Finite Group Databasebeta

← #36365616

Subgroups — #36365616

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
1#26012181C211yesmaximal, characteristic
2D4×Dic346C211yesmaximal, characteristic
3#26011860C211yesmaximal, characteristic, complemented
4#26011833C211yesmaximal, characteristic, complemented
5#26011822C211yesmaximal, characteristic, complemented
6#26011799C211yesmaximal, characteristic, complemented
7#26011749C211yesmaximal, characteristic, complemented
8#26011732C211yesmaximal, characteristic, complemented
9#26011702C211yesmaximal, characteristic, complemented
10#26011649C211yesmaximal, characteristic, complemented
11#26011649C211yesmaximal, characteristic, complemented
12#26011599C211yesmaximal, characteristic, complemented
13#26011501C211yesmaximal, characteristic, complemented
14C2×#19149604C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic
16C22×Dic346C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×Dic692C2211yescharacteristic
20C2×Dic692C2211yescharacteristic
21#19149679C2211yescharacteristic
22#19149678—21no—
23#19149672—21no—
24#19149671C2211yescharacteristic
25#19149669—21no—
26#19149660—21no—
27#19149658C2211yescharacteristic
28#19149658C2211yescharacteristic
29#19149658C2211yescharacteristic
30#19149658C2211yescharacteristic
31#19149656—21no—
32#19149656—21no—
33C4×Dic346C2211yescharacteristic
34C4×Dic346C2211yescharacteristic
35D4×Dic173—21no—
36#19149638C2211yescharacteristic
37#19149635—21no—
38C2×C4⋊Dic173C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149604C2211yescharacteristic
50#19149604C2211yescharacteristic
51#19149604C2211yescharacteristic
52C2×M16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic
58#97×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
68C2×Dic346C2311yescharacteristic
69C22×Dic173—21no—
70C22×Dic173—21no—
71C22×Dic173—21no—
72Dic692—22no—
73Dic692—41no—
74Dic692—41no—
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C2×D4×C173D411yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101M16×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
111Dic346—21no—
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—42no—
119Dic346—42no—
120Dic346#4112yes—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—42no—
130C2×Dic173—42no—
131C2×Dic173—42no—
132C23×C173C2×D411yescharacteristic
133C23×C173C2×D411yescharacteristic
134D4×C173—22no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2411yescharacteristic
144C8×C173—21no—
145C8×C173—21no—
146C8×C173—21no—
147C8×C173—21no—
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×C173C2×#4111yescharacteristic
160C4×C173H(5,𝔽2)11yescharacteristic
161Dic173—22no—
162Dic173—41no—
163Dic173—41no—
164Dic173—41no—
165Dic173—41no—
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#552—1731no—
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#435D17311yescharacteristic
189#419—1731no—
190C2×#103—1731no—
191C22×Q8—1731no—
192C22×Q8—1731no—
193C2×Q16—1731no—
194C2×Q16—1731no—
195C2×Q16—1731no—
196C2×M16C2×D17311yescharacteristic
197C22×C8C2×D17311yescharacteristic
198C4⋊Q8—3461no—
199C4⋊Q8—1731no—
200#126—3461no—
201#125—3461no—
202#125—1731no—
203#123—3461no—
204C22⋊Q8—3461no—
205C22⋊Q8—3461no—
206C22⋊Q8—1731no—
207C22⋊Q8—1731no—
208C22⋊Q8—1731no—
209C22⋊Q8—1731no—
210C4×Q8—1731no—
211C4×Q8—1731no—
212C4×D4—3461no—
213C4×D4C2×D17311yescharacteristic
214#117—3461no—
215#117—1731no—
216C2×C4⋊C4—1731no—
217#107—1731no—
218#106—1731no—
219#106—1731no—
220C4⋊C8C2×D17311yescharacteristic
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⋊C8C2×D17311yescharacteristic
232C22⋊C8C2×D17311yescharacteristic
233#97C2×D17311yescharacteristic
234C2×Q8—3462no—
235C2×Q8—3462no—
236C2×Q8—3461no—
237C2×Q8—3461no—
238C2×Q8—1731no—
239C2×Q8—1731no—
240C2×Q8—1731no—
241C2×Q8—1731no—
242C2×Q8—6921no—
243C2×Q8—6921no—
244C2×D4D69211yescharacteristic
245C22×C4—3461no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4C22×D17311yescharacteristic
249C22×C4C22×D17311yescharacteristic
250Q16—3462no—
251Q16—6921no—
252Q16—6921no—
253M16—41no—
254C2×C8—41no—
255C2×C8C22×D17311yescharacteristic
256C2×C8C22×D17311yescharacteristic
257C2×C8C22×D17311yescharacteristic
258C2×C8C22×D17311yescharacteristic
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—3461no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—1731no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4D69211yescharacteristic
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4D69211yescharacteristic
281C22⋊C4D69211yescharacteristic
282C42—3461no—
283C42—3461no—
284C42—3461no—
285C42C22×D17311yescharacteristic
286C23C2×D69211yescharacteristic
287C23C2×D69211yescharacteristic
288Q8—3461no—
289Q8—3461no—
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—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—6922no—
312C2×C4—6922no—
313C2×C4—6922no—
314C2×C4C2×D69211yescharacteristic
315C2×C4C2×D69211yescharacteristic
316C2×C4C2×D69211yescharacteristic
317C2×C4C2×D69211yescharacteristic
318C2×C4C23×D17311yescharacteristic
319C8—21no—
320C8—21no—
321C8—21no—
322C8—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1914971212yes—
329C22C22×D69211yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—3462no—
335C4—6921no—
336C4—6921no—
337C4—6921no—
338C4—6921no—
339C4—6921no—
340C4—6921no—
341C4—6921no—
342C4#41×D17311yescharacteristic
343C4#1914972411yescharacteristic
344C2—22no—
345C2—41no—
346C2#2601213611yescharacteristic
347C2C2×#1914971211yescharacteristic
348C2#2601230811yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#435×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×C173D4
hypercenter#435D173
hypocenterC173#3312
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
3. #26011860C2semidirect
4. #26011833C2semidirect
5. #26011822C2semidirect
6. #26011799C2semidirect
7. #26011749C2semidirect
8. #26011732C2semidirect
9. #26011702C2semidirect
10. #26011649C2semidirect
11. #26011649C2semidirect
12. #26011599C2semidirect
13. #26011501C2semidirect
14. C2×#19149604C2semidirect
174. C173#3312semidirect

Maximal subgroups

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

subgroupindex
#260121812
D4×Dic3462
#260118602
#260118332
#260118222
#260117992
#260117492
#260117322
#260117022
#260116492
#260115992
#260115012
C2×#191496042
#435×C1732
#3312173