Finite Group Databasebeta

← #37581737

Subgroups — #37581737

Order 24448.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15690288C211yesmaximal, characteristic, complemented
2C2×#20259343C212yesmaximal, complemented
3#27133173C211yesmaximal, characteristic, complemented
4#27132828C212yesmaximal, complemented
5#27132819C211yesmaximal, characteristic, complemented
6#27132816C211yesmaximal, characteristic, complemented
7#27132807C212yesmaximal, complemented
8#27132796C211yesmaximal, characteristic, complemented
9#27132658C212yesmaximal, complemented
10C22⋊C8×D191C211yesmaximal, characteristic, complemented
11#448×C191C211yesmaximal, characteristic, complemented
12C2×#15690288—28no—
13C2×#15690288—21no—
14C2×#15690288—21no—
15C2×#15690288C2212yes—
16C2×#15690288C2212yes—
17C2×#15690286C2211yescharacteristic, complemented
18C22×D764—21no—
19C22×D764C2211yescharacteristic
20C22×C4×D191—21no—
21C22×C4×D191C2211yescharacteristic
22#20259343—28no—
23C2×#15690277C2212yescomplemented
24C2×#15690276C2212yescomplemented
25C2×#15690272C2212yescomplemented
26C2×C8×D191C2212yescomplemented
27#20259304—21no—
28#20259302C2211yescharacteristic, complemented
29#20259301C2211yescharacteristic
30#20259293C2211yescharacteristic, complemented
31#20259290—21no—
32D4×Dic191—21no—
33(C22⋊D191)⋊C4—21no—
34#20259280C2211yescharacteristic, complemented
35#20259246C2212yescomplemented
36#20259244C2212yescomplemented
37#20259242C2212yescomplemented
38C22⋊(C191⋊C8)C2211yescharacteristic
39(C2×D191)⋊C8C2212yescomplemented
40C22×Q8×C191C2211yescharacteristic
41C2×SD16×C191C2212yescomplemented
42#121×C191C2211yescharacteristic
43#103×C191C2212yescomplemented
44C22⋊C8×C191C2211yescharacteristic
45C23×D191—21no—
46C23×D191D411yescharacteristic
47#15690288—28no—
48#15690288—28no—
49#15690288—24no—
50#15690288—24no—
51#15690288—24no—
52#15690288D418yescomplemented
53#15690286—24no—
54C2×Dic382C2311yescharacteristic
55C2×D764—28no—
56C2×D764—22no—
57C2×D764—22no—
58C2×D764—24no—
59C2×D764—21no—
60C2×D764C2312yes—
61C2×C22⋊D191—21no—
62C2×C22⋊D191—21no—
63C22×Dic191—21no—
64C22×Dic191D411yescharacteristic
65C2×C4×D191—28no—
66C2×C4×D191—22no—
67C2×C4×D191—22no—
68C2×C4×D191—24no—
69C2×C4×D191D411yescharacteristic
70C2×C4×D191D411yescharacteristic
71C2×C4×D191C2311yescharacteristic
72C2×C4×D191C2311yescharacteristic
73#15690277—24no—
74#15690276—24no—
75#15690272—24no—
76C2×C191⋊C8C2312yes—
77C8×D191—24no—
78C4⋊Dic191—21no—
79C4⋊Dic191C2311yescharacteristic
80Dic191⋊C4—21no—
81C22⋊Dic191—21no—
82C22⋊Dic191—21no—
83(C2×D191)⋊C4—21no—
84(C2×D191)⋊C4—21no—
85C4×Dic191—21no—
86C2×Q8×C191—24no—
87C2×Q8×C191—21no—
88C2×Q8×C191C2312yes—
89C2×D4×C191—21no—
90C2×D4×C191C2311yescharacteristic
91C22×C4×C191—21no—
92C22×C4×C191C2311yescharacteristic
93SD16×C191—24no—
94C2×C8×C191C2312yes—
95C4⋊C4×C191C2311yescharacteristic
96C22⋊C4×C191—21no—
97C22×D191—24no—
98C22×D191—22no—
99C22×D191—22no—
100C22×D191—22no—
101C22×D191—41no—
102C22×D191—42no—
103C22×D191—42no—
104C22×D191C2×D411yescharacteristic
105C22×D191C2×D411yescharacteristic
106Dic382—22no—
107D764—28no—
108D764—24no—
109D764—28no—
110D764—22no—
111D764C2×D414yes—
112C22⋊D191—42no—
113C22⋊D191—42no—
114C2×Dic191—22no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—42no—
118C2×Dic191C2×D411yescharacteristic
119C2×Dic191C2×D411yescharacteristic
120C4×D191—28no—
121C4×D191—22no—
122C4×D191—22no—
123C4×D191—28no—
124C4×D191—22no—
125C4×D191C2×D414yes—
126C191⋊C8—22no—
127C23×C191—21no—
128C23×C191C2×D411yescharacteristic
129Q8×C191—24no—
130Q8×C191—22no—
131Q8×C191C2×D414yes—
132D4×C191—22no—
133D4×C191—42no—
134C2×C4×C191—24no—
135C2×C4×C191—22no—
136C2×C4×C191—22no—
137C2×C4×C191—21no—
138C2×C4×C191C2×D411yescharacteristic
139C2×C4×C191C2411yescharacteristic
140C8×C191—22no—
141C22×C191—21no—
142C22×C191—21no—
143C22×C191—41no—
144C22×C191—41no—
145C22×C191—41no—
146C22×C191#13512yescomplemented
147C22×C191C22×D411yescharacteristic
148C2×D191—21no—
149C2×D191—21no—
150C2×D191—21no—
151C2×D191—24no—
152C2×D191—21no—
153C2×D191—21no—
154C2×D191—44no—
155C2×D191—48no—
156C2×D191—42no—
157C2×D191—42no—
158C2×D191#12012yes—
159C4×C191—24no—
160C4×C191—21no—
161C4×C191—41no—
162C4×C191C22×D412yes—
163Dic191—21no—
164Dic191—41no—
165Dic191—41no—
166Dic191#12012yes—
167C2×C191—22no—
168C2×C191—81no—
169C2×C191C2×#12011yescharacteristic
170C2×C191C2×#13511yescharacteristic
171C2×C191#57511yescharacteristic
172D191—22no—
173D191—41no—
174D191—44no—
175C191#300211yescharacteristic, complemented, Sylow
176#3002—1911nomaximal, Sylow
177C22×#41—1911no—
178C2×#135—1912no—
179#525—1911no—
180#452—1912no—
181#451—1911no—
182#450—1911no—
183#449—1912no—
184#448D19111yescharacteristic, complemented
185#447—1911no—
186#416—1912no—
187C2×C22⋊C8—1911no—
188C2×#41—3828no—
189C2×#41—3821no—
190C2×#41—3821no—
191C2×#41—1912no—
192C2×#41—1912no—
193C2×#41—1911no—
194C22×Q8C2×D19111yescharacteristic
195C22×D4—3821no—
196C22×D4—1911no—
197C23×C4—3821no—
198C23×C4—1911no—
199#135—3828no—
200C2×Q16—1912no—
201C2×SD16—1912no—
202C2×SD16C2×D19112yescomplemented
203C2×D8—1912no—
204C22×C8—1912no—
205#123—3821no—
206C22⋊Q8—1911no—
207C22⋊Q8—1911no—
208#121—1911no—
209#121C2×D19111yescharacteristic
210#120—3821no—
211C4×D4—3821no—
212C4×D4—3821no—
213#117—1911no—
214#103—1912no—
215#103C2×D19112yescomplemented
216#102—1912no—
217#102—1912no—
218C22⋊C8—1911no—
219C22⋊C8—1912no—
220C22⋊C8C2×D19111yescharacteristic
221C24—3821no—
222C24—1911no—
223#41—3828no—
224#41—3828no—
225#41—3824no—
226#41—3824no—
227#41—3824no—
228#41—3824no—
229#41—1918no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8—1911no—
233C2×Q8C22×D19112yes—
234C2×D4—21no—
235C2×D4—3828no—
236C2×D4—3822no—
237C2×D4—3822no—
238C2×D4—3824no—
239C2×D4—3821no—
240C2×D4—3821no—
241C2×D4—3821no—
242C2×D4—1912no—
243C2×D4C22×D19111yescharacteristic
244C22×C4—21no—
245C22×C4—3828no—
246C22×C4—3822no—
247C22×C4—3822no—
248C22×C4—3824no—
249C22×C4—3821no—
250C22×C4—1911no—
251C22×C4—1911no—
252C22×C4—1911no—
253C22×C4—1911no—
254C22×C4—1911no—
255C22×C4C22×D19111yescharacteristic
256Q16—3824no—
257SD16—24no—
258SD16—3824no—
259D8—3824no—
260C2×C8—3824no—
261C2×C8—1912no—
262C2×C8C22×D19112yes—
263C4⋊C4—3821no—
264C4⋊C4—3821no—
265C4⋊C4—1911no—
266C4⋊C4C22×D19111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3821no—
269C22⋊C4—3821no—
270C22⋊C4—3821no—
271C22⋊C4—3821no—
272C42—3821no—
273C23—21no—
274C23—3824no—
275C23—3822no—
276C23—3822no—
277C23—3822no—
278C23—1911no—
279C23—1911no—
280C23—7641no—
281C23—7642no—
282C23—7642no—
283C23D4×D19111yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—3822no—
287Q8D4×D19114yes—
288D4—22no—
289D4—42no—
290D4—3828no—
291D4—3824no—
292D4—3828no—
293D4—3822no—
294D4—1914no—
295D4—7642no—
296D4—7642no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—3828no—
302C2×C4—3822no—
303C2×C4—3828no—
304C2×C4—3822no—
305C2×C4—3822no—
306C2×C4—3822no—
307C2×C4—3821no—
308C2×C4—3821no—
309C2×C4—1911no—
310C2×C4—1911no—
311C2×C4—1914no—
312C2×C4—7642no—
313C2×C4D4×D19111yescharacteristic
314C2×C4C23×D19111yescharacteristic
315C8—22no—
316C8—3822no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—3821no—
323C22—3821no—
324C22—3821no—
325C22—3821no—
326C22—3821no—
327C22—3824no—
328C22—1912no—
329C22—7644no—
330C22—7648no—
331C22—7642no—
332C22—7642no—
333C22#2025934312yescomplemented
334C22C2×D4×D19111yescharacteristic
335C4—24no—
336C4—21no—
337C4—41no—
338C4—3821no—
339C4—1912no—
340C4—7641no—
341C4—7641no—
342C4C2×D4×D19112yes—
343C2—22no—
344C2—81no—
345C2—3822no—
346C2—7641no—
347C2—7644no—
348C2#120×D19111yescharacteristic
349C2C2×#2025934311yescharacteristic
350C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#448×C191C2
socleC22×C191C22×D4
hypercenter#448D191
hypocenterC191#3002
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#15690288C2semidirect
2. C2×#20259343C2semidirect
3. #27133173C2semidirect
4. #27132828C2semidirect
5. #27132819C2semidirect
6. #27132816C2semidirect
7. #27132807C2semidirect
8. #27132796C2semidirect
9. #27132658C2semidirect
10. C22⋊C8×D191C2semidirect
11. #448×C191C2semidirect
17. C2×#15690286C22semidirect
23. C2×#15690277C22semidirect
24. C2×#15690276C22semidirect
25. C2×#15690272C22semidirect
26. C2×C8×D191C22semidirect
28. #20259302C22semidirect
30. #20259293C22semidirect
34. #20259280C22semidirect
35. #20259246C22semidirect
36. #20259244C22semidirect
37. #20259242C22semidirect
39. (C2×D191)⋊C8C22semidirect
41. C2×SD16×C191C22semidirect
43. #103×C191C22semidirect
52. #15690288D4semidirect
146. C22×C191#135semidirect
175. C191#3002semidirect
184. #448D191semidirect
202. C2×SD16C2×D191semidirect
215. #103C2×D191semidirect
333. C22#20259343semidirect

Maximal subgroups

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

subgroupindex
C22×#156902882
C2×#202593432
#271331732
#271328282
#271328192
#271328162
#271328072
#271327962
#271326582
C22⋊C8×D1912
#448×C1912
#3002191