Finite Group Databasebeta

← #36365607

Subgroups — #36365607

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#26012175C211yesmaximal, characteristic
2Q8×C22⋊D173C211yesmaximal, characteristic
3#26011858C211yesmaximal, characteristic, complemented
4#26011833C211yesmaximal, characteristic, complemented
5#26011820C211yesmaximal, characteristic, complemented
6#26011800C211yesmaximal, characteristic, complemented
7#26011747C211yesmaximal, characteristic, complemented
8#26011729C211yesmaximal, characteristic, complemented
9#26011699C211yesmaximal, characteristic, complemented
10#26011645C211yesmaximal, characteristic, complemented
11#26011598C211yesmaximal, characteristic, complemented
12#26011565C211yesmaximal, characteristic
13#26011500C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#451×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583810C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149677—21no—
21#19149677C2211yescharacteristic
22#19149672—21no—
23#19149670C2211yescharacteristic
24#19149669—21no—
25#19149661—21no—
26#19149658—21no—
27#19149658C2211yescharacteristic
28#19149655—21no—
29#19149655C2211yescharacteristic
30#19149655C2211yescharacteristic
31Q8×Dic173C2211yescharacteristic
32Q8×Dic173C2211yescharacteristic
33C4×C22⋊D173—21no—
34C4×C22⋊D173C2211yescharacteristic
35#19149638C2211yescharacteristic
36#19149637—21no—
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149603C2211yescharacteristic
48#19149603C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61Q8×D173—22no—
62Q8×D173—41no—
63C2×C22⋊D173D411yescharacteristic
64C2×C4×D173—21no—
65C2×C4×D173—21no—
66C2×C4×D173C2311yescharacteristic
67#14583816—41no—
68#14583810—41no—
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C8×D173—41no—
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173—21no—
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173D411yescharacteristic
89#14583803—21no—
90#14583803—21no—
91#14583803D411yescharacteristic
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173—21no—
97C2×Q8×C173—22no—
98C2×Q8×C173—41no—
99C2×Q8×C173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104Q16×C173—22no—
105Q16×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—42no—
115C22⋊D173—22no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—42no—
126C4×D173—42no—
127#13063436—21no—
128#13063436—21no—
129C23×C173C2×D411yescharacteristic
130Q8×C173—21no—
131Q8×C173—21no—
132Q8×C173—21no—
133Q8×C173—21no—
134Q8×C173—21no—
135Q8×C173—42no—
136Q8×C173#4112yes—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—42no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173—21no—
148C22×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173C22×D411yescharacteristic
152C2×D173—21no—
153C2×D173—21no—
154C2×D173#13712yes—
155C4×C173—21no—
156C4×C173—22no—
157C4×C173—41no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173C2×#4111yescharacteristic
161C4×C173H(5,𝔽2)11yescharacteristic
162Dic173—21no—
163Dic173—21no—
164Dic173—21no—
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169C2×C173—41no—
170C2×C173#54611yescharacteristic
171C2×C173C2×#13711yescharacteristic
172C2×C173#57811yescharacteristic
173D173—22no—
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#451D17311yescharacteristic, complemented
187#441—1731no—
188#435—1731no—
189#419—1731no—
190C2×#103—1731no—
191C22×Q8—1731no—
192C22×Q8C2×D17311yescharacteristic
193C2×Q16—1731no—
194C2×Q16C2×D17311yescharacteristic
195C2×Q16C2×D17311yescharacteristic
196C2×M16—1731no—
197C22×C8—1731no—
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⋊Q8C2×D17311yescharacteristic
210C4×Q8—1731no—
211C4×Q8—1731no—
212C4×D4—3461no—
213C4×D4—1731no—
214#117—3461no—
215#117—1731no—
216C2×C4⋊C4—1731no—
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#103C2×D17311yescharacteristic
230#103C2×D17311yescharacteristic
231C22⋊C8—1731no—
232C22⋊C8C2×D17311yescharacteristic
233#97—1731no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—3461no—
238C2×Q8—3462no—
239C2×Q8—1731no—
240C2×Q8—6921no—
241C2×Q8C22×D17311yescharacteristic
242C2×Q8C22×D17311yescharacteristic
243C2×Q8C22×D17311yescharacteristic
244C2×D4—1731no—
245C22×C4—21no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4—1731no—
249C22×C4C22×D17311yescharacteristic
250Q16—22no—
251Q16—41no—
252Q16—6921no—
253M16—6921no—
254C2×C8—1731no—
255C2×C8—1731no—
256C2×C8—6921no—
257C2×C8C22×D17311yescharacteristic
258C2×C8C22×D17311yescharacteristic
259C4⋊C4—21no—
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⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—1731no—
281C22⋊C4—1731no—
282C42—3461no—
283C42—3461no—
284C42—3461no—
285C42—1731no—
286C23—1731no—
287C23D4×D17311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—3461no—
295Q8—3461no—
296Q8—6922no—
297Q8#1458382512yes—
298D4—3462no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—1731no—
313C2×C4—1731no—
314C2×C4—1731no—
315C2×C4—6922no—
316C2×C4—6922no—
317C2×C4D4×D17311yescharacteristic
318C2×C4C23×D17311yescharacteristic
319C8—21no—
320C8—21no—
321C8—3461no—
322C8—3461no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—3461no—
327C22—3461no—
328C22—1732no—
329C22C2×D4×D17311yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—3461no—
336C4—3461no—
337C4—3461no—
338C4—6921no—
339C4—6921no—
340C4—6921no—
341C4—6921no—
342C4C2×#1458382511yescharacteristic
343C4#1914972411yescharacteristic
344C2—41no—
345C2—3462no—
346C2#2601212811yescharacteristic
347C2#137×D17311yescharacteristic
348C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#451×C173C2
socleC22×C173C22×D4
prime radicalC2×C22⋊D173D4
hypercenter#451D173
hypocenterC173#3312
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #26011858C2semidirect
4. #26011833C2semidirect
5. #26011820C2semidirect
6. #26011800C2semidirect
7. #26011747C2semidirect
8. #26011729C2semidirect
9. #26011699C2semidirect
10. #26011645C2semidirect
11. #26011598C2semidirect
13. #26011500C2semidirect
14. #103×D173C2semidirect
15. #451×C173C2semidirect
174. C173#3312semidirect
186. #451D173semidirect

Maximal subgroups

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

subgroupindex
#260121752
Q8×C22⋊D1732
#260118582
#260118332
#260118202
#260118002
#260117472
#260117292
#260116992
#260116452
#260115982
#260115652
#260115002
#103×D1732
#451×C1732
#3312173