Finite Group Databasebeta

← #27181302

Subgroups — #27181302

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#20265706C211yesmaximal, characteristic
2C4×Q8×D101C211yesmaximal, characteristic, complemented
3#20265511C211yesmaximal, characteristic, complemented
4#20265494C211yesmaximal, characteristic
5#20265485C211yesmaximal, characteristic, complemented
6#20265484C211yesmaximal, characteristic, complemented
7#20265396C211yesmaximal, characteristic, complemented
8#20265331C211yesmaximal, characteristic, complemented
9#20265314C211yesmaximal, characteristic
10#20265284C211yesmaximal, characteristic, complemented
11#20265281C211yesmaximal, characteristic, complemented
12#20265278C211yesmaximal, characteristic, complemented
13#20265274C211yesmaximal, characteristic, complemented
14#20265233C211yesmaximal, characteristic, complemented
15#464×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19#15692248—21no—
20#15692247C2211yescharacteristic
21#15692237—21no—
22#15692231C2211yescharacteristic
23#15692226C2211yescharacteristic
24Q8×Dic101C2211yescharacteristic
25Q8×Dic101C2211yescharacteristic
26C4×Dic202—21no—
27C4×Dic202C2211yescharacteristic
28#15692220—21no—
29#15692220C2211yescharacteristic
30D4×Dic101C2211yescharacteristic
31#15692215—21no—
32#15692211—21no—
33C4⋊C4×D101—21no—
34C4⋊C4×D101C2211yescharacteristic
35C42×D101—21no—
36C42×D101C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41C4⋊#11717579C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692167C2211yescharacteristic
49#15692167C2211yescharacteristic
50#15692165C2211yescharacteristic
51#15692165C2211yescharacteristic
52C2×Q16×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54#124×C101C2211yescharacteristic
55C4×Q8×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61Q8×D101—22no—
62Q8×D101—22no—
63C2×C22⋊D101—21no—
64C22×Dic101—21no—
65C2×C4×D101—21no—
66C2×C4×D101—21no—
67C2×C4×D101D411yescharacteristic
68C2×C4×D101D411yescharacteristic
69C2×C4×D101C2311yescharacteristic
70#13108064—22no—
71#13108062—22no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C4⋊Dic101—21no—
75C4⋊Dic101—21no—
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87#13108051—21no—
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101D411yescharacteristic
93C4×Dic101D411yescharacteristic
94C4×Dic101C2311yescharacteristic
95C2×Q8×C101C2311yescharacteristic
96C2×Q8×C101C2311yescharacteristic
97C2×D4×C101C2311yescharacteristic
98Q16×C101—22no—
99SD16×C101—22no—
100C2×C8×C101C2311yescharacteristic
101C2×C8×C101C2311yescharacteristic
102C4⋊C4×C101—21no—
103C4⋊C4×C101C2311yescharacteristic
104C22⋊C4×C101—21no—
105C22⋊C4×C101—21no—
106C42×C101—21no—
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C22×D101C2×D411yescharacteristic
110Q8×C101—21no—
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×C101#4112yes—
114D4×C101—21no—
115D4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101C2×D411yescharacteristic
120C2×C4×C101C2×D411yescharacteristic
121C2×C4×C101C2411yescharacteristic
122C22⋊D101—42no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—21no—
127C2×Dic101—42no—
128C2×Dic101C2×D411yescharacteristic
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131C4×D101—24no—
132C4×D101—22no—
133C4×D101—21no—
134C4×D101—21no—
135C4×D101—41no—
136C4×D101—42no—
137Dic202—24no—
138Dic202—21no—
139Dic202#4112yes—
140C8×C101—21no—
141C8×C101—21no—
142#11717579—21no—
143#11717579—21no—
144C22×C101—41no—
145C22×C101—41no—
146C22×C101—41no—
147C22×C101C22×D411yescharacteristic
148C2×D101—21no—
149C2×D101—21no—
150C2×D101#13712yes—
151C4×C101—22no—
152C4×C101—21no—
153C4×C101—21no—
154C4×C101—41no—
155C4×C101—41no—
156C4×C101C2×#4111yescharacteristic
157C4×C101C2×#4111yescharacteristic
158Dic101—21no—
159Dic101—22no—
160Dic101—21no—
161Dic101—21no—
162Dic101—41no—
163Dic101—41no—
164Dic101—41no—
165C2×C101—81no—
166C2×C101#52511yescharacteristic
167C2×C101C2×#13711yescharacteristic
168C2×C101#57511yescharacteristic
169D101—22no—
170#3117—1011nomaximal, Sylow
171C101#311711yescharacteristic, complemented, Sylow
172#529—1011no—
173C2×C4×Q8—1011no—
174#484—1011no—
175#481—1011no—
176#479—1011no—
177#479—1011no—
178#464—1011no—
179#464D10111yescharacteristic, complemented
180#451—1011no—
181#448—1011no—
182#441—1011no—
183#441—1011no—
184#440—1011no—
185#440—1011no—
186#432—1011no—
187C22×Q8—1011no—
188C2×Q16—1011no—
189C2×Q16C2×D10111yescharacteristic
190C2×SD16—1011no—
191C2×SD16C2×D10111yescharacteristic
192#126—2021no—
193#125—1011no—
194#124C2×D10111yescharacteristic
195#123—2021no—
196C22⋊Q8—1011no—
197#121—1011no—
198C4×Q8—1011no—
199C4×Q8—1011no—
200C4×Q8—1011no—
201C4×Q8—1011no—
202C4×Q8—2021no—
203C4×Q8—2021no—
204C4×Q8C2×D10111yescharacteristic
205C4×D4—1011no—
206C4×D4—2021no—
207#117—2021no—
208C2×C4⋊C4—1011no—
209C2×C4⋊C4—2021no—
210C2×C42—1011no—
211C2×C42—2021no—
212#107—1011no—
213#107—1011no—
214#106—1011no—
215#106—1011no—
216C4⋊C8—1011no—
217C4⋊C8C2×D10111yescharacteristic
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⋊C8—1011no—
228#97—1011no—
229#97—1011no—
230C2×Q8—1011no—
231C2×Q8—2021no—
232C2×Q8—2022no—
233C2×Q8—2022no—
234C2×Q8C22×D10111yescharacteristic
235C2×Q8C22×D10111yescharacteristic
236C2×D4—2021no—
237C2×D4C22×D10111yescharacteristic
238C22×C4—1011no—
239C22×C4—1011no—
240C22×C4—1011no—
241C22×C4—2021no—
242C22×C4—2021no—
243C22×C4—2021no—
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—21no—
267C22⋊C4—2021no—
268C22⋊C4—2021no—
269C22⋊C4—2021no—
270C42—21no—
271C42—1011no—
272C42—1011no—
273C42—1011no—
274C42—2021no—
275C42—2021no—
276C42—2021no—
277C42—2021no—
278C42C22×D10111yescharacteristic
279C23—21no—
280C23—1011no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—1012no—
285Q8—2024no—
286Q8—2021no—
287Q8#1310807312yes—
288D4—21no—
289D4—21no—
290D4—4042no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—1011no—
295C2×C4—1011no—
296C2×C4—1011no—
297C2×C4—4041no—
298C2×C4—4042no—
299C2×C4—4042no—
300C2×C4—2022no—
301C2×C4—2024no—
302C2×C4—2021no—
303C2×C4—2021no—
304C2×C4—2021no—
305C2×C4—2021no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4D4×D10111yescharacteristic
309C2×C4D4×D10111yescharacteristic
310C2×C4C23×D10111yescharacteristic
311C8—21no—
312C8—21no—
313C8—2021no—
314C8—2021no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—1012no—
319C22—2021no—
320C22—2021no—
321C22C2×D4×D10111yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—4041no—
328C4—4041no—
329C4—4041no—
330C4—2021no—
331C4—2022no—
332C4—2021no—
333C4—2021no—
334C4#41×D10111yescharacteristic
335C4C2×#1310807311yescharacteristic
336C2—81no—
337C2—2022no—
338C2#2026568711yescharacteristic
339C2#137×D10111yescharacteristic
340C2#2026597411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#464×C101C2
socleC22×C101C22×D4
prime radical#15692226C22
hypercenter#464D101
hypocenterC101#3117
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D101C2semidirect
3. #20265511C2semidirect
5. #20265485C2semidirect
6. #20265484C2semidirect
7. #20265396C2semidirect
8. #20265331C2semidirect
10. #20265284C2semidirect
11. #20265281C2semidirect
12. #20265278C2semidirect
13. #20265274C2semidirect
14. #20265233C2semidirect
15. #464×C101C2semidirect
171. C101#3117semidirect
179. #464D101semidirect

Maximal subgroups

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

subgroupindex
#202657062
C4×Q8×D1012
#202655112
#202654942
#202654852
#202654842
#202653962
#202653312
#202653142
#202652842
#202652812
#202652782
#202652742
#202652332
#464×C1012
#3117101