4⋊C4×C173C22semidirect
57. #102×C173C22semidirect
58. #102×C173C22semidirect
131. C22⋊D173D8semidirect
154. D4×C173#41semidirect
204. C173#3294semidirect
220. C2×#102D173semidirect
229. C22×C8C2×D173semidirect
246. C2×C4⋊C4C2×D173semidirect
259. #102C2×D173semidirect
260. #102C2×D173semidirect
347. D4#14583826semidirect

Maximal subgroups

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

subgroupindex
#260121422
D4×C22⋊D1732
#260118362
#260118042
#260118032
#260117692
#260117272
#260117262
#260116852
#260116262
#260116252
#260115712
C8×C22⋊D1732
#102×D1732
C2×#102×C1732
#3294173