Finite Group Databasebeta

← #36365138

Subgroups — #36365138

Order 22144.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149712C211yesmaximal, characteristic, complemented
2C22×Dic692C211yesmaximal, characteristic, complemented
3D4×Dic346C211yesmaximal, characteristic, complemented
4#26012150C211yesmaximal, characteristic
5#26011885C211yesmaximal, characteristic, complemented
6#26011749C211yesmaximal, characteristic, complemented
7#26011720C211yesmaximal, characteristic, complemented
8#26011714C211yesmaximal, characteristic, complemented
9#26011702C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011649C211yesmaximal, characteristic, complemented
12#26011649C211yesmaximal, characteristic, complemented
13#26011632C211yesmaximal, characteristic, complemented
14#26011599C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C22×Dic346C2211yescharacteristic
19C22×Dic346C2211yescharacteristic
20#19149712—24no—
21C2×Dic692—24no—
22C2×Dic692C2211yescharacteristic
23C2×Dic692C2211yescharacteristic
24C2×Dic692C2211yescharacteristic
25C2×Dic692C2211yescharacteristic
26C2×Dic692C2211yescharacteristic
27C2×Dic692C2211yescharacteristic
28C2×#14583815C2211yescharacteristic
29C2×#14583815C2211yescharacteristic
30C2×#14583815C2211yescharacteristic
31#19149679C2211yescharacteristic
32#19149678—21no—
33#19149667C2211yescharacteristic
34#19149665—21no—
35#19149658C2211yescharacteristic
36#19149658C2211yescharacteristic
37#19149658C2211yescharacteristic
38#19149657—21no—
39#19149656—21no—
40#19149656—21no—
41#19149653C2211yescharacteristic
42#19149651—21no—
43C4×Dic346C2211yescharacteristic
44C4×Dic346C2211yescharacteristic
45D4×Dic173—21no—
46#19149639—21no—
47#19149611C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149604C2211yescharacteristic
50#19149604C2211yescharacteristic
51#19149604C2211yescharacteristic
52#19149604C2211yescharacteristic
53#19149601C2211yescharacteristic
54C2×M16×C173C2211yescharacteristic
55C22×C8×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic
57C4⋊C8×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C22⋊C8×C173C2211yescharacteristic
60#97×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—41no—
63C2×Dic346—22no—
64C2×Dic346—22no—
65C2×Dic346—22no—
66C2×Dic346—21no—
67C2×Dic346—21no—
68C2×Dic346—21no—
69C2×Dic346—41no—
70C2×Dic346—41no—
71C2×Dic346—41no—
72C2×Dic346C2311yescharacteristic
73C2×Dic346C2311yescharacteristic
74C2×Dic346C2311yescharacteristic
75C2×Dic346C2311yescharacteristic
76C2×Dic346C2311yescharacteristic
77C2×D692C2311yescharacteristic
78C2×C22⋊D173—21no—
79C2×C22⋊D173—21no—
80C22×Dic173—21no—
81C22×Dic173—21no—
82C22×Dic173—21no—
83C2×C4×D173—21no—
84Dic692—22no—
85Dic692—22no—
86Dic692—22no—
87Dic692—22no—
88Dic692—41no—
89Dic692D414yescomplemented
90#14583815—22no—
91#14583815—22no—
92#14583815—41no—
93C4⋊Dic173—21no—
94C4⋊Dic173—21no—
95C4⋊Dic173C2311yescharacteristic
96C4⋊Dic173C2311yescharacteristic
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100#14583805—21no—
101#14583805—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173—21no—
104C22⋊Dic173—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108C4×Dic173—21no—
109C4×Dic173—21no—
110C2×D4×C173D411yescharacteristic
111C22×C4×C173C2311yescharacteristic
112C22×C4×C173C2311yescharacteristic
113M16×C173—22no—
114C2×C8×C173—22no—
115C2×C8×C173C2311yescharacteristic
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C2×C8×C173C2311yescharacteristic
119C4⋊C4×C173D411yescharacteristic
120C22⋊C4×C173D411yescharacteristic
121C22⋊C4×C173D411yescharacteristic
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—21no—
130Dic346—21no—
131Dic346—21no—
132Dic346—42no—
133Dic346—42no—
134Dic346—42no—
135Dic346C2×D412yes—
136Dic346C2×D412yes—
137D692—21no—
138D692—21no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C2×Dic173—21no—
146C2×Dic173—21no—
147C2×Dic173—21no—
148C2×Dic173—42no—
149C2×Dic173—42no—
150C2×Dic173—42no—
151C4×D173—42no—
152C23×C173C2×D411yescharacteristic
153C23×C173C2×D411yescharacteristic
154D4×C173—22no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2×D411yescharacteristic
162C2×C4×C173C2×D411yescharacteristic
163C2×C4×C173C2411yescharacteristic
164C8×C173—21no—
165C8×C173—21no—
166C8×C173—21no—
167C8×C173C2×D412yes—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173#13712yescomplemented
174C22×C173C22×D411yescharacteristic
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173—21no—
181C4×C173—21no—
182C4×C173C22×D411yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—22no—
185Dic173—22no—
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189Dic173—41no—
190Dic173—41no—
191C2×C173—22no—
192C2×C173—41no—
193C2×C173D4211yescharacteristic
194C2×C173C2×#13711yescharacteristic
195C2×C173#57811yescharacteristic
196D173—81no—
197C173#327011yescharacteristic, complemented, Sylow
198#3270—1731nomaximal, Sylow
199C2×#137—1731no—
200C22×Q16—1731no—
201D4×Q8—1731no—
202#548—1731no—
203#497—1731no—
204#470—1731no—
205#465—1731no—
206#464—1731no—
207#462—1731no—
208#452—1731no—
209#451—1731no—
210#451—1731no—
211#448—1731no—
212#441—1731no—
213#435D17311yescharacteristic, complemented
214C2×#41—1731no—
215C22×Q8—1731no—
216C22×Q8—1731no—
217C22×Q8—1731no—
218#137—3464no—
219C2×Q16—3464no—
220C2×Q16—1731no—
221C2×Q16—1731no—
222C2×Q16—1731no—
223C2×Q16—1731no—
224C2×Q16—1731no—
225C2×Q16—1731no—
226C2×SD16—1731no—
227C2×SD16—1731no—
228C2×SD16—1731no—
229C2×M16C2×D17311yescharacteristic
230C22×C8C2×D17311yescharacteristic
231C4⋊Q8—3461no—
232C4⋊Q8—1731no—
233#124—3461no—
234#124—1731no—
235C22⋊Q8—3461no—
236C22⋊Q8—3461no—
237C22⋊Q8—3461no—
238C22⋊Q8—1731no—
239C22⋊Q8—1731no—
240C22⋊Q8—1731no—
241#121—3461no—
242#121—1731no—
243C4×Q8—1731no—
244C4×Q8—1731no—
245C4×D4—3461no—
246C4×D4—3461no—
247C4×D4C2×D17311yescharacteristic
248#106—1731no—
249C4⋊C8C2×D17311yescharacteristic
250#103—1731no—
251#103—1731no—
252#103—1731no—
253#103—1731no—
254#103—1731no—
255#102—1731no—
256C22⋊C8C2×D17311yescharacteristic
257C22⋊C8C2×D17311yescharacteristic
258#97C2×D17311yescharacteristic
259#41—3462no—
260#41—6921no—
261C2×Q8—3462no—
262C2×Q8—3462no—
263C2×Q8—3462no—
264C2×Q8—3461no—
265C2×Q8—3461no—
266C2×Q8—3461no—
267C2×Q8—1731no—
268C2×Q8—1731no—
269C2×Q8—1731no—
270C2×Q8—1731no—
271C2×Q8—1731no—
272C2×Q8—6921no—
273C2×Q8—6921no—
274C2×Q8—6921no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—1731no—
278C2×D4D69211yescharacteristic
279C22×C4—3461no—
280C22×C4—3461no—
281C22×C4—3461no—
282C22×C4—3461no—
283C22×C4C22×D17311yescharacteristic
284C22×C4C22×D17311yescharacteristic
285Q16—3462no—
286Q16—3462no—
287Q16—3462no—
288Q16—3462no—
289Q16—1734no—
290Q16—6921no—
291SD16—3462no—
292SD16—3462no—
293SD16—6921no—
294M16—22no—
295C2×C8—22no—
296C2×C8C22×D17311yescharacteristic
297C2×C8C22×D17311yescharacteristic
298C2×C8C22×D17311yescharacteristic
299C2×C8C22×D17311yescharacteristic
300C4⋊C4—3461no—
301C4⋊C4—3461no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—3461no—
305C4⋊C4—3461no—
306C4⋊C4—3461no—
307C4⋊C4—1731no—
308C4⋊C4—1731no—
309C4⋊C4D69211yescharacteristic
310C22⋊C4—3461no—
311C22⋊C4—3461no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4D69211yescharacteristic
317C22⋊C4D69211yescharacteristic
318C42—3461no—
319C42—3461no—
320C42C22×D17311yescharacteristic
321C23—3461no—
322C23C2×D69211yescharacteristic
323C23C2×D69211yescharacteristic
324Q8—3461no—
325Q8—3461no—
326Q8—3461no—
327Q8—3461no—
328Q8—3461no—
329Q8—3461no—
330Q8—3461no—
331Q8—3461no—
332Q8—1732no—
333Q8—1732no—
334Q8—6922no—
335Q8—6922no—
336Q8—6922no—
337D4—22no—
338D4—3461no—
339D4—3461no—
340D4—6922no—
341D4—6922no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—6922no—
354C2×C4—6922no—
355C2×C4—6922no—
356C2×C4—6922no—
357C2×C4C2×D69211yescharacteristic
358C2×C4C2×D69211yescharacteristic
359C2×C4C2×D69211yescharacteristic
360C2×C4C2×D69211yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8D4×D17312yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22#1914971212yescomplemented
375C22C22×D69211yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—3462no—
381C4—3462no—
382C4—6921no—
383C4—6921no—
384C4—6921no—
385C4—6921no—
386C4—6921no—
387C4C2×D4×D17311yescharacteristic
388C4#1914972511yescharacteristic
389C2—22no—
390C2—41no—
391C2—13841no—
392C2D4×D69211yescharacteristic
393C2C2×#1914971211yescharacteristic
394C2#2601230811yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#435×C173C2
socleC22×C173C22×D4
prime radical#26012150C2
hypercenter#435D173
hypocenterC173#3270
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
1. C2×#19149712C2semidirect
2. C22×Dic692C2semidirect
3. D4×Dic346C2semidirect
5. #26011885C2semidirect
6. #26011749C2semidirect
7. #26011720C2semidirect
8. #26011714C2semidirect
9. #26011702C2semidirect
10. #26011655C2semidirect
11. #26011649C2semidirect
12. #26011649C2semidirect
13. #26011632C2semidirect
14. #26011599C2semidirect
15. #435×C173C2semidirect
89. Dic692D4semidirect
173. C22×C173#137semidirect
197. C173#3270semidirect
213. #435D173semidirect
374. C22#19149712semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C22×Dic6922
D4×Dic3462
#260121502
#260118852
#260117492
#260117202
#260117142
#260117022
#260116552
#260116492
#260116322
#260115992
#435×C1732
#3270173