Finite Group Databasebeta

← #36362011

Subgroups — #36362011

Order 22144.

774 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D173C211yesmaximal, characteristic
2C2×#19149687C211yesmaximal, characteristic, complemented
3C2×#19149685C211yesmaximal, characteristic, complemented
4#26011458C211yesmaximal, characteristic, complemented
5#26011456C212yesmaximal, complemented
6#26011456C212yesmaximal, complemented
7#26011454C211yesmaximal, characteristic, complemented
8#26011453C212yesmaximal, complemented
9C2×#19149591C211yesmaximal, characteristic, complemented
10C22⋊C8×D173C212yesmaximal, complemented
11#408×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C411yescharacteristic
13C2×#14583826C2212yes—
14C2×#14583826C2211yescharacteristic
15C2×#14583825C411yescharacteristic
16C2×#14583825C412yes—
17#41×D173—24no—
18#41×D173—24no—
19C2×Q8×D173C411yescharacteristic
20C2×D4×D173C411yescharacteristic
21C2×D4×D173C412yes—
22C22×C4×D173C2212yes—
23C22×C4×D173C2211yescharacteristic
24#19149687—24no—
25#19149685—24no—
26C2×#14583810C2211yescharacteristic, complemented
27C2×#14583809C2212yescomplemented
28C2×#14583809C2212yescomplemented
29C22×#13063436C2211yescharacteristic, complemented
30C2×C8×D173C2212yescomplemented
31C2×C8×D173C2212yescomplemented
32C22⋊#13063436C2212yescomplemented
33C22⋊#13063436C2212yescomplemented
34#19149591C2211yescharacteristic
35#19149591C2212yes—
36#19149591C2211yescharacteristic
37#19149591C2211yescharacteristic, complemented
38#19149591C2211yescharacteristic, complemented
39#19149591C2212yescomplemented
40C2×#41×C173C2211yescharacteristic
41C2×M16×C173C2211yescharacteristic, complemented
42C22×C8×C173C2211yescharacteristic, complemented
43C22⋊C8×C173C2212yescomplemented
44C22⋊C8×C173C2212yescomplemented
45C23×D173C2×C412yes—
46C23×D173C2×C411yescharacteristic
47#14583827—24no—
48#14583826—24no—
49#14583826—24no—
50#14583826D414yes—
51#14583826D414yes—
52#14583825—24no—
53#14583825—28no—
54C2×Dic346C2×C412yes—
55C2×Dic346C2×C411yescharacteristic
56Q8×D173—24no—
57C2×D692C2×C412yes—
58C2×D692C2×C411yescharacteristic
59C2×C22⋊D173C2×C412yes—
60C2×C22⋊D173C2×C412yes—
61C2×C22⋊D173C2×C412yes—
62D4×D173—24no—
63D4×D173—28no—
64C22×Dic173C2×C412yes—
65C22×Dic173C2×C411yescharacteristic
66C2×C4×D173—24no—
67C2×C4×D173—24no—
68C2×C4×D173—22no—
69C2×C4×D173—22no—
70C2×C4×D173C2×C412yes—
71C2×C4×D173C2×C411yescharacteristic
72C2×C4×D173C2×C412yes—
73C2×C4×D173C2×C411yescharacteristic
74C2×C4×D173C2311yescharacteristic
75C2×C4×D173C2311yescharacteristic
76C2×C4×D173C2312yes—
77#14583810—22no—
78#14583809—24no—
79#14583809—24no—
80C2×#13063436—22no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2312yes—
83C2×#13063436C2311yescharacteristic
84C8×D173—24no—
85C8×D173—24no—
86#41×C173—22no—
87#41×C173—22no—
88C2×Q8×C173C2×C411yescharacteristic
89C2×D4×C173C2×C412yes—
90C2×D4×C173C2×C411yescharacteristic
91C22×C4×C173C2312yes—
92C22×C4×C173C2311yescharacteristic
93M16×C173—22no—
94C2×C8×C173—22no—
95C2×C8×C173C2311yescharacteristic
96C2×C8×C173C2311yescharacteristic
97C2×C8×C173C2312yes—
98C22×D173—22no—
99C22×D173—24no—
100C22×D173—24no—
101C22×D173—22no—
102C22×D173—42no—
103C22×D173—41no—
104C22×D173C22×C411yescharacteristic
105C22×D173C22×C412yes—
106C22×D173C22×C411yescharacteristic
107Dic346—24no—
108Dic346C22⋊C414yes—
109D692—24no—
110D692C22⋊C414yes—
111C22⋊D173—24no—
112C22⋊D173—24no—
113C22⋊D173C22⋊C418yes—
114C2×Dic173—22no—
115C2×Dic173—24no—
116C2×Dic173C22×C411yescharacteristic
117C2×Dic173C22×C411yescharacteristic
118C2×Dic173C22×C412yes—
119C4×D173—24no—
120C4×D173—22no—
121C4×D173—22no—
122C4×D173—21no—
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—21no—
126C4×D173—24no—
127C4×D173C2×D414yes—
128C4×D173C2×D414yes—
129#13063436—21no—
130#13063436—21no—
131#13063436—22no—
132C23×C173C22×C412yes—
133C23×C173C22×C411yescharacteristic
134Q8×C173—22no—
135D4×C173—24no—
136D4×C173—22no—
137C2×C4×C173—22no—
138C2×C4×C173—22no—
139C2×C4×C173C22×C412yes—
140C2×C4×C173C22×C411yescharacteristic
141C2×C4×C173C2×D412yes—
142C2×C4×C173C2×D412yes—
143C2×C4×C173C2411yescharacteristic
144C8×C173—21no—
145C8×C173—22no—
146C8×C173—21no—
147C22×C173—21no—
148C22×C173—21no—
149C22×C173—22no—
150C22×C173—22no—
151C22×C173—22no—
152C22×C173C2×C22⋊C412yes—
153C22×C173C23×C411yescharacteristic
154C2×D173—21no—
155C2×D173—22no—
156C2×D173—21no—
157C2×D173—22no—
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—48no—
162C2×D173—44no—
163C2×D173C2×C22⋊C414yes—
164C2×D173#13112yescomplemented
165C4×C173—22no—
166C4×C173C2×C22⋊C412yes—
167C4×C173C22×D411yescharacteristic
168C4×C173C22×D411yescharacteristic
169Dic173—21no—
170Dic173—21no—
171Dic173C2×C22⋊C414yes—
172C2×C173—22no—
173C2×C173—42no—
174C2×C173C22×C22⋊C411yescharacteristic
175C2×C173C2×#13111yescharacteristic
176C2×C173#56811yescharacteristic
177D173—22no—
178D173—24no—
179D173—41no—
180C173#288011yescharacteristic, complemented, Sylow
181#2880—1731nomaximal, Sylow
182C22×#41—1731no—
183C2×#131—1731no—
184C2×#131—1731no—
185#408—1731no—
186#408—1732no—
187#408—1732no—
188#408D17311yescharacteristic, complemented
189#407—1731no—
190#407—1732no—
191C2×C22⋊C8—1731no—
192C2×C22⋊C8—1732no—
193C2×#41—3464no—
194C2×#41—3464no—
195C2×#41—1731no—
196C2×#41—1731no—
197C2×#41—1732no—
198C2×#41—1732no—
199C2×#41—1731no—
200C2×#41C2×D17311yescharacteristic
201C22×Q8—1731no—
202C22×D4—1731no—
203C22×D4—1732no—
204C23×C4—1732no—
205C23×C4—1731no—
206#131—3464no—
207#131—3464no—
208C2×M16—1731no—
209C2×M16—1732no—
210C2×M16—1732no—
211C2×M16C2×D17311yescharacteristic, complemented
212C22×C8—1731no—
213C22×C8—1732no—
214C22×C8—1732no—
215C22×C8C2×D17311yescharacteristic, complemented
216C22⋊C8—1732no—
217C22⋊C8—1732no—
218C22⋊C8—1731no—
219C22⋊C8—1732no—
220C22⋊C8—1731no—
221C22⋊C8—1731no—
222C22⋊C8—1731no—
223C22⋊C8—1732no—
224C22⋊C8C2×D17312yescomplemented
225C22⋊C8C2×D17312yescomplemented
226C24—1732no—
227C24—1731no—
228#41—22no—
229#41—22no—
230#41—3464no—
231#41—3464no—
232#41—3464no—
233#41—3468no—
234#41—3464no—
235#41—1734no—
236#41—1734no—
237C2×Q8—3464no—
238C2×Q8—1732no—
239C2×Q8—1731no—
240C2×Q8C4×D17311yescharacteristic
241C2×D4—3468no—
242C2×D4—3464no—
243C2×D4—1732no—
244C2×D4—1732no—
245C2×D4—1732no—
246C2×D4—1732no—
247C2×D4—1731no—
248C2×D4C4×D17312yes—
249C2×D4C4×D17311yescharacteristic
250C22×C4—3464no—
251C22×C4—3464no—
252C22×C4—3462no—
253C22×C4—3462no—
254C22×C4—1731no—
255C22×C4—1732no—
256C22×C4—1732no—
257C22×C4—1731no—
258C22×C4—1732no—
259C22×C4—1731no—
260C22×C4—1731no—
261C22×C4—1731no—
262C22×C4—1732no—
263C22×C4C22×D17312yes—
264C22×C4C22×D17311yescharacteristic
265M16—22no—
266M16—3464no—
267M16—3464no—
268M16—3462no—
269C2×C8—22no—
270C2×C8—3464no—
271C2×C8—3464no—
272C2×C8—3462no—
273C2×C8—1731no—
274C2×C8—1732no—
275C2×C8—1731no—
276C2×C8C22×D17311yescharacteristic
277C2×C8C22×D17312yes—
278C2×C8C22×D17311yescharacteristic
279C23—3464no—
280C23—3462no—
281C23—3464no—
282C23—3462no—
283C23—1731no—
284C23—1731no—
285C23—1732no—
286C23—6921no—
287C23—6922no—
288C23C2×C4×D17312yes—
289C23C2×C4×D17311yescharacteristic
290Q8—22no—
291Q8—3464no—
292Q8—1734no—
293D4—24no—
294D4—22no—
295D4—3464no—
296D4—3464no—
297D4—3464no—
298D4—1738no—
299D4—1734no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—3464no—
303C2×C4—3464no—
304C2×C4—3462no—
305C2×C4—3464no—
306C2×C4—3462no—
307C2×C4—3462no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—1731no—
313C2×C4—1731no—
314C2×C4—1732no—
315C2×C4—1734no—
316C2×C4—1734no—
317C2×C4C2×C4×D17312yes—
318C2×C4C2×C4×D17311yescharacteristic
319C2×C4D4×D17312yes—
320C2×C4D4×D17312yes—
321C2×C4C23×D17311yescharacteristic
322C8—21no—
323C8—21no—
324C8—22no—
325C8—3461no—
326C8—3461no—
327C8—3462no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—3461no—
334C22—3462no—
335C22—3461no—
336C22—3462no—
337C22—3461no—
338C22—3461no—
339C22—3461no—
340C22—1734no—
341C22—1732no—
342C22—6924no—
343C22—6928no—
344C22C22⋊C4×D17312yes—
345C22C22×C4×D17311yescharacteristic
346C4—22no—
347C4—3461no—
348C4—3461no—
349C4—1734no—
350C4C22⋊C4×D17312yes—
351C4C2×D4×D17311yescharacteristic
352C4C2×D4×D17311yescharacteristic
353C2—22no—
354C2—42no—
355C2—3464no—
356C2—3462no—
357C2—6921no—
358C2C2×C22⋊C4×D17311yescharacteristic
359C2#131×D17311yescharacteristic
360C2#2601225611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D173
commutator, socleC22×C173C23×C4
Fitting#408×C173C2
prime radicalC2×#41×D173C2
cosocleC2×C4×C173C24
hypercenter#408D173
hypocenterC173#2880
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#19149687C2semidirect
3. C2×#19149685C2semidirect
4. #26011458C2semidirect
5. #26011456C2semidirect
6. #26011456C2semidirect
7. #26011454C2semidirect
8. #26011453C2semidirect
9. C2×#19149591C2semidirect
10. C22⋊C8×D173C2semidirect
11. #408×C173C2semidirect
26. C2×#14583810C22semidirect
27. C2×#14583809C22semidirect
28. C2×#14583809C22semidirect
29. C22×#13063436C22semidirect
30. C2×C8×D173C22semidirect
31. C2×C8×D173C22semidirect
32. C22⋊#13063436C22semidirect
33. C22⋊#13063436C22semidirect
37. #19149591C22semidirect
38. #19149591C22semidirect
39. #19149591C22semidirect
41. C2×M16×C173C22semidirect
42. C22×C8×C173C22semidirect
43. C22⋊C8×C173C22semidirect
44. C22⋊C8×C173C22semidirect
164. C2×D173#131semidirect
180. C173#2880semidirect
188. #408D173semidirect
211. C2×M16C2×D173semidirect
215. C22×C8C2×D173semidirect
224. C22⋊C8C2×D173semidirect
225. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1732
C2×#191496872
C2×#191496852
#260114582
#260114562
#260114542
#260114532
C2×#191495912
C22⋊C8×D1732
#408×C1732
#2880173