Finite Group Databasebeta

← #36365608

Subgroups — #36365608

Order 22144.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic346C211yesmaximal, characteristic
2#26011861C211yesmaximal, characteristic, complemented
3#26011832C211yesmaximal, characteristic, complemented
4#26011821C211yesmaximal, characteristic, complemented
5#26011795C211yesmaximal, characteristic, complemented
6#26011747C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011701C211yesmaximal, characteristic, complemented
9#26011646C211yesmaximal, characteristic, complemented
10#26011646C211yesmaximal, characteristic, complemented
11#26011596C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic
13#26011499C211yesmaximal, characteristic, complemented
14C2×#19149602C211yesmaximal, characteristic, complemented
15#552×C173C211yesmaximal, characteristic
16C22×Dic346C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583816C2211yescharacteristic
21C2×#14583810C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149679C2211yescharacteristic
24#19149678—21no—
25#19149658C2211yescharacteristic
26#19149658C2211yescharacteristic
27#19149656—21no—
28#19149656—21no—
29C4×Dic346C2211yescharacteristic
30D4×Dic173—21no—
31#19149612C2211yescharacteristic
32#19149610C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149603C2211yescharacteristic
36#19149602C2211yescharacteristic
37#19149602C2211yescharacteristic
38#19149602C2211yescharacteristic
39#19149602C2211yescharacteristic
40#19149602C2211yescharacteristic
41#19149602C2211yescharacteristic
42#19149602C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149602C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47#19149590C2211yescharacteristic
48#126×C173—21no—
49#125×C173—21no—
50#125×C173C2211yescharacteristic
51#123×C173—21no—
52C22⋊Q8×C173C2211yescharacteristic
53C22⋊Q8×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic
56#117×C173—21no—
57#117×C173C2211yescharacteristic
58C2×C4⋊C4×C173C2211yescharacteristic
59C2×Dic346—22no—
60C2×Dic346—22no—
61C2×Dic346—21no—
62C2×Dic346—21no—
63C2×Dic346—41no—
64C2×Dic346—41no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C22×Dic173—21no—
69C22×Dic173—21no—
70#14583816—22no—
71#14583816—41no—
72#14583816—41no—
73#14583810—41no—
74C2×#13063436—41no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C4×Dic173—21no—
88C2×Q8×C173C2311yescharacteristic
89C2×D4×C173D411yescharacteristic
90C22×C4×C173—21no—
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C4⋊C4×C173—21no—
94C4⋊C4×C173—21no—
95C4⋊C4×C173—21no—
96C4⋊C4×C173—21no—
97C4⋊C4×C173—21no—
98C4⋊C4×C173—21no—
99C4⋊C4×C173D411yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173D411yescharacteristic
107C22⋊C4×C173D411yescharacteristic
108C42×C173—21no—
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111Dic346—21no—
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117Dic346—42no—
118Dic346#4112yes—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—42no—
124C2×Dic173—42no—
125#13063436—21no—
126#13063436—21no—
127#13063436—21no—
128#13063436—21no—
129C23×C173C2×D411yescharacteristic
130C23×C173C2×D411yescharacteristic
131Q8×C173—21no—
132Q8×C173—21no—
133D4×C173—22no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—42no—
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C22×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173—21no—
153C22×C173#13712yes—
154C22×C173C22×D411yescharacteristic
155C4×C173—21no—
156C4×C173—21no—
157C4×C173—21no—
158C4×C173—21no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173C2×#4111yescharacteristic
164C4×C173H(5,𝔽2)11yescharacteristic
165Dic173—22no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169C2×C173—22no—
170C2×C173—41no—
171C2×C173#54611yescharacteristic
172C2×C173C2×#13711yescharacteristic
173C2×C173#57811yescharacteristic
174C173#331211yescharacteristic, complemented, Sylow
175#3312—1731nomaximal, Sylow
176#552D17311yescharacteristic
177D4×Q8—1731no—
178#491—1731no—
179#485—1731no—
180#483—1731no—
181#479—1731no—
182#470—1731no—
183#467—1731no—
184#462—1731no—
185#451—1731no—
186#451—1731no—
187#441—1731no—
188#435—1731no—
189#419—1731no—
190C2×#103—1731no—
191C22×Q8—1731no—
192C22×Q8—1731no—
193C2×Q16—1731no—
194C2×Q16—1731no—
195C2×Q16—1731no—
196C2×M16—1731no—
197C22×C8—1731no—
198C4⋊Q8—3461no—
199C4⋊Q8—1731no—
200#126—21no—
201#125—21no—
202#125C2×D17311yescharacteristic
203#123—21no—
204C22⋊Q8—3461no—
205C22⋊Q8—3461no—
206C22⋊Q8—1731no—
207C22⋊Q8—1731no—
208C22⋊Q8C2×D17311yescharacteristic
209C22⋊Q8C2×D17311yescharacteristic
210C4×Q8—1731no—
211C4×Q8C2×D17311yescharacteristic
212C4×D4—3461no—
213C4×D4C2×D17311yescharacteristic
214#117—21no—
215#117C2×D17311yescharacteristic
216C2×C4⋊C4C2×D17311yescharacteristic
217#107—1731no—
218#106—1731no—
219#106—1731no—
220C4⋊C8—1731no—
221#103—1731no—
222#103—1731no—
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103—1731no—
231C22⋊C8—1731no—
232C22⋊C8—1731no—
233#97—1731no—
234C2×Q8—3462no—
235C2×Q8—3462no—
236C2×Q8—3461no—
237C2×Q8—3461no—
238C2×Q8—1731no—
239C2×Q8—1731no—
240C2×Q8—1731no—
241C2×Q8—6921no—
242C2×Q8—6921no—
243C2×Q8C22×D17311yescharacteristic
244C2×D4C22⋊D17311yescharacteristic
245C22×C4—21no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4C22×D17311yescharacteristic
249C22×C4C22×D17311yescharacteristic
250Q16—3462no—
251Q16—6921no—
252Q16—6921no—
253M16—6921no—
254C2×C8—1731no—
255C2×C8—1731no—
256C2×C8—1731no—
257C2×C8—1731no—
258C2×C8—6921no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—1731no—
271C4⋊C4C22⋊D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
273C4⋊C4C22×D17311yescharacteristic
274C4⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4C22⋊D17311yescharacteristic
281C22⋊C4C22⋊D17311yescharacteristic
282C42—21no—
283C42—21no—
284C42—3461no—
285C42C22×D17311yescharacteristic
286C23C2×C22⋊D17311yescharacteristic
287C23C2×C22⋊D17311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—3461no—
291Q8—3461no—
292Q8—3461no—
293Q8—3461no—
294Q8—3461no—
295Q8—1732no—
296Q8—6922no—
297Q8—6922no—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—6922no—
313C2×C4—6922no—
314C2×C4C2×C22⋊D17311yescharacteristic
315C2×C4C2×C22⋊D17311yescharacteristic
316C2×C4C2×C22⋊D17311yescharacteristic
317C2×C4C2×C22⋊D17311yescharacteristic
318C2×C4C23×D17311yescharacteristic
319C8—3461no—
320C8—3461no—
321C8—3461no—
322C8—3461no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1914970912yes—
329C22C22×C22⋊D17311yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—3462no—
339C4—6921no—
340C4—6921no—
341C4—6921no—
342C4#41×D17311yescharacteristic
343C4#1914972511yescharacteristic
344C2—22no—
345C2—41no—
346C2#2601212911yescharacteristic
347C2C2×#1914970911yescharacteristic
348C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#552×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×C173D4
hypercenter#552D173
hypocenterC173#3312
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #26011861C2semidirect
3. #26011832C2semidirect
4. #26011821C2semidirect
5. #26011795C2semidirect
6. #26011747C2semidirect
7. #26011730C2semidirect
8. #26011701C2semidirect
9. #26011646C2semidirect
10. #26011646C2semidirect
11. #26011596C2semidirect
13. #26011499C2semidirect
14. C2×#19149602C2semidirect
174. C173#3312semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3462
#260118612
#260118322
#260118212
#260117952
#260117472
#260117302
#260117012
#260116462
#260115962
#260115632
#260114992
C2×#191496022
#552×C1732
#3312173