Finite Group Databasebeta

← #27181276

Subgroups — #27181276

Order 12928.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265722C211yesmaximal, characteristic, complemented
2#41×Dic101C211yesmaximal, characteristic, complemented
3#20265502C211yesmaximal, characteristic, complemented
4#20265498C211yesmaximal, characteristic, complemented
5#20265460C211yesmaximal, characteristic, complemented
6#20265455C211yesmaximal, characteristic, complemented
7#20265375C211yesmaximal, characteristic, complemented
8#20265370C211yesmaximal, characteristic, complemented
9#20265322C211yesmaximal, characteristic, complemented
10#20265289C211yesmaximal, characteristic, complemented
11#20265286C211yesmaximal, characteristic, complemented
12#20265280C211yesmaximal, characteristic, complemented
13#20265278C211yesmaximal, characteristic, complemented
14#20265231C211yesmaximal, characteristic, complemented
15#449×C101C211yesmaximal, characteristic, complemented
16C2×#13108073—21no—
17C2×#13108061C2211yescharacteristic
18C2×#13108059C2211yescharacteristic
19#15692254C2211yescharacteristic
20#15692251C2211yescharacteristic
21#15692241—21no—
22#15692229C2211yescharacteristic
23#15692227—21no—
24Q8×Dic101C2211yescharacteristic
25Q8×Dic101C2211yescharacteristic
26D4×Dic101—21no—
27D4×Dic101—21no—
28D4×Dic101C2211yescharacteristic
29D4×Dic101C2211yescharacteristic
30D4×Dic101C2211yescharacteristic
31#15692217C2211yescharacteristic
32#15692215—21no—
33#15692214—21no—
34#15692214C2211yescharacteristic
35C2×C4×Dic101—21no—
36C2×C4×Dic101C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692165C2211yescharacteristic
51#15692165C2211yescharacteristic
52C2×#41×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C2×D8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59#13108073—44no—
60C2×Dic202—21no—
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C22×Dic101—21no—
66C22×Dic101—21no—
67C22×Dic101—21no—
68C22×Dic101D411yescharacteristic
69C22×Dic101D411yescharacteristic
70C2×C4×D101—21no—
71#13108061—22no—
72#13108059—22no—
73C2×#11717579C2311yescharacteristic
74C2×#11717579C2311yescharacteristic
75C4⋊Dic101—21no—
76C4⋊Dic101—21no—
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81C22⋊Dic101—21no—
82C22⋊Dic101—21no—
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85#13108051—21no—
86#13108051—21no—
87C4×Dic101—21no—
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101D411yescharacteristic
91C4×Dic101D411yescharacteristic
92C4×Dic101C2311yescharacteristic
93C4×Dic101C2311yescharacteristic
94#41×C101—22no—
95#41×C101—22no—
96C2×Q8×C101C2311yescharacteristic
97C2×D4×C101—21no—
98C2×D4×C101—21no—
99C2×D4×C101C2311yescharacteristic
100C2×D4×C101C2311yescharacteristic
101C22×C4×C101—21no—
102C22×C4×C101C2311yescharacteristic
103SD16×C101—22no—
104D8×C101—22no—
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101C2311yescharacteristic
108C22⋊C4×C101—21no—
109C23×C101—21no—
110C23×C101—21no—
111C23×C101C2×D411yescharacteristic
112C22×D101—21no—
113Q8×C101—21no—
114Q8×C101#4112yescomplemented
115D4×C101—24no—
116D4×C101—21no—
117D4×C101—21no—
118D4×C101—21no—
119D4×C101—42no—
120D4×C101#4112yescomplemented
121C2×C4×C101—24no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101C2×D411yescharacteristic
127C2×C4×C101C2411yescharacteristic
128C22⋊D101—42no—
129C22⋊D101—42no—
130C22⋊D101—42no—
131C2×Dic101—22no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—41no—
136C2×Dic101—42no—
137C2×Dic101—42no—
138C2×Dic101—42no—
139C2×Dic101C2×D411yescharacteristic
140C2×Dic101C2×D411yescharacteristic
141C2×Dic101C2×D411yescharacteristic
142C2×Dic101C2×D411yescharacteristic
143D404—21no—
144D404—21no—
145C4×D101—42no—
146Dic202—42no—
147C8×C101—21no—
148C8×C101—21no—
149#11717579—21no—
150#11717579—21no—
151C22×C101—21no—
152C22×C101—22no—
153C22×C101—21no—
154C22×C101—21no—
155C22×C101—41no—
156C22×C101—41no—
157C22×C101—41no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101C22×D411yescharacteristic
161C2×D101—41no—
162C2×D101—41no—
163C2×D101—41no—
164C4×C101—21no—
165C4×C101—22no—
166C4×C101—41no—
167C4×C101C2×#4111yescharacteristic
168C4×C101C2×#4111yescharacteristic
169Dic101—21no—
170Dic101—21no—
171Dic101—21no—
172Dic101—41no—
173Dic101—41no—
174Dic101—41no—
175Dic101AGL(1,ℤ/8)12yes—
176C2×C101—41no—
177C2×C101—42no—
178C2×C101—81no—
179C2×C101#52511yescharacteristic
180C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C101#57511yescharacteristic
182D101—81no—
183#3115—1011nomaximal, Sylow
184C101#311511yescharacteristic, complemented, Sylow
185#532—1011no—
186C4×#41—1011no—
187#482—1011no—
188#482—1011no—
189#475—1011no—
190#474—1011no—
191#460—1011no—
192#459—1011no—
193#449—1011no—
194#449D10111yescharacteristic, complemented
195#442—1011no—
196#442—1011no—
197#440—1011no—
198#440—1011no—
199#432—1011no—
200C2×#41—2021no—
201C2×#41C2×D10111yescharacteristic
202C2×SD16—1011no—
203C2×SD16C2×D10111yescharacteristic
204C2×D8—1011no—
205C2×D8C2×D10111yescharacteristic
206C4⋊Q8—1011no—
207#127—1011no—
208#124—2021no—
209#121—1011no—
210#121—2021no—
211#121C2×D10111yescharacteristic
212C4×Q8—1011no—
213C4×Q8—1011no—
214C4×D4—1011no—
215C4×D4—1011no—
216C4×D4—1011no—
217C4×D4—1011no—
218C4×D4—2021no—
219C4×D4—2021no—
220C4×D4—2021no—
221#117—1011no—
222#117—2021no—
223C2×C42—1011no—
224C2×C42—2021no—
225#107—1011no—
226#107—1011no—
227#106—1011no—
228#106—1011no—
229C4⋊C8—1011no—
230C4⋊C8—1011no—
231#103—1011no—
232#103C2×D10111yescharacteristic
233#102—1011no—
234#102—1011no—
235#102—1011no—
236#102—1011no—
237#102—1011no—
238#102C2×D10111yescharacteristic
239C22⋊C8—1011no—
240C22⋊C8C2×D10111yescharacteristic
241#97—1011no—
242#97—1011no—
243#41—22no—
244#41—22no—
245#41—4044no—
246C2×Q8—2021no—
247C2×Q8C22×D10111yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—1011no—
251C2×D4—2021no—
252C2×D4—2021no—
253C2×D4—2021no—
254C2×D4C22×D10111yescharacteristic
255C2×D4C22×D10111yescharacteristic
256C22×C4—21no—
257C22×C4—1011no—
258C22×C4—1011no—
259C22×C4—2021no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4—2021no—
263C22×C4C22×D10111yescharacteristic
264SD16—22no—
265SD16—2022no—
266D8—22no—
267D8—2022no—
268C2×C8—1011no—
269C2×C8—1011no—
270C2×C8C22×D10111yescharacteristic
271C2×C8C22×D10111yescharacteristic
272C4⋊C4—1011no—
273C4⋊C4—1011no—
274C4⋊C4—1011no—
275C4⋊C4—2021no—
276C4⋊C4—2021no—
277C4⋊C4—2021no—
278C4⋊C4C22×D10111yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C42—1011no—
287C42—1011no—
288C42—1011no—
289C42—1011no—
290C42—2021no—
291C42—2021no—
292C42—2021no—
293C23—21no—
294C23—21no—
295C23—2021no—
296C23D4×D10111yescharacteristic
297Q8—21no—
298Q8—4042no—
299Q8#1310807312yescomplemented
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—4042no—
306D4—4042no—
307D4—4042no—
308D4—2021no—
309D4—2021no—
310D4#1310807312yescomplemented
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—1011no—
317C2×C4—1011no—
318C2×C4—1011no—
319C2×C4—1011no—
320C2×C4—4042no—
321C2×C4—4042no—
322C2×C4—4042no—
323C2×C4—4042no—
324C2×C4—4041no—
325C2×C4—2022no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4D4×D10111yescharacteristic
330C2×C4C23×D10111yescharacteristic
331C8—21no—
332C8—21no—
333C8—2021no—
334C8—2021no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—4041no—
345C22—4041no—
346C22—4041no—
347C22C2×D4×D10111yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—1012no—
352C4—4041no—
353C4—4041no—
354C4—4041no—
355C4—2021no—
356C4—2021no—
357C4—2021no—
358C4C2×#1310807311yescharacteristic
359C4C2×#1310807311yescharacteristic
360C2—41no—
361C2—42no—
362C2—81no—
363C2—8081no—
364C2#2026568311yescharacteristic
365C2AGL(1,ℤ/8)×D10111yescharacteristic
366C2#2026597411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#449×C101C2
socleC22×C101C22×D4
hypercenter#449D101
hypocenterC101#3115
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #20265722C2semidirect
2. #41×Dic101C2semidirect
3. #20265502C2semidirect
4. #20265498C2semidirect
5. #20265460C2semidirect
6. #20265455C2semidirect
7. #20265375C2semidirect
8. #20265370C2semidirect
9. #20265322C2semidirect
10. #20265289C2semidirect
11. #20265286C2semidirect
12. #20265280C2semidirect
13. #20265278C2semidirect
14. #20265231C2semidirect
15. #449×C101C2semidirect
114. Q8×C101#41semidirect
120. D4×C101#41semidirect
184. C101#3115semidirect
194. #449D101semidirect
299. Q8#13108073semidirect
310. D4#13108073semidirect

Maximal subgroups

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

subgroupindex
#202657222
#41×Dic1012
#202655022
#202654982
#202654602
#202654552
#202653752
#202653702
#202653222
#202652892
#202652862
#202652802
#202652782
#202652312
#449×C1012
#3115101