Finite Group Databasebeta

← #27173664

Subgroups — #27173664

Order 12928.

656 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692232C211yesmaximal, characteristic
2C2×#15692220C211yesmaximal, characteristic, complemented
3C22×C4×Dic101C211yesmaximal, characteristic
4#20265081C211yesmaximal, characteristic, complemented
5#20265072C211yesmaximal, characteristic, complemented
6#20265062C211yesmaximal, characteristic, complemented
7#20265058C211yesmaximal, characteristic, complemented
8#20265050C211yesmaximal, characteristic
9#20265035C211yesmaximal, characteristic
10C4⋊C4×Dic101C211yesmaximal, characteristic, complemented
11#20265028C211yesmaximal, characteristic, complemented
12#20265028C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic101C211yesmaximal, characteristic
14C22⋊C4×Dic101C211yesmaximal, characteristic
15#385×C101C211yesmaximal, characteristic
16C22×Dic202C2211yescharacteristic
17C23×Dic101C2211yescharacteristic
18C23×Dic101C2211yescharacteristic
19#15692232C418yescomplemented
20#15692220—24no—
21C2×C4⋊Dic101C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C4×Dic101—24no—
31C2×C4×Dic101C2211yescharacteristic
32C2×C4×Dic101C2211yescharacteristic
33C2×C4×Dic101C2211yescharacteristic
34C2×C4×Dic101C2211yescharacteristic
35C2×C4×Dic101C2211yescharacteristic
36C2×C4×Dic101C2211yescharacteristic
37C2×C4×Dic101C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39#15692162C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47C23×C4×C101C2211yescharacteristic
48C2×C4⋊C4×C101C2211yescharacteristic
49C2×C4⋊C4×C101C2211yescharacteristic
50C2×C22⋊C4×C101C2211yescharacteristic
51C2×C22⋊C4×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53#95×C101C2211yescharacteristic
54C2×Dic202—22no—
55C2×Dic202—22no—
56C2×Dic202C2×C414yescomplemented
57C22×Dic101—22no—
58C22×Dic101—22no—
59C22×Dic101—22no—
60C22×Dic101—22no—
61C22×Dic101C2×C414yes—
62C22×Dic101Q814yescomplemented
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C22×Dic101C2311yescharacteristic
69C22×Dic101C2311yescharacteristic
70C22×Dic101C2311yescharacteristic
71C4⋊Dic101C2×C414yescomplemented
72#13108053—22no—
73#13108053—22no—
74#13108053—22no—
75#13108053—22no—
76#13108053C2×C414yescomplemented
77#13108053C2×C414yescomplemented
78C22⋊Dic101—22no—
79C22⋊Dic101C2×C414yes—
80C4×Dic101—22no—
81C4×Dic101—22no—
82C4×Dic101—22no—
83C4×Dic101—22no—
84C4×Dic101—22no—
85C4×Dic101—22no—
86C4×Dic101—22no—
87C4×Dic101D414yes—
88C24×C101C2311yescharacteristic
89C22×C4×C101—22no—
90C22×C4×C101—22no—
91C22×C4×C101—22no—
92C22×C4×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C22×C4×C101C2311yescharacteristic
95C22×C4×C101C2311yescharacteristic
96C22×C4×C101C2311yescharacteristic
97C22×C4×C101C2311yescharacteristic
98C4⋊C4×C101—22no—
99C4⋊C4×C101—22no—
100C22⋊C4×C101—22no—
101C22⋊C4×C101C2×C414yes—
102C23×C101—21no—
103C23×C101—21no—
104C23×C101—21no—
105C23×C101—21no—
106C23×C101C22×C412yes—
107C23×C101C2×Q812yes—
108C23×C101#4112yes—
109C23×C101C2411yescharacteristic
110C2×C4×C101—28no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101C22×C412yes—
126C2×C4×C101C22×C412yes—
127C2×C4×C101C2×D412yes—
128C2×C4×C101#4112yes—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—28no—
138C2×Dic101—28no—
139C2×Dic101—21no—
140C2×Dic101—21no—
141C2×Dic101—21no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101C22×C412yes—
146C2×Dic101C22×C412yes—
147C2×Dic101C22×C412yes—
148C2×Dic101C22×C412yes—
149C2×Dic101C2×D412yes—
150C2×Dic101C2×D412yes—
151C2×Dic101C2×Q812yes—
152C2×Dic101C2×Q812yes—
153C2×Dic101#4112yes—
154C2×Dic101#4112yes—
155Dic202—28no—
156C22×C101—22no—
157C22×C101—22no—
158C22×C101—22no—
159C22×C101—22no—
160C22×C101—22no—
161C22×C101—22no—
162C22×C101C4×Q814yescomplemented
163C22×C101C23×C411yescharacteristic
164C22×C101C22×D411yescharacteristic
165C22×C101C22×Q811yescharacteristic
166C22×C101C2×#4111yescharacteristic
167C22×C101C2×#4111yescharacteristic
168C22×C101C2×#4111yescharacteristic
169C22×C101C2×#4111yescharacteristic
170C4×C101—22no—
171C4×C101—22no—
172C4×C101—22no—
173C4×C101—22no—
174C4×C101—41no—
175C4×C101—41no—
176Dic101—22no—
177Dic101—22no—
178Dic101—22no—
179Dic101—22no—
180Dic101—41no—
181Dic101—41no—
182Dic101C4×D414yes—
183Dic101C22⋊Q814yescomplemented
184C2×C101—24no—
185C2×C101C2×C4×D411yescharacteristic
186C2×C101C2×C4×Q811yescharacteristic
187C2×C101C4×#4111yescharacteristic
188C2×C101C2×C22⋊Q811yescharacteristic
189C2×C101#52511yescharacteristic
190C2×C101#52911yescharacteristic
191C2×C101#53311yescharacteristic
192C4×C22⋊Q8—1011nomaximal, Sylow
193C101C4×C22⋊Q811yescharacteristic, complemented, Sylow
194C2×C22⋊Q8—1011no—
195C2×C4×Q8—1011no—
196C22×C42—1011no—
197#391—1011no—
198#389—1011no—
199#387—1011no—
200#387—1011no—
201#385—1011no—
202#385D10111yescharacteristic
203#380—1011no—
204C4×C4⋊C4—1011no—
205C4×C4⋊C4—1011no—
206C4×C4⋊C4—1011no—
207C4×C22⋊C4—1011no—
208C4×C22⋊C4—1011no—
209C22×Q8—1011no—
210C23×C4—1011no—
211C23×C4—1011no—
212C23×C4C2×D10111yescharacteristic
213C22⋊Q8—1018no—
214C4×Q8—2024no—
215C2×C4⋊C4—1011no—
216C2×C4⋊C4—1011no—
217C2×C4⋊C4—1011no—
218C2×C4⋊C4—1011no—
219C2×C4⋊C4—1011no—
220C2×C4⋊C4—1011no—
221C2×C4⋊C4—1011no—
222C2×C4⋊C4C2×D10111yescharacteristic
223C2×C4⋊C4C2×D10111yescharacteristic
224C2×C22⋊C4—1011no—
225C2×C22⋊C4—1011no—
226C2×C22⋊C4C2×D10111yescharacteristic
227C2×C22⋊C4C2×D10111yescharacteristic
228C2×C42—1011no—
229C2×C42—1011no—
230C2×C42—1011no—
231C2×C42—1011no—
232C2×C42—1011no—
233C2×C42—1011no—
234C2×C42—1011no—
235C2×C42—1011no—
236C2×C42—2024no—
237#95—1011no—
238#95—1011no—
239#95—1011no—
240#95—1011no—
241#95—1011no—
242#95—1011no—
243#95—1011no—
244#95—1011no—
245#95C2×D10111yescharacteristic
246#95C2×D10111yescharacteristic
247C24C22×D10111yescharacteristic
248C2×Q8—1014no—
249C2×Q8—2022no—
250C2×Q8—2022no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—1011no—
255C22×C4—1011no—
256C22×C4—1014no—
257C22×C4—1014no—
258C22×C4—1011no—
259C22×C4—1011no—
260C22×C4—1011no—
261C22×C4—1011no—
262C22×C4—1011no—
263C22×C4—1011no—
264C22×C4—2022no—
265C22×C4—2022no—
266C22×C4—2022no—
267C22×C4—2022no—
268C22×C4C22×D10111yescharacteristic
269C22×C4C22×D10111yescharacteristic
270C22×C4C22×D10111yescharacteristic
271C22×C4C22×D10111yescharacteristic
272C22×C4C22×D10111yescharacteristic
273C22×C4C22×D10111yescharacteristic
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—1014no—
277C4⋊C4—1014no—
278C4⋊C4—1014no—
279C4⋊C4—2022no—
280C4⋊C4—2022no—
281C4⋊C4—2022no—
282C4⋊C4—2022no—
283C22⋊C4—22no—
284C22⋊C4—1014no—
285C22⋊C4—2022no—
286C22⋊C4C4×D10114yes—
287C42—1014no—
288C42—2022no—
289C42—2022no—
290C42—2022no—
291C42—2022no—
292C42—2022no—
293C42—2022no—
294C42—2022no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23C2×C4×D10112yes—
300C23Q8×D10112yes—
301C23#1310807312yes—
302C23C23×D10111yescharacteristic
303Q8—2028no—
304C2×C4—28no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—1012no—
320C2×C4—1012no—
321C2×C4—1012no—
322C2×C4—1012no—
323C2×C4—1012no—
324C2×C4—1012no—
325C2×C4—1012no—
326C2×C4—1012no—
327C2×C4—1012no—
328C2×C4—1012no—
329C2×C4—2028no—
330C2×C4—2028no—
331C2×C4—2021no—
332C2×C4—2021no—
333C2×C4—2021no—
334C2×C4—2021no—
335C2×C4—2021no—
336C2×C4—2021no—
337C2×C4—2021no—
338C2×C4—2021no—
339C2×C4—2021no—
340C2×C4—2021no—
341C2×C4—2021no—
342C2×C4—2021no—
343C2×C4—2021no—
344C2×C4—2021no—
345C2×C4C2×C4×D10112yes—
346C2×C4C2×C4×D10112yes—
347C2×C4D4×D10112yes—
348C2×C4#1310807312yes—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22#1569222014yescomplemented
356C22C22×C4×D10111yescharacteristic
357C22C2×D4×D10111yescharacteristic
358C22C2×Q8×D10111yescharacteristic
359C22#41×D10111yescharacteristic
360C22#41×D10111yescharacteristic
361C22C2×#1310807311yescharacteristic
362C22C2×#1310807311yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—1014no—
370C4—1014no—
371C4—4041no—
372C4—4041no—
373C4—2022no—
374C4—2022no—
375C4—2022no—
376C4—2022no—
377C2—24no—
378C2C4×D4×D10111yescharacteristic
379C2C2×#1569222011yescharacteristic
380C2C4×#1310807311yescharacteristic
381C2C22⋊Q8×D10111yescharacteristic
382C2#2026568311yescharacteristic
383C2#2026570411yescharacteristic
384C2#2026572911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#385×C101C2
socle, cosocleC23×C101C24
prime radicalC24×C101C23
hypercenter#385D101
hypocenterC101C4×C22⋊Q8
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#15692220C2semidirect
4. #20265081C2semidirect
5. #20265072C2semidirect
6. #20265062C2semidirect
7. #20265058C2semidirect
10. C4⋊C4×Dic101C2semidirect
11. #20265028C2semidirect
12. #20265028C2semidirect
19. #15692232C4semidirect
56. C2×Dic202C2×C4semidirect
62. C22×Dic101Q8semidirect
71. C4⋊Dic101C2×C4semidirect
76. #13108053C2×C4semidirect
77. #13108053C2×C4semidirect
162. C22×C101C4×Q8semidirect
183. Dic101C22⋊Q8semidirect
193. C101C4×C22⋊Q8semidirect
355. C22#15692220semidirect

Maximal subgroups

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

subgroupindex
C2×#156922322
C2×#156922202
C22×C4×Dic1012
#202650812
#202650722
#202650622
#202650582
#202650502
#202650352
C4⋊C4×Dic1012
#202650282
C22⋊C4×Dic1012
#385×C1012
C4×C22⋊Q8101