Finite Group Databasebeta

← #36365308

Subgroups — #36365308

Order 22144.

472 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
1C22×#14583816C211yesmaximal, characteristic, complemented
2C2×Q16×D173C211yesmaximal, characteristic, complemented
3Q8×C22⋊D173C211yesmaximal, characteristic
4#26012154C211yesmaximal, characteristic
5#26011869C211yesmaximal, characteristic, complemented
6#26011731C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011700C211yesmaximal, characteristic, complemented
9#26011699C211yesmaximal, characteristic, complemented
10#26011649C211yesmaximal, characteristic, complemented
11#26011648C211yesmaximal, characteristic, complemented
12#26011645C211yesmaximal, characteristic, complemented
13#26011583C211yesmaximal, characteristic, complemented
14#26011554C211yesmaximal, characteristic
15#451×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19C2×Dic692C2211yescharacteristic
20C2×Dic692C2211yescharacteristic
21C2×#14583816—24no—
22C2×#14583816C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583816C2211yescharacteristic
25C2×#14583816C2211yescharacteristic
26C2×#14583816C2211yescharacteristic
27Q16×D173—24no—
28C22×#13063436C2211yescharacteristic
29C2×C8×D173C2211yescharacteristic
30#19149678—21no—
31#19149677—21no—
32#19149677C2211yescharacteristic
33#19149677C2211yescharacteristic
34#19149659C2211yescharacteristic
35#19149658—21no—
36#19149658C2211yescharacteristic
37#19149657—21no—
38#19149656—21no—
39#19149655—21no—
40#19149655C2211yescharacteristic
41Q8×Dic173C2211yescharacteristic
42#19149644C2211yescharacteristic
43C4×C22⋊D173—21no—
44C4×C22⋊D173C2211yescharacteristic
45#19149639—21no—
46#19149612C2211yescharacteristic
47#19149608C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149604C2211yescharacteristic
50#19149603C2211yescharacteristic
51#19149602C2211yescharacteristic
52#19149591C2211yescharacteristic
53C8×Dic173C2211yescharacteristic
54C22×Q8×C173C2211yescharacteristic
55C2×Q16×C173C2211yescharacteristic
56C2×Q16×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60C22⋊C8×C173C2211yescharacteristic
61C2×Dic346—21no—
62C2×Dic346—21no—
63C2×Dic346—22no—
64C2×Dic346—21no—
65C2×Dic346—41no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×Dic346C2311yescharacteristic
69Q8×D173—22no—
70Q8×D173—22no—
71Q8×D173—41no—
72Q8×D173—41no—
73C2×C22⋊D173D411yescharacteristic
74C22×Dic173—21no—
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173C2311yescharacteristic
78Dic692—22no—
79Dic692—41no—
80#14583816—22no—
81#14583816—22no—
82#14583816—22no—
83#14583816—22no—
84#14583816D414yescomplemented
85C2×#13063436—22no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C8×D173—22no—
89C4⋊Dic173—21no—
90C4⋊Dic173—21no—
91C4⋊Dic173C2311yescharacteristic
92#14583805—21no—
93#14583805—21no—
94#14583805—21no—
95#14583805—21no—
96#14583805—21no—
97#14583805—21no—
98#14583805—21no—
99#14583805D411yescharacteristic
100C22⋊Dic173—21no—
101C22⋊Dic173D411yescharacteristic
102#14583803—21no—
103#14583803—21no—
104#14583803D411yescharacteristic
105C4×Dic173—21no—
106C4×Dic173—21no—
107C4×Dic173C2311yescharacteristic
108C2×Q8×C173—21no—
109C2×Q8×C173—22no—
110C2×Q8×C173—41no—
111C2×Q8×C173C2311yescharacteristic
112C2×Q8×C173C2311yescharacteristic
113C2×Q8×C173C2311yescharacteristic
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116Q16×C173—22no—
117Q16×C173—41no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173—21no—
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—42no—
130Dic346—42no—
131Dic346—42no—
132Dic346C2×D412yes—
133C22⋊D173Q1614yescomplemented
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—42no—
139C2×Dic173C2×D411yescharacteristic
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C4×D173—21no—
143C4×D173—21no—
144C4×D173—42no—
145C4×D173—42no—
146#13063436—21no—
147#13063436C2×D412yes—
148C23×C173C2×D411yescharacteristic
149Q8×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—21no—
154Q8×C173—42no—
155Q8×C173C2×D412yes—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—42no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2411yescharacteristic
165C8×C173—21no—
166C8×C173—21no—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173C2×Q1612yescomplemented
170C22×C173C22×D411yescharacteristic
171C2×D173—21no—
172C2×D173—21no—
173C2×D173C2×Q1612yescomplemented
174C4×C173—21no—
175C4×C173—22no—
176C4×C173—41no—
177C4×C173—41no—
178C4×C173—41no—
179C4×C173C22×D411yescharacteristic
180C4×C173H(5,𝔽2)11yescharacteristic
181Dic173—21no—
182Dic173—22no—
183Dic173—21no—
184Dic173—41no—
185Dic173—41no—
186Dic173—41no—
187Dic173C2×Q1612yescomplemented
188C2×C173—22no—
189C2×C173D4211yescharacteristic
190C2×C173C22×Q1611yescharacteristic
191C2×C173#57811yescharacteristic
192D173—22no—
193C173D4×Q1611yescharacteristic, complemented, Sylow
194D4×Q16—1731nomaximal, Sylow
195C22×Q16—1731no—
196C22×Q16—1731no—
197D4×Q8—1731no—
198D4×Q8—1731no—
199#494—1731no—
200#467—1731no—
201#467—1731no—
202#462—1731no—
203#462—1731no—
204#451—1731no—
205#451—1731no—
206#451—1731no—
207#451D17311yescharacteristic, complemented
208C4×Q16—1731no—
209C8×D4—1731no—
210C22×Q8—1731no—
211C22×Q8—1731no—
212C22×Q8—1731no—
213C22×Q8C2×D17311yescharacteristic
214C2×Q16—3464no—
215C2×Q16—3464no—
216C2×Q16—1731no—
217C2×Q16—1731no—
218C2×Q16—1731no—
219C2×Q16—1731no—
220C2×Q16—1731no—
221C2×Q16—1731no—
222C2×Q16—1731no—
223C2×Q16C2×D17311yescharacteristic
224C2×Q16C2×D17311yescharacteristic
225C22×C8—1731no—
226C22×C8—1731no—
227C4⋊Q8—3461no—
228C4⋊Q8—3461no—
229C4⋊Q8—1731no—
230C4⋊Q8—1731no—
231C22⋊Q8—3461no—
232C22⋊Q8—3461no—
233C22⋊Q8—3461no—
234C22⋊Q8—3461no—
235C22⋊Q8—1731no—
236C22⋊Q8—1731no—
237C22⋊Q8—1731no—
238C22⋊Q8C2×D17311yescharacteristic
239C4×Q8—1731no—
240C4×Q8—1731no—
241C4×D4—3461no—
242C4×D4—3461no—
243C4×D4—1731no—
244#107—1731no—
245C4⋊C8—1731no—
246#103—1731no—
247#103—1731no—
248#103—1731no—
249#103—1731no—
250#103C2×D17311yescharacteristic
251#103C2×D17311yescharacteristic
252C22⋊C8—1731no—
253C22⋊C8C2×D17311yescharacteristic
254C4×C8—1731no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—3461no—
259C2×Q8—3461no—
260C2×Q8—3462no—
261C2×Q8—3461no—
262C2×Q8—3462no—
263C2×Q8—3462no—
264C2×Q8—1731no—
265C2×Q8—1731no—
266C2×Q8—1731no—
267C2×Q8—6921no—
268C2×Q8—6921no—
269C2×Q8—6921no—
270C2×Q8C22×D17311yescharacteristic
271C2×Q8C22×D17311yescharacteristic
272C2×Q8C22×D17311yescharacteristic
273C2×D4—1731no—
274C22×C4—21no—
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4—3461no—
278C22×C4—1731no—
279C22×C4C22×D17311yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—3462no—
283Q16—3462no—
284Q16—3462no—
285Q16—3462no—
286Q16—3462no—
287Q16—1734no—
288Q16—6921no—
289C2×C8—3462no—
290C2×C8—3462no—
291C2×C8—1731no—
292C2×C8—1731no—
293C2×C8C22×D17311yescharacteristic
294C2×C8C22×D17311yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—3461no—
297C4⋊C4—3461no—
298C4⋊C4—3461no—
299C4⋊C4—3461no—
300C4⋊C4—3461no—
301C4⋊C4—3461no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—3461no—
305C4⋊C4—1731no—
306C4⋊C4—1731no—
307C4⋊C4C22×D17311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C22⋊C4—3461no—
312C22⋊C4—1731no—
313C22⋊C4—1731no—
314C42—3461no—
315C42—3461no—
316C42—1731no—
317C23—1731no—
318C23D4×D17311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—3461no—
326Q8—3461no—
327Q8—3461no—
328Q8—3461no—
329Q8—3461no—
330Q8—1732no—
331Q8—6922no—
332Q8—6922no—
333Q8—6922no—
334Q8D4×D17312yes—
335D4—1734no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—1731no—
352C2×C4—6922no—
353C2×C4—6922no—
354C2×C4—6922no—
355C2×C4D4×D17311yescharacteristic
356C2×C4C23×D17311yescharacteristic
357C8—21no—
358C8—21no—
359C8—3461no—
360C8—1732no—
361C22—21no—
362C22—21no—
363C22—3461no—
364C22—3461no—
365C22—1732no—
366C22Q16×D17312yescomplemented
367C22C2×D4×D17311yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—3461no—
374C4—3461no—
375C4—3462no—
376C4—1732no—
377C4—6921no—
378C4—6921no—
379C4—6921no—
380C4C2×D4×D17311yescharacteristic
381C4#1914972411yescharacteristic
382C2—22no—
383C2—3462no—
384C2#2601212311yescharacteristic
385C2C2×Q16×D17311yescharacteristic
386C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#451×C173C2
socleC22×C173C22×D4
prime radicalC2×C22⋊D173D4
hypercenter#451D173
hypocenterC173D4×Q16
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
1. C22×#14583816C2semidirect
2. C2×Q16×D173C2semidirect
5. #26011869C2semidirect
6. #26011731C2semidirect
7. #26011730C2semidirect
8. #26011700C2semidirect
9. #26011699C2semidirect
10. #26011649C2semidirect
11. #26011648C2semidirect
12. #26011645C2semidirect
13. #26011583C2semidirect
15. #451×C173C2semidirect
84. #14583816D4semidirect
133. C22⋊D173Q16semidirect
169. C22×C173C2×Q16semidirect
173. C2×D173C2×Q16semidirect
187. Dic173C2×Q16semidirect
193. C173D4×Q16semidirect
207. #451D173semidirect
366. C22Q16×D173semidirect

Maximal subgroups

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

subgroupindex
C22×#145838162
C2×Q16×D1732
Q8×C22⋊D1732
#260121542
#260118692
#260117312
#260117302
#260117002
#260116992
#260116492
#260116482
#260116452
#260115832
#260115542
#451×C1732
D4×Q16173