Finite Group Databasebeta

← #27174461

Subgroups — #27174461

Order 12928.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×D404C211yesmaximal, characteristic
2C2×#15692240C212yesmaximal, complemented
3C2×C22⋊C4×D101C211yesmaximal, characteristic, complemented
4#20265095C212yesmaximal, complemented
5#20265094C212yesmaximal, complemented
6#20265055C212yesmaximal, complemented
7#20265050C211yesmaximal, characteristic, complemented
8#20265032C211yesmaximal, characteristic, complemented
9#20265032C211yesmaximal, characteristic, complemented
10#20265032C211yesmaximal, characteristic, complemented
11#385×C101C211yesmaximal, characteristic, complemented
12C24×D101C2211yescharacteristic
13C24×D101C2211yescharacteristic
14C22×D404—24no—
15C22×D404—28no—
16C22×D404C2212yes—
17C22×D404C2212yes—
18C22×C4×D101C2211yescharacteristic, complemented
19C22×C4×D101C2211yescharacteristic, complemented
20#15692240—28no—
21C2×#13108053C2212yescomplemented
22C2×C22⋊Dic101C2211yescharacteristic
23C2×C22⋊Dic101C2211yescharacteristic
24C2×#13108051C2212yescomplemented
25C2×#13108051C2212yescomplemented
26C2×#13108051C2212yescomplemented
27C2×#13108051C2211yescharacteristic, complemented
28C2×#13108051C2211yescharacteristic, complemented
29C2×#13108051C2212yescomplemented
30C2×#13108051C2212yescomplemented
31C2×#13108051C2212yescomplemented
32C22⋊C4×D101—24no—
33#15692162C2212yescomplemented
34C23×C4×C101C2211yescharacteristic
35C2×C4⋊C4×C101C2212yescomplemented
36C2×C22⋊C4×C101C2211yescharacteristic
37C2×C22⋊C4×C101C2211yescharacteristic
38#95×C101C2212yescomplemented
39C23×D101—22no—
40C23×D101—22no—
41C23×D101—22no—
42C23×D101—24no—
43C23×D101—24no—
44C23×D101—22no—
45C23×D101—22no—
46C23×D101—42no—
47C23×D101—42no—
48C23×D101D414yes—
49C23×D101C2311yescharacteristic
50C23×D101C2312yes—
51C23×D101C2311yescharacteristic
52C2×D404—232no—
53C2×D404—24no—
54C2×D404—24no—
55C2×D404—24no—
56C2×D404—24no—
57C2×D404—24no—
58C2×D404D418yes—
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2312yes—
62C2×C4×D101—22no—
63C2×C4×D101—22no—
64C2×C4×D101—24no—
65C2×C4×D101—42no—
66#13108053—24no—
67C22⋊Dic101—22no—
68C22⋊Dic101—22no—
69#13108051—22no—
70#13108051—22no—
71#13108051—24no—
72#13108051—24no—
73#13108051—24no—
74#13108051—42no—
75#13108051—42no—
76#13108051—42no—
77C24×C101C2311yescharacteristic
78C22×C4×C101—22no—
79C22×C4×C101—24no—
80C22×C4×C101C2311yescharacteristic
81C22×C4×C101C2311yescharacteristic
82C22×C4×C101C2312yes—
83C22×C4×C101C2312yes—
84C4⋊C4×C101—24no—
85C22⋊C4×C101—22no—
86C22⋊C4×C101—22no—
87C23×C101—22no—
88C23×C101—22no—
89C23×C101C2×D412yes—
90C23×C101#4112yes—
91C23×C101#4112yes—
92C23×C101C2411yescharacteristic
93C22×D101—21no—
94C22×D101—21no—
95C22×D101—22no—
96C22×D101—28no—
97C22×D101—28no—
98C22×D101—22no—
99C22×D101—22no—
100C22×D101—22no—
101C22×D101—21no—
102C22×D101—22no—
103C22×D101—21no—
104C22×D101—22no—
105C22×D101—22no—
106C22×D101—22no—
107C22×D101—22no—
108C22×D101—28no—
109C22×D101—44no—
110C22×D101—48no—
111C22×D101—48no—
112C22×D101—48no—
113C22×D101—48no—
114C22×D101—44no—
115C22×D101—44no—
116C22×D101C2×D412yes—
117C22×D101C2×D414yes—
118C22×D101C2×D414yes—
119C22×D101C2×D412yes—
120C2×C4×C101—28no—
121C2×C4×C101—22no—
122C2×C4×C101—22no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—22no—
126C2×C4×C101—22no—
127C2×C4×C101—22no—
128C2×C4×C101—22no—
129C2×C4×C101—22no—
130C2×C4×C101C2×D414yes—
131C2×Dic101—22no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—22no—
135C2×Dic101—22no—
136C2×Dic101—22no—
137C2×Dic101—22no—
138D404—216no—
139D404—216no—
140C4×D101—44no—
141C4×D101—44no—
142C22×C101—22no—
143C22×C101—22no—
144C22×C101—24no—
145C22×C101—24no—
146C22×C101#12314yescomplemented
147C22×C101C22×D411yescharacteristic
148C22×C101C22×D412yes—
149C22×C101C2×#4111yescharacteristic
150C22×C101C2×#4111yescharacteristic
151C22×C101H(5,𝔽2)12yes—
152C2×D101—22no—
153C2×D101—22no—
154C2×D101—24no—
155C2×D101—24no—
156C2×D101—22no—
157C2×D101—24no—
158C2×D101—22no—
159C2×D101—22no—
160C2×D101—24no—
161C2×D101—24no—
162C2×D101—416no—
163C2×D101—416no—
164C2×D101—42no—
165C2×D101—42no—
166C2×D101—42no—
167C2×D101—42no—
168C2×D101—42no—
169C2×D101—42no—
170C2×D101#12014yes—
171C4×C101—24no—
172C4×C101—41no—
173C4×C101—41no—
174C4×C101—42no—
175Dic101—41no—
176Dic101—41no—
177Dic101—42no—
178C2×C101—24no—
179C2×C101C2×#12011yescharacteristic
180C2×C101C2×#12311yescharacteristic
181C2×C101#53411yescharacteristic
182C2×C101#54612yes—
183C2×C101#54612yes—
184D101—24no—
185D101—42no—
186D101—44no—
187#2405—1011nomaximal, Sylow
188C101#240511yescharacteristic, complemented, Sylow
189C23×D4—1011no—
190C2×#123—1012no—
191C22×C22⋊C4—1011no—
192#394—1012no—
193#394—1012no—
194#386—1012no—
195#385—1011no—
196#385D10111yescharacteristic, complemented
197#379—1011no—
198#379—1011no—
199#379—1011no—
200C25—1011no—
201C25—1011no—
202C22×D4—1012no—
203C22×D4—1012no—
204C22×D4—2028no—
205C22×D4—2024no—
206C23×C4—1011no—
207C23×C4—1011no—
208C23×C4C2×D10111yescharacteristic
209#123—2028no—
210C2×C4⋊C4—1012no—
211C2×C4⋊C4C2×D10112yescomplemented
212C2×C22⋊C4—1012no—
213C2×C22⋊C4—1012no—
214C2×C22⋊C4—1012no—
215C2×C22⋊C4—1011no—
216C2×C22⋊C4—1011no—
217C2×C22⋊C4—1011no—
218C2×C22⋊C4—1011no—
219C2×C22⋊C4—1012no—
220C2×C22⋊C4—1012no—
221C2×C22⋊C4—1012no—
222C2×C22⋊C4—2024no—
223C2×C22⋊C4C2×D10111yescharacteristic
224C2×C22⋊C4C2×D10111yescharacteristic
225#95—1012no—
226#95C2×D10112yescomplemented
227C24—1011no—
228C24—1014no—
229C24—1012no—
230C24—1011no—
231C24—4042no—
232C24—4042no—
233C24—2024no—
234C24—2024no—
235C24—2022no—
236C24—2022no—
237C24—2022no—
238C24—2022no—
239C24—2022no—
240C24C22×D10111yescharacteristic
241C2×D4—1018no—
242C2×D4—2024no—
243C2×D4—2024no—
244C2×D4—2024no—
245C2×D4—2024no—
246C2×D4—20232no—
247C2×D4—2024no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—1011no—
251C22×C4—1012no—
252C22×C4—1011no—
253C22×C4—4042no—
254C22×C4—2022no—
255C22×C4—2022no—
256C22×C4—2024no—
257C22×C4C22×D10111yescharacteristic
258C22×C4C22×D10112yes—
259C22×C4C22×D10111yescharacteristic
260C22×C4C22×D10112yes—
261C4⋊C4—24no—
262C4⋊C4—2024no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—4042no—
266C22⋊C4—4042no—
267C22⋊C4—4042no—
268C22⋊C4—2022no—
269C22⋊C4—2022no—
270C22⋊C4—2024no—
271C22⋊C4—2024no—
272C22⋊C4—2024no—
273C22⋊C4—2022no—
274C22⋊C4—2022no—
275C23—22no—
276C23—22no—
277C23—1014no—
278C23—1012no—
279C23—1014no—
280C23—1012no—
281C23—4044no—
282C23—4048no—
283C23—4048no—
284C23—4048no—
285C23—4048no—
286C23—4044no—
287C23—4044no—
288C23—2028no—
289C23—2028no—
290C23—2022no—
291C23—2022no—
292C23—2022no—
293C23—2021no—
294C23—2022no—
295C23—2021no—
296C23—2022no—
297C23—2022no—
298C23—2022no—
299C23—2022no—
300C23—2028no—
301C23—2021no—
302C23—2021no—
303C23—2022no—
304C23D4×D10112yes—
305C23#1310807412yes—
306C23#1310807512yes—
307C23C23×D10111yescharacteristic
308D4—20216no—
309D4—20216no—
310C2×C4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—4044no—
321C2×C4—4044no—
322C2×C4—2022no—
323C2×C4—2021no—
324C2×C4—2021no—
325C2×C4—2022no—
326C2×C4—2022no—
327C2×C4—2022no—
328C2×C4—2022no—
329C2×C4D4×D10114yes—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—22no—
334C22—1014no—
335C22—40416no—
336C22—40416no—
337C22—4042no—
338C22—4042no—
339C22—4042no—
340C22—4042no—
341C22—4042no—
342C22—4042no—
343C22—2022no—
344C22—2024no—
345C22—2024no—
346C22—2024no—
347C22—2022no—
348C22—2022no—
349C22—2022no—
350C22—2024no—
351C22—2024no—
352C22—2022no—
353C22#1569224014yescomplemented
354C22C2×D4×D10111yescharacteristic
355C22C2×D4×D10112yes—
356C22C2×#1310807411yescharacteristic
357C22C2×#1310807511yescharacteristic
358C22#1569230112yes—
359C4—24no—
360C4—41no—
361C4—42no—
362C4—41no—
363C4—4041no—
364C4—4042no—
365C4—4041no—
366C2—24no—
367C2—4044no—
368C2—4042no—
369C2—2024no—
370C2#120×D10111yescharacteristic
371C2C2×#1569224011yescharacteristic
372C2#2026573811yescharacteristic
373C2#2026581812yes—
374C2#2026582312yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#385×C101C2
prime radicalC23×D404C2
hypercenter#385D101
hypocenterC101#2405
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#15692240C2semidirect
3. C2×C22⋊C4×D101C2semidirect
4. #20265095C2semidirect
5. #20265094C2semidirect
6. #20265055C2semidirect
7. #20265050C2semidirect
8. #20265032C2semidirect
9. #20265032C2semidirect
10. #20265032C2semidirect
11. #385×C101C2semidirect
18. C22×C4×D101C22semidirect
19. C22×C4×D101C22semidirect
21. C2×#13108053C22semidirect
24. C2×#13108051C22semidirect
25. C2×#13108051C22semidirect
26. C2×#13108051C22semidirect
27. C2×#13108051C22semidirect
28. C2×#13108051C22semidirect
29. C2×#13108051C22semidirect
30. C2×#13108051C22semidirect
31. C2×#13108051C22semidirect
33. #15692162C22semidirect
35. C2×C4⋊C4×C101C22semidirect
38. #95×C101C22semidirect
146. C22×C101#123semidirect
188. C101#2405semidirect
196. #385D101semidirect
211. C2×C4⋊C4C2×D101semidirect
226. #95C2×D101semidirect
353. C22#15692240semidirect

Maximal subgroups

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

subgroupindex
C23×D4042
C2×#156922402
C2×C22⋊C4×D1012
#202650952
#202650942
#202650552
#202650502
#202650322
#385×C1012
#2405101