Finite Group Databasebeta

← #27181287

Subgroups — #27181287

Order 12928.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265729C211yesmaximal, characteristic, complemented
2#41×Dic101C211yesmaximal, characteristic
3#20265519C211yesmaximal, characteristic, complemented
4#20265515C211yesmaximal, characteristic, complemented
5#20265470C211yesmaximal, characteristic, complemented
6#20265465C211yesmaximal, characteristic, complemented
7#20265385C211yesmaximal, characteristic, complemented
8#20265380C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic
10#20265284C211yesmaximal, characteristic, complemented
11#20265281C211yesmaximal, characteristic, complemented
12#20265278C211yesmaximal, characteristic, complemented
13#20265274C211yesmaximal, characteristic, complemented
14#20265231C211yesmaximal, characteristic, complemented
15#452×C101C211yesmaximal, characteristic
16C2×#13108064C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18#15692254C2211yescharacteristic
19#15692253C2211yescharacteristic
20#15692245—21no—
21#15692234C2211yescharacteristic
22#15692232—21no—
23Q8×Dic101C2211yescharacteristic
24Q8×Dic101C2211yescharacteristic
25Q8×Dic101C2211yescharacteristic
26#15692220—21no—
27#15692220C2211yescharacteristic
28D4×Dic101—21no—
29D4×Dic101—21no—
30D4×Dic101C2211yescharacteristic
31D4×Dic101C2211yescharacteristic
32#15692214—21no—
33#15692214C2211yescharacteristic
34#15692211—21no—
35C2×C4×Dic101—21no—
36C2×C4×Dic101C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692179C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692165C2211yescharacteristic
51#15692165C2211yescharacteristic
52C2×#41×C101C2211yescharacteristic
53C2×Q16×C101C2211yescharacteristic
54C2×SD16×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C22×Dic101—21no—
62C22×Dic101—21no—
63C22×Dic101D411yescharacteristic
64C22×Dic101D411yescharacteristic
65#13108064—22no—
66#13108062—22no—
67C2×#11717579C2311yescharacteristic
68C2×#11717579C2311yescharacteristic
69C4⋊Dic101—21no—
70C4⋊Dic101—21no—
71C4⋊Dic101—21no—
72C4⋊Dic101C2311yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80C22⋊Dic101—21no—
81C22⋊Dic101—21no—
82C22⋊Dic101—21no—
83C22⋊Dic101—21no—
84C4×Dic101—21no—
85C4×Dic101—21no—
86C4×Dic101—21no—
87C4×Dic101—21no—
88C4×Dic101—21no—
89C4×Dic101D411yescharacteristic
90C4×Dic101D411yescharacteristic
91C4×Dic101C2311yescharacteristic
92C4×Dic101C2311yescharacteristic
93#41×C101—22no—
94#41×C101—22no—
95C2×Q8×C101C2311yescharacteristic
96C2×Q8×C101C2311yescharacteristic
97C2×D4×C101—21no—
98C2×D4×C101C2311yescharacteristic
99C22×C4×C101—21no—
100C22×C4×C101C2311yescharacteristic
101Q16×C101—22no—
102SD16×C101—22no—
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C4⋊C4×C101—21no—
106C4⋊C4×C101C2311yescharacteristic
107C22⋊C4×C101—21no—
108C23×C101—21no—
109C23×C101C2×D411yescharacteristic
110Q8×C101—21no—
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×C101#4112yes—
114D4×C101—24no—
115D4×C101—21no—
116D4×C101#4112yes—
117C2×C4×C101—24no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2411yescharacteristic
125C2×Dic101—22no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—41no—
131C2×Dic101—42no—
132C2×Dic101—42no—
133C2×Dic101C2×D411yescharacteristic
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137Dic202—21no—
138Dic202—21no—
139Dic202—42no—
140C8×C101—21no—
141C8×C101—21no—
142#11717579—21no—
143#11717579—21no—
144C22×C101—21no—
145C22×C101—22no—
146C22×C101—21no—
147C22×C101—21no—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101C22×D411yescharacteristic
151C4×C101—21no—
152C4×C101—22no—
153C4×C101—41no—
154C4×C101—41no—
155C4×C101C2×#4111yescharacteristic
156C4×C101C2×#4111yescharacteristic
157Dic101—21no—
158Dic101—21no—
159Dic101—21no—
160Dic101—41no—
161Dic101—41no—
162Dic101—41no—
163Dic101—41no—
164Dic101#13712yes—
165C2×C101—41no—
166C2×C101—42no—
167C2×C101#52511yescharacteristic
168C2×C101C2×#13711yescharacteristic
169C2×C101#57511yescharacteristic
170#3116—1011nomaximal, Sylow
171C101#311611yescharacteristic, complemented, Sylow
172#533—1011no—
173C4×#41—1011no—
174#485—1011no—
175#485—1011no—
176#477—1011no—
177#476—1011no—
178#462—1011no—
179#461—1011no—
180#452—1011no—
181#452D10111yescharacteristic
182#441—1011no—
183#441—1011no—
184#440—1011no—
185#440—1011no—
186#432—1011no—
187C2×#41C2×D10111yescharacteristic
188C2×Q16—1011no—
189C2×Q16C2×D10111yescharacteristic
190C2×SD16—1011no—
191C2×SD16C2×D10111yescharacteristic
192C4⋊Q8—1011no—
193C4⋊Q8—1011no—
194#125—2021no—
195C22⋊Q8—1011no—
196C22⋊Q8—2021no—
197C22⋊Q8C2×D10111yescharacteristic
198C4×Q8—1011no—
199C4×Q8—1011no—
200C4×Q8—1011no—
201C4×Q8—1011no—
202C4×Q8—2021no—
203C4×D4—1011no—
204C4×D4—1011no—
205C4×D4—2021no—
206C4×D4—2021no—
207#117—1011no—
208#117—2021no—
209#117—2021no—
210C2×C42—1011no—
211C2×C42—2021no—
212#107—1011no—
213#107—1011no—
214#106—1011no—
215#106—1011no—
216C4⋊C8—1011no—
217C4⋊C8—1011no—
218#103—1011no—
219#103—1011no—
220#103—1011no—
221#103—1011no—
222#103—1011no—
223#103C2×D10111yescharacteristic
224#102—1011no—
225#102C2×D10111yescharacteristic
226C22⋊C8—1011no—
227C22⋊C8C2×D10111yescharacteristic
228#97—1011no—
229#97—1011no—
230#41—22no—
231#41—22no—
232C2×Q8—1011no—
233C2×Q8—2021no—
234C2×Q8C22×D10111yescharacteristic
235C2×Q8C22×D10111yescharacteristic
236C2×D4—21no—
237C2×D4C22×D10111yescharacteristic
238C22×C4—21no—
239C22×C4—1011no—
240C22×C4—1011no—
241C22×C4—2021no—
242C22×C4—2021no—
243C22×C4C22×D10111yescharacteristic
244Q16—22no—
245Q16—2022no—
246SD16—22no—
247SD16—2022no—
248C2×C8—1011no—
249C2×C8—1011no—
250C2×C8C22×D10111yescharacteristic
251C2×C8C22×D10111yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—1011no—
254C4⋊C4—1011no—
255C4⋊C4—1011no—
256C4⋊C4—2021no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
259C4⋊C4—2021no—
260C4⋊C4—2021no—
261C4⋊C4—2021no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4C22×D10111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2021no—
267C22⋊C4—2021no—
268C22⋊C4—2021no—
269C22⋊C4—2021no—
270C42—1011no—
271C42—1011no—
272C42—1011no—
273C42—1011no—
274C42—2021no—
275C42—2021no—
276C42—2021no—
277C42—2021no—
278C42—2021no—
279C23—21no—
280C23D4×D10111yescharacteristic
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—4042no—
285Q8—2021no—
286Q8—2021no—
287Q8#1310807312yes—
288D4—24no—
289D4—21no—
290D4#1310807312yes—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—1011no—
298C2×C4—1011no—
299C2×C4—1011no—
300C2×C4—1011no—
301C2×C4—4042no—
302C2×C4—4042no—
303C2×C4—4041no—
304C2×C4—2022no—
305C2×C4—2021no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4D4×D10111yescharacteristic
310C2×C4C23×D10111yescharacteristic
311C8—21no—
312C8—21no—
313C8—2021no—
314C8—2021no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22C2×D4×D10111yescharacteristic
322C4—21no—
323C4—22no—
324C4—41no—
325C4—41no—
326C4—1012no—
327C4—4041no—
328C4—4041no—
329C4—4041no—
330C4—4041no—
331C4—2021no—
332C4—2021no—
333C4—2021no—
334C4C2×#1310807311yescharacteristic
335C4C2×#1310807311yescharacteristic
336C2—41no—
337C2—42no—
338C2#2026568311yescharacteristic
339C2#137×D10111yescharacteristic
340C2#2026597411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#452×C101C2
socleC22×C101C22×D4
prime radicalC2×#41×C101C22
hypercenter#452D101
hypocenterC101#3116
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #20265729C2semidirect
3. #20265519C2semidirect
4. #20265515C2semidirect
5. #20265470C2semidirect
6. #20265465C2semidirect
7. #20265385C2semidirect
8. #20265380C2semidirect
10. #20265284C2semidirect
11. #20265281C2semidirect
12. #20265278C2semidirect
13. #20265274C2semidirect
14. #20265231C2semidirect
171. C101#3116semidirect

Maximal subgroups

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

subgroupindex
#202657292
#41×Dic1012
#202655192
#202655152
#202654702
#202654652
#202653852
#202653802
#202653392
#202652842
#202652812
#202652782
#202652742
#202652312
#452×C1012
#3116101