Finite Group Databasebeta

← #27174479

Subgroups — #27174479

Order 12928.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108075C211yesmaximal, characteristic, complemented
2C2×#15692229C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D101C211yesmaximal, characteristic, complemented
4#20265094C212yesmaximal, complemented
5#20265086C212yesmaximal, complemented
6#20265084C211yesmaximal, characteristic, complemented
7#20265078C211yesmaximal, characteristic, complemented
8#20265053C212yesmaximal, complemented
9#20265048C212yesmaximal, complemented
10#20265034C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C101C211yesmaximal, characteristic, complemented
12C2×#13108075—24no—
13C2×#13108075—24no—
14C2×#13108075—24no—
15C22×D404C2212yes—
16C22×D404C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C22⋊D101C2212yes—
19C22×C4×D101C2212yes—
20C22×C4×D101C2211yescharacteristic
21#15692229—24no—
22C4×C22⋊D101—24no—
23C2×C4⋊Dic101C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×C22⋊Dic101C2211yescharacteristic
26C2×#13108051C2212yes—
27C2×#13108051C2212yes—
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2212yes—
30C2×#13108051C2212yes—
31C2×C4×Dic101C2211yescharacteristic
32#15692162C2212yes—
33#15692162C2212yes—
34#15692162C2212yes—
35C22×Q8×C101C2211yescharacteristic
36C23×C4×C101C2211yescharacteristic
37C22⋊Q8×C101—24no—
38C2×C4⋊C4×C101C2211yescharacteristic
39C2×C4⋊C4×C101C2212yes—
40C2×C22⋊C4×C101C2212yes—
41C23×D101C2312yes—
42C23×D101C2311yescharacteristic
43#13108075—232no—
44C2×D404—24no—
45C2×D404—24no—
46C2×D404—22no—
47C2×D404—24no—
48C2×D404—22no—
49C2×D404D414yescomplemented
50C2×C22⋊D101—22no—
51C2×C22⋊D101—22no—
52C2×C22⋊D101—24no—
53C2×C22⋊D101—24no—
54C2×C22⋊D101—22no—
55C2×C22⋊D101—42no—
56C22×Dic101C2312yes—
57C22×Dic101C2311yescharacteristic
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C2×C4×D101—24no—
61C2×C4×D101—22no—
62C2×C4×D101—24no—
63C2×C4×D101—22no—
64C2×C4×D101—24no—
65C2×C4×D101D414yescomplemented
66C4⋊Dic101—22no—
67#13108053—22no—
68C22⋊Dic101—22no—
69#13108051—22no—
70#13108051—24no—
71#13108051—24no—
72#13108051—42no—
73#13108051—42no—
74C4×Dic101—22no—
75C24×C101C2311yescharacteristic
76C2×Q8×C101—22no—
77C2×Q8×C101—22no—
78C2×Q8×C101D414yescomplemented
79C22×C4×C101—22no—
80C22×C4×C101—22no—
81C22×C4×C101—22no—
82C22×C4×C101C2312yes—
83C22×C4×C101C2311yescharacteristic
84C22×C4×C101C2312yes—
85C22×C4×C101C2311yescharacteristic
86C4⋊C4×C101—22no—
87C4⋊C4×C101—24no—
88C22⋊C4×C101—24no—
89C23×C101—21no—
90C23×C101—21no—
91C23×C101—21no—
92C23×C101—21no—
93C23×C101—21no—
94C23×C101—21no—
95C23×C101#4112yescomplemented
96C23×C101C2411yescharacteristic
97C22×D101—22no—
98C22×D101—21no—
99C22×D101—22no—
100C22×D101—21no—
101C22×D101—22no—
102C22×D101—21no—
103C22×D101—21no—
104C22×D101—22no—
105C22×D101—21no—
106C22×D101—22no—
107C22×D101—21no—
108C22×D101—22no—
109C22×D101C2×D414yes—
110C22×D101C2×D412yes—
111Q8×C101—28no—
112C2×C4×C101—22no—
113C2×C4×C101—22no—
114C2×C4×C101—22no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—22no—
119C2×C4×C101—22no—
120C2×C4×C101—44no—
121C2×C4×C101C2×D412yes—
122C2×C4×C101C2×D412yes—
123C2×C4×C101C2×D414yes—
124C2×C4×C101C2×D412yes—
125C22⋊D101—28no—
126C22⋊D101—48no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—22no—
133C2×Dic101—22no—
134C2×Dic101—21no—
135C2×Dic101—22no—
136C2×Dic101—21no—
137C2×Dic101C2×D412yes—
138C2×Dic101#4112yescomplemented
139D404—216no—
140D404—28no—
141C4×D101—216no—
142C4×D101—28no—
143C22×C101—22no—
144C22×C101—22no—
145C22×C101—22no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101C22×D411yescharacteristic
151C22×C101C22×D411yescharacteristic
152C22×C101C22×D411yescharacteristic
153C22×C101C2×#4111yescharacteristic
154C22×C101C2×#4111yescharacteristic
155C22×C101H(5,𝔽2)11yescharacteristic
156C22×C1012−1+411yescharacteristic
157C2×D101—24no—
158C2×D101—22no—
159C2×D101—22no—
160C2×D101—22no—
161C2×D101—24no—
162C2×D101—24no—
163C2×D101—42no—
164C2×D101—42no—
165C2×D101—41no—
166C2×D101—41no—
167C2×D101—41no—
168C2×D101—42no—
169C2×D101—42no—
170C2×D101—41no—
171C4×C101—24no—
172C4×C101—41no—
173C4×C101—42no—
174C4×C101#12014yes—
175Dic101—22no—
176Dic101—22no—
177Dic101—41no—
178Dic101—42no—
179C2×C101—42no—
180C2×C101C2×#12011yescharacteristic
181C2×C101#52511yescharacteristic
182C2×C101#53711yescharacteristic
183C2×C101#54611yescharacteristic
184C2×C101#54711yescharacteristic
185C2×C101#54811yescharacteristic
186C2×C101#55011yescharacteristic
187D101—42no—
188D101—44no—
189#2407—1011nomaximal, Sylow
190C101#240711yescharacteristic, complemented, Sylow
191C22×#41—1011no—
192C2×C22⋊Q8D10111yescharacteristic, complemented
193C2×#121—1011no—
194C2×C4×D4—1011no—
195#394—1012no—
196#392—1012no—
197#391—1011no—
198#390—1011no—
199#386—1012no—
200#385—1012no—
201#380—1011no—
202C2×#41—2024no—
203C2×#41—2024no—
204C2×#41—2024no—
205C22×Q8C2×D10111yescharacteristic
206C22×D4—1012no—
207C22×D4—1011no—
208C22×D4—1011no—
209C22×D4—1012no—
210C23×C4—1011no—
211C23×C4—1012no—
212C23×C4C2×D10111yescharacteristic
213C22⋊Q8—24no—
214#121—2024no—
215C4×D4—2024no—
216C2×C4⋊C4—1011no—
217C2×C4⋊C4—1011no—
218C2×C4⋊C4C2×D10111yescharacteristic
219C2×C4⋊C4C2×D10112yes—
220C2×C22⋊C4—1012no—
221C2×C22⋊C4—1012no—
222C2×C22⋊C4—1011no—
223C2×C22⋊C4—1011no—
224C2×C22⋊C4—1012no—
225C2×C22⋊C4—1012no—
226C2×C22⋊C4C2×D10112yes—
227C2×C42—1011no—
228#95—1012no—
229#95—1012no—
230#95—1012no—
231C24—1011no—
232C24—1012no—
233C24C22×D10111yescharacteristic
234#41—20232no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8C22⋊D10114yescomplemented
238C2×D4—1014no—
239C2×D4—4042no—
240C2×D4—2024no—
241C2×D4—2024no—
242C2×D4—2022no—
243C2×D4—2024no—
244C2×D4—2024no—
245C2×D4—2022no—
246C2×D4—2022no—
247C2×D4—2022no—
248C2×D4—2024no—
249C2×D4—2022no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—1011no—
254C22×C4—1011no—
255C22×C4—1014no—
256C22×C4—1012no—
257C22×C4—1011no—
258C22×C4—2024no—
259C22×C4—2024no—
260C22×C4—2022no—
261C22×C4—2024no—
262C22×C4—2022no—
263C22×C4C22×D10112yes—
264C22×C4C22×D10111yescharacteristic
265C22×C4C22×D10112yes—
266C22×C4C22×D10111yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—2022no—
270C4⋊C4—2022no—
271C22⋊C4—24no—
272C22⋊C4—4042no—
273C22⋊C4—4042no—
274C22⋊C4—2024no—
275C22⋊C4—2022no—
276C22⋊C4—2024no—
277C22⋊C4—2022no—
278C42—2022no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—1012no—
286C23—1014no—
287C23—2022no—
288C23—2021no—
289C23—2022no—
290C23—2021no—
291C23—2022no—
292C23—2021no—
293C23—2021no—
294C23—2022no—
295C23—2021no—
296C23—2022no—
297C23—2022no—
298C23—2021no—
299C23#1310807512yescomplemented
300C23C23×D10111yescharacteristic
301Q8—28no—
302D4—4048no—
303D4—20216no—
304D4—2028no—
305D4—2028no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—44no—
315C2×C4—1012no—
316C2×C4—1012no—
317C2×C4—20216no—
318C2×C4—2021no—
319C2×C4—2022no—
320C2×C4—2022no—
321C2×C4—2021no—
322C2×C4—2022no—
323C2×C4—2021no—
324C2×C4—2021no—
325C2×C4—2021no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4—2028no—
329C2×C4D4×D10112yes—
330C2×C4D4×D10112yes—
331C2×C4C2×C22⋊D10114yes—
332C2×C4C2×C22⋊D10112yes—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—4042no—
341C22—4041no—
342C22—4041no—
343C22—4042no—
344C22—4041no—
345C22—4042no—
346C22—4041no—
347C22—4042no—
348C22—2022no—
349C22—2024no—
350C22—2024no—
351C22—2024no—
352C22—2022no—
353C22—2022no—
354C22C2×D4×D10111yescharacteristic
355C22C2×D4×D10111yescharacteristic
356C22C22×C22⋊D10111yescharacteristic
357C22#41×D10111yescharacteristic
358C22C2×#1310807511yescharacteristic
359C22#1569230111yescharacteristic
360C22#1569230311yescharacteristic
361C4—24no—
362C4—42no—
363C4—41no—
364C4—4041no—
365C4—4042no—
366C4—2022no—
367C4—2022no—
368C4#1569222314yes—
369C2—42no—
370C2—4044no—
371C2—4042no—
372C2C2×#1569222311yescharacteristic
373C2#2026569011yescharacteristic
374C2#2026575711yescharacteristic
375C2#2026581811yescharacteristic
376C2#2026582911yescharacteristic
377C2#2026583811yescharacteristic
378C2#2026584611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
FittingC2×C22⋊Q8×C101C2
hypercenterC2×C22⋊Q8D101
hypocenterC101#2407
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13108075C2semidirect
2. C2×#15692229C2semidirect
3. C2×C4×C22⋊D101C2semidirect
4. #20265094C2semidirect
5. #20265086C2semidirect
6. #20265084C2semidirect
7. #20265078C2semidirect
8. #20265053C2semidirect
9. #20265048C2semidirect
10. #20265034C2semidirect
11. C2×C22⋊Q8×C101C2semidirect
49. C2×D404D4semidirect
65. C2×C4×D101D4semidirect
78. C2×Q8×C101D4semidirect
95. C23×C101#41semidirect
138. C2×Dic101#41semidirect
190. C101#2407semidirect
192. C2×C22⋊Q8D101semidirect
237. C2×Q8C22⋊D101semidirect
299. C23#13108075semidirect

Maximal subgroups

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

subgroupindex
C22×#131080752
C2×#156922292
C2×C4×C22⋊D1012
#202650942
#202650862
#202650842
#202650782
#202650532
#202650482
#202650342
C2×C22⋊Q8×C1012
#2407101