Finite Group Databasebeta

← #27174483

Subgroups — #27174483

Order 12928.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 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×D4×D101C211yesmaximal, characteristic, complemented
2C2×#15692228C212yesmaximal, complemented
3C2×#15692225C212yesmaximal, complemented
4C2×#15692217C211yesmaximal, characteristic, complemented
5#20265094C211yesmaximal, characteristic, complemented
6#20265087C211yesmaximal, characteristic, complemented
7#20265085C211yesmaximal, characteristic, complemented
8#20265077C212yesmaximal, complemented
9#20265036C211yesmaximal, characteristic, complemented
10#20265032C212yesmaximal, complemented
11#394×C101C211yesmaximal, characteristic, complemented
12C24×D101C2212yes—
13C22×D404C2211yescharacteristic
14C22×D404C2211yescharacteristic
15C22×D404C2212yescomplemented
16C22×C22⋊D101C2212yes—
17C22×C22⋊D101C2212yescomplemented
18C2×D4×D101—28no—
19C2×D4×D101—24no—
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic, complemented
22#15692228—28no—
23#15692225—28no—
24#15692217—24no—
25C2×#13108053C2212yescomplemented
26C2×C22⋊Dic101C2212yescomplemented
27C2×#13108051C2211yescharacteristic
28C2×#13108051C2212yes—
29C2×#13108051C2212yescomplemented
30C2×#13108051C2212yescomplemented
31C2×#13108051C2211yescharacteristic, complemented
32C2×#13108051C2212yescomplemented
33C2×C4×Dic101C2211yescharacteristic, complemented
34#15692162C2211yescharacteristic, complemented
35C22×D4×C101C2211yescharacteristic
36C2×C4⋊C4×C101C2211yescharacteristic, complemented
37C2×C22⋊C4×C101C2212yescomplemented
38C2×C22⋊C4×C101C2212yescomplemented
39#95×C101C2211yescharacteristic, complemented
40C23×D101—24no—
41C23×D101—24no—
42C23×D101—24no—
43C23×D101—24no—
44C23×D101—24no—
45C23×D101—42no—
46C23×D101—42no—
47C23×D101C2312yes—
48C23×D101C2311yescharacteristic
49C23×D101C2311yescharacteristic
50C2×D404—24no—
51C2×D404—22no—
52C2×D404—22no—
53C2×D404—22no—
54C2×D404—24no—
55C2×D404—24no—
56C2×D404—42no—
57C2×D404D414yescomplemented
58C2×C22⋊D101—24no—
59C2×C22⋊D101—24no—
60C2×C22⋊D101—24no—
61C2×C22⋊D101—24no—
62C2×C22⋊D101—42no—
63C2×C22⋊D101D418yescomplemented
64D4×D101—232no—
65C22×Dic101C2312yes—
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C2×C4×D101—24no—
69C2×C4×D101—22no—
70C2×C4×D101—24no—
71C2×C4×D101D414yescomplemented
72#13108053—24no—
73C22⋊Dic101—42no—
74#13108051—22no—
75#13108051—24no—
76#13108051—24no—
77#13108051—22no—
78#13108051—24no—
79#13108051—42no—
80C4×Dic101—22no—
81C24×C101C2312yes—
82C2×D4×C101—24no—
83C2×D4×C101—22no—
84C22×C4×C101C2311yescharacteristic
85C22×C4×C101C2312yes—
86C22×C4×C101C2311yescharacteristic
87C22×C4×C101C2311yescharacteristic
88C4⋊C4×C101—22no—
89C22⋊C4×C101—24no—
90C22⋊C4×C101—24no—
91C23×C101—22no—
92C23×C101—22no—
93C23×C101—22no—
94C23×C101—22no—
95C23×C101—22no—
96C23×C101—22no—
97C23×C101C2×D414yes—
98C23×C101C2411yescharacteristic
99C22×D101—22no—
100C22×D101—22no—
101C22×D101—216no—
102C22×D101—21no—
103C22×D101—22no—
104C22×D101—22no—
105C22×D101—22no—
106C22×D101—22no—
107C22×D101—21no—
108C22×D101—22no—
109C22×D101—22no—
110C22×D101—22no—
111C22×D101—22no—
112C22×D101—48no—
113C22×D101—48no—
114C22×D101—48no—
115C22×D101—48no—
116C22×D101—48no—
117C22×D101—48no—
118C22×D101C2×D414yes—
119C22×D101C2×D412yes—
120C22×D101C2×D412yes—
121C22×D101C2×D412yescomplemented
122C22×D101#4112yes—
123D4×C101—28no—
124C2×C4×C101—21no—
125C2×C4×C101—22no—
126C2×C4×C101—21no—
127C2×C4×C101—22no—
128C2×C4×C101—22no—
129C2×C4×C101—22no—
130C2×C4×C101—21no—
131C2×C4×C101—22no—
132C2×C4×C101C2×D412yes—
133C2×C4×C101C2×D412yescomplemented
134C22⋊D101—216no—
135C22⋊D101—48no—
136C2×Dic101—21no—
137C2×Dic101—22no—
138C2×Dic101—22no—
139C2×Dic101—22no—
140C2×Dic101—22no—
141C2×Dic101C2×D412yes—
142C2×Dic101C2×D414yes—
143D404—28no—
144D404—28no—
145D404—48no—
146C4×D101—28no—
147C4×D101—44no—
148C22×C101—24no—
149C22×C101—24no—
150C22×C101—24no—
151C22×C101—42no—
152C22×C101—42no—
153C22×C101—42no—
154C22×C101—42no—
155C22×C101C22×D411yescharacteristic
156C22×C101C22×D412yes—
157C22×C101C22×D411yescharacteristic
158C22×C101C2×#4111yescharacteristic
159C22×C101H(5,𝔽2)12yes—
160C2×D101—22no—
161C2×D101—22no—
162C2×D101—24no—
163C2×D101—24no—
164C2×D101—24no—
165C2×D101—22no—
166C2×D101—24no—
167C2×D101—24no—
168C2×D101—24no—
169C2×D101—432no—
170C2×D101—41no—
171C2×D101—42no—
172C2×D101—41no—
173C2×D101—42no—
174C2×D101—42no—
175C2×D101—42no—
176C2×D101—42no—
177C2×D101#12114yescomplemented
178C4×C101—22no—
179C4×C101—22no—
180C4×C101—41no—
181C4×C101—42no—
182Dic101—41no—
183Dic101—42no—
184Dic101#12014yescomplemented
185C2×C101—44no—
186C2×C101C2×#12011yescharacteristic
187C2×C101C2×#12111yescharacteristic
188C2×C101#53511yescharacteristic
189C2×C101D4212yes—
190C2×C101#54612yes—
191D101—24no—
192D101—42no—
193D101—44no—
194#2408—1011nomaximal, Sylow
195C101#240811yescharacteristic, complemented, Sylow
196C23×D4—1011no—
197C2×#121—1012no—
198C2×#120—1012no—
199C2×C4×D4—1011no—
200#394—1011no—
201#394D10111yescharacteristic, complemented
202#392—1011no—
203#392—1011no—
204#390—1012no—
205#380—1011no—
206#379—1012no—
207C25—1012no—
208C22×D4—1012no—
209C22×D4—1012no—
210C22×D4—1011no—
211C22×D4—1012no—
212C22×D4—1011no—
213C22×D4—2024no—
214C22×D4—2028no—
215C22×D4C2×D10111yescharacteristic
216C23×C4—1011no—
217C23×C4—1011no—
218#121—2028no—
219#120—2028no—
220C4×D4—2024no—
221C2×C4⋊C4—1012no—
222C2×C4⋊C4C2×D10111yescharacteristic, complemented
223C2×C22⋊C4—1012no—
224C2×C22⋊C4—1012no—
225C2×C22⋊C4—1011no—
226C2×C22⋊C4—1011no—
227C2×C22⋊C4—1012no—
228C2×C22⋊C4—1012no—
229C2×C22⋊C4—1012no—
230C2×C22⋊C4C2×D10112yescomplemented
231C2×C22⋊C4C2×D10112yescomplemented
232C2×C42—1011no—
233#95—1011no—
234#95C2×D10111yescharacteristic, complemented
235C24—1011no—
236C24—1012no—
237C24—1011no—
238C24—4042no—
239C24—4042no—
240C24—2024no—
241C24—2024no—
242C24—2024no—
243C24—2024no—
244C24—2024no—
245C24C22×D10112yes—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—1018no—
249C2×D4—1014no—
250C2×D4—4042no—
251C2×D4—4042no—
252C2×D4—2024no—
253C2×D4—2024no—
254C2×D4—2022no—
255C2×D4—2022no—
256C2×D4—2024no—
257C2×D4—2024no—
258C2×D4—2024no—
259C2×D4—20232no—
260C2×D4—2024no—
261C2×D4—2024no—
262C2×D4—2022no—
263C22×C4—1011no—
264C22×C4—1014no—
265C22×C4—1012no—
266C22×C4—1011no—
267C22×C4—2022no—
268C22×C4—2024no—
269C22×C4—2024no—
270C22×C4C22×D10112yes—
271C22×C4C22×D10111yescharacteristic
272C22×C4C22×D10111yescharacteristic
273C22×C4C22×D10111yescharacteristic
274C4⋊C4—22no—
275C4⋊C4—2024no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—4042no—
279C22⋊C4—4042no—
280C22⋊C4—2022no—
281C22⋊C4—2024no—
282C22⋊C4—2024no—
283C22⋊C4—2022no—
284C22⋊C4—2024no—
285C42—2022no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—1012no—
293C23—1012no—
294C23—1012no—
295C23—1014no—
296C23—1012no—
297C23—4048no—
298C23—4048no—
299C23—4048no—
300C23—4048no—
301C23—4048no—
302C23—4048no—
303C23—20216no—
304C23—2021no—
305C23—2022no—
306C23—2022no—
307C23—2022no—
308C23—2022no—
309C23—2021no—
310C23—2022no—
311C23—2022no—
312C23—2022no—
313C23—2022no—
314C23—2022no—
315C23—2022no—
316C23D4×D10114yes—
317C23C23×D10111yescharacteristic
318D4—28no—
319D4—4048no—
320D4—4048no—
321D4—2028no—
322D4—2028no—
323D4—20216no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—1012no—
333C2×C4—1014no—
334C2×C4—4044no—
335C2×C4—2028no—
336C2×C4—2021no—
337C2×C4—2022no—
338C2×C4—2022no—
339C2×C4—2022no—
340C2×C4—2022no—
341C2×C4D4×D10112yes—
342C2×C4D4×D10112yescomplemented
343C22—24no—
344C22—24no—
345C22—24no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—1014no—
351C22—40432no—
352C22—4042no—
353C22—4041no—
354C22—4042no—
355C22—4042no—
356C22—4042no—
357C22—4041no—
358C22—4042no—
359C22—2024no—
360C22—2024no—
361C22—2024no—
362C22—2024no—
363C22—2022no—
364C22—2022no—
365C22—2024no—
366C22—2024no—
367C22—2022no—
368C22C2×D4×D10111yescharacteristic
369C22C2×D4×D10111yescharacteristic
370C22C2×D4×D10112yes—
371C22#41×D10111yescharacteristic
372C22#1569230112yes—
373C4—22no—
374C4—22no—
375C4—42no—
376C4—41no—
377C4—1014no—
378C4—4041no—
379C4—4042no—
380C2—44no—
381C2—4044no—
382C2—4042no—
383C2—2024no—
384C2#120×D10111yescharacteristic
385C2#121×D10111yescharacteristic
386C2#2026574511yescharacteristic
387C2#2026581112yes—
388C2#2026581912yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#394×C101C2
hypercenter#394D101
hypocenterC101#2408
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×D4×D101C2semidirect
2. C2×#15692228C2semidirect
3. C2×#15692225C2semidirect
4. C2×#15692217C2semidirect
5. #20265094C2semidirect
6. #20265087C2semidirect
7. #20265085C2semidirect
8. #20265077C2semidirect
9. #20265036C2semidirect
10. #20265032C2semidirect
11. #394×C101C2semidirect
15. C22×D404C22semidirect
17. C22×C22⋊D101C22semidirect
21. C22×C4×D101C22semidirect
25. C2×#13108053C22semidirect
26. C2×C22⋊Dic101C22semidirect
29. C2×#13108051C22semidirect
30. C2×#13108051C22semidirect
31. C2×#13108051C22semidirect
32. C2×#13108051C22semidirect
33. C2×C4×Dic101C22semidirect
34. #15692162C22semidirect
36. C2×C4⋊C4×C101C22semidirect
37. C2×C22⋊C4×C101C22semidirect
38. C2×C22⋊C4×C101C22semidirect
39. #95×C101C22semidirect
57. C2×D404D4semidirect
63. C2×C22⋊D101D4semidirect
71. C2×C4×D101D4semidirect
121. C22×D101C2×D4semidirect
133. C2×C4×C101C2×D4semidirect
177. C2×D101#121semidirect
184. Dic101#120semidirect
195. C101#2408semidirect
201. #394D101semidirect
222. C2×C4⋊C4C2×D101semidirect
230. C2×C22⋊C4C2×D101semidirect
231. C2×C22⋊C4C2×D101semidirect
234. #95C2×D101semidirect
342. C2×C4D4×D101semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1012
C2×#156922282
C2×#156922252
C2×#156922172
#202650942
#202650872
#202650852
#202650772
#202650362
#202650322
#394×C1012
#2408101