Finite Group Databasebeta

← #27173864

Subgroups — #27173864

Order 12928.

528 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692231C211yesmaximal, characteristic
2#20265084C211yesmaximal, characteristic, complemented
3#20265074C211yesmaximal, characteristic, complemented
4#20265072C211yesmaximal, characteristic, complemented
5#20265062C211yesmaximal, characteristic, complemented
6#20265061C211yesmaximal, characteristic, complemented
7#20265060C211yesmaximal, characteristic, complemented
8#20265059C211yesmaximal, characteristic, complemented
9#20265051C211yesmaximal, characteristic
10#20265048C211yesmaximal, characteristic
11#20265036C211yesmaximal, characteristic
12C4×#13108051C211yesmaximal, characteristic
13#20265024C211yesmaximal, characteristic
14#95×D101C211yesmaximal, characteristic
15#391×C101C211yesmaximal, characteristic, complemented
16C22×C4×D101C2211yescharacteristic
17C22×C4×D101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19#15692231C418yescomplemented
20C2×C4⋊Dic101C2211yescharacteristic
21C2×C4⋊Dic101C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×C4×Dic101C2211yescharacteristic
33C2×C4×Dic101C2211yescharacteristic
34C2×C4×Dic101C2211yescharacteristic
35#15692162C2211yescharacteristic
36#15692162C2211yescharacteristic
37#15692162C2211yescharacteristic
38#15692162C2211yescharacteristic
39#15692162C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45C22×Q8×C101C2211yescharacteristic
46C2×C4⋊C4×C101C2211yescharacteristic
47C2×C42×C101C2211yescharacteristic
48#95×C101C2211yescharacteristic
49#95×C101C2211yescharacteristic
50#95×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52C23×D101C2311yescharacteristic
53C22×Dic101C2311yescharacteristic
54C22×Dic101C2311yescharacteristic
55C22×Dic101C2311yescharacteristic
56C22×Dic101C2311yescharacteristic
57C22×Dic101C2311yescharacteristic
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C2×C4×D101—22no—
61C2×C4×D101—41no—
62C2×C4×D101—41no—
63C2×C4×D101—41no—
64C2×C4×D101—41no—
65C2×C4×D101—41no—
66C2×C4×D101—41no—
67C2×C4×D101C2×C414yes—
68C2×C4×D101Q814yescomplemented
69C4⋊Dic101—41no—
70C4⋊Dic101C2×C414yescomplemented
71#13108053—41no—
72#13108053—41no—
73#13108053—41no—
74#13108053—41no—
75#13108053C2×C414yescomplemented
76#13108053C2×C414yescomplemented
77#13108051—22no—
78#13108051—22no—
79#13108051C2×C414yes—
80#13108051C2×C414yes—
81C4×Dic101—41no—
82C4×Dic101—41no—
83C4×Dic101—41no—
84C2×Q8×C101—41no—
85C2×Q8×C101—41no—
86C2×Q8×C101C2×C414yescomplemented
87C22×C4×C101C2311yescharacteristic
88C22×C4×C101C2311yescharacteristic
89C22×C4×C101C2311yescharacteristic
90C22×C4×C101C2311yescharacteristic
91C22×C4×C101C2311yescharacteristic
92C22×C4×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C4⋊C4×C101—41no—
95C42×C101—41no—
96C23×C101C2411yescharacteristic
97C22×D101—21no—
98C22×D101—21no—
99C22×D101—21no—
100C22×D101—21no—
101C22×D101C22×C412yes—
102C22×D101C2×Q812yes—
103C22×D101#4112yes—
104Q8×C101—44no—
105C2×C4×C101—21no—
106C2×C4×C101—21no—
107C2×C4×C101—21no—
108C2×C4×C101—21no—
109C2×C4×C101—21no—
110C2×C4×C101—21no—
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×C101C22×C412yes—
123C2×C4×C101C22×C412yes—
124C2×C4×C101C22×C412yes—
125C2×C4×C101C2×Q812yes—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
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—21no—
138C2×Dic101—21no—
139C2×Dic101—21no—
140C2×Dic101—21no—
141C2×Dic101—21no—
142C2×Dic101—21no—
143C2×Dic101C22×C412yes—
144C2×Dic101C22×C412yes—
145C2×Dic101C22×C412yes—
146C2×Dic101C2×Q812yes—
147C4×D101—44no—
148C4×D101—44no—
149C4×D101—44no—
150C22×C101C23×C411yescharacteristic
151C22×C101C22×Q811yescharacteristic
152C22×C101C2×#4111yescharacteristic
153C22×C101H(5,𝔽2)11yescharacteristic
154C22×C101H(5,𝔽2)11yescharacteristic
155C22×C101H(5,𝔽2)11yescharacteristic
156C22×C1012−1+411yescharacteristic
157C2×D101—22no—
158C2×D101—22no—
159C2×D101—22no—
160C2×D101—22no—
161C2×D101—22no—
162C2×D101—22no—
163C2×D101C4×Q814yes—
164C4×C101—22no—
165C4×C101—22no—
166C4×C101—22no—
167C4×C101—22no—
168C4×C101—41no—
169C4×C101—41no—
170C4×C101—41no—
171Dic101—22no—
172Dic101—22no—
173Dic101—22no—
174Dic101—22no—
175Dic101—41no—
176Dic101—41no—
177Dic101—41no—
178C2×C101C2×C4×Q811yescharacteristic
179C2×C101#51811yescharacteristic
180C2×C101#52011yescharacteristic
181C2×C101#53811yescharacteristic
182C2×C101#53911yescharacteristic
183C2×C101#54311yescharacteristic
184C2×C101#54411yescharacteristic
185D101—24no—
186#2329—1011nomaximal, Sylow
187C101#232911yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—1011no—
189#391—1011no—
190#391D10111yescharacteristic, complemented
191#389—1011no—
192#389—1011no—
193#387—1011no—
194#387—1011no—
195#387—1011no—
196#387—1011no—
197#385—1011no—
198#385—1011no—
199#380—1011no—
200C4×C22⋊C4—1011no—
201C4×C22⋊C4—1011no—
202C2×#95—1011no—
203C22×Q8C2×D10111yescharacteristic
204C23×C4—1011no—
205C23×C4—1011no—
206C23×C4—1011no—
207C22⋊Q8—1018no—
208C2×C4⋊C4—1011no—
209C2×C4⋊C4—1011no—
210C2×C4⋊C4—1011no—
211C2×C4⋊C4—1011no—
212C2×C4⋊C4—1011no—
213C2×C4⋊C4—1011no—
214C2×C4⋊C4—1011no—
215C2×C4⋊C4—1011no—
216C2×C4⋊C4C2×D10111yescharacteristic
217C2×C22⋊C4—1011no—
218C2×C22⋊C4—1011no—
219C2×C22⋊C4—1011no—
220C2×C22⋊C4—1011no—
221C2×C42—1011no—
222C2×C42—1011no—
223C2×C42—1011no—
224C2×C42C2×D10111yescharacteristic
225#95—1011no—
226#95—1011no—
227#95—1011no—
228#95—1011no—
229#95—1011no—
230#95—1011no—
231#95—1011no—
232#95—1011no—
233#95—1011no—
234#95—1011no—
235#95C2×D10111yescharacteristic
236#95C2×D10111yescharacteristic
237#95C2×D10111yescharacteristic
238#95C2×D10111yescharacteristic
239C24—1011no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8C4×D10114yescomplemented
243C22×C4—1011no—
244C22×C4—1011no—
245C22×C4—1011no—
246C22×C4—1011no—
247C22×C4—1014no—
248C22×C4—1014no—
249C22×C4—1011no—
250C22×C4—1011no—
251C22×C4—1011no—
252C22×C4—4041no—
253C22×C4—4041no—
254C22×C4—4041no—
255C22×C4—4041no—
256C22×C4—4041no—
257C22×C4—4041no—
258C22×C4—2022no—
259C22×C4C22×D10111yescharacteristic
260C22×C4C22×D10111yescharacteristic
261C22×C4C22×D10111yescharacteristic
262C22×C4C22×D10111yescharacteristic
263C22×C4C22×D10111yescharacteristic
264C22×C4C22×D10111yescharacteristic
265C22×C4C22×D10111yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—1014no—
268C4⋊C4—1014no—
269C4⋊C4—1014no—
270C4⋊C4—4041no—
271C4⋊C4—4041no—
272C4⋊C4—4041no—
273C4⋊C4—4041no—
274C4⋊C4—4041no—
275C22⋊C4—1014no—
276C22⋊C4—1014no—
277C22⋊C4—2022no—
278C22⋊C4—2022no—
279C42—41no—
280C42—4041no—
281C42—4041no—
282C42—4041no—
283C23—1012no—
284C23—1012no—
285C23—1012no—
286C23—2021no—
287C23—2021no—
288C23—2021no—
289C23—2021no—
290C23C23×D10111yescharacteristic
291Q8—44no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—1012no—
310C2×C4—1012no—
311C2×C4—1012no—
312C2×C4—1012no—
313C2×C4—4044no—
314C2×C4—4044no—
315C2×C4—4044no—
316C2×C4—2021no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4—2021no—
323C2×C4—2021no—
324C2×C4—2021no—
325C2×C4—2021no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4—2021no—
330C2×C4—2021no—
331C2×C4—2021no—
332C2×C4—2021no—
333C2×C4C2×C4×D10112yes—
334C2×C4C2×C4×D10112yes—
335C2×C4C2×C4×D10112yes—
336C2×C4Q8×D10112yes—
337C22—1014no—
338C22—2022no—
339C22—2022no—
340C22—2022no—
341C22—2022no—
342C22—2022no—
343C22—2022no—
344C22C22×C4×D10111yescharacteristic
345C22C2×Q8×D10111yescharacteristic
346C22#41×D10111yescharacteristic
347C22#1569230011yescharacteristic
348C22#1569230011yescharacteristic
349C22#1569230111yescharacteristic
350C22#1569230311yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—4041no—
359C4—4041no—
360C4—4041no—
361C4—2022no—
362C4—2022no—
363C4—2022no—
364C4—2022no—
365C2—2024no—
366C2C4×Q8×D10111yescharacteristic
367C2#2026565011yescharacteristic
368C2#2026565911yescharacteristic
369C2#2026576111yescharacteristic
370C2#2026577311yescharacteristic
371C2#2026580211yescharacteristic
372C2#2026580411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#391×C101C2
socle, cosocleC23×C101C24
prime radicalC23×D101C23
hypercenter#391D101
hypocenterC101#2329
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #20265084C2semidirect
3. #20265074C2semidirect
4. #20265072C2semidirect
5. #20265062C2semidirect
6. #20265061C2semidirect
7. #20265060C2semidirect
8. #20265059C2semidirect
15. #391×C101C2semidirect
19. #15692231C4semidirect
68. C2×C4×D101Q8semidirect
70. C4⋊Dic101C2×C4semidirect
75. #13108053C2×C4semidirect
76. #13108053C2×C4semidirect
86. C2×Q8×C101C2×C4semidirect
187. C101#2329semidirect
190. #391D101semidirect
242. C2×Q8C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922312
#202650842
#202650742
#202650722
#202650622
#202650612
#202650602
#202650592
#202650512
#202650482
#202650362
C4×#131080512
#202650242
#95×D1012
#391×C1012
#2329101