Finite Group Databasebeta

← #27181340

Subgroups — #27181340

Order 12928.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265657C211yesmaximal, characteristic
2#20265510C211yesmaximal, characteristic, complemented
3#20265505C211yesmaximal, characteristic, complemented
4#20265481C211yesmaximal, characteristic, complemented
5#20265474C211yesmaximal, characteristic, complemented
6#20265384C211yesmaximal, characteristic, complemented
7#20265379C211yesmaximal, characteristic, complemented
8#20265337C211yesmaximal, characteristic
9#20265337C211yesmaximal, characteristic
10#20265282C211yesmaximal, characteristic, complemented
11#20265275C211yesmaximal, characteristic, complemented
12C4×#13108064C211yesmaximal, characteristic, complemented
13C4×#13108061C211yesmaximal, characteristic, complemented
14#20265238C211yesmaximal, characteristic, complemented
15#541×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108064C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20C2×#13108061C2211yescharacteristic
21C4×Dic202C2211yescharacteristic
22C4×Dic202C2211yescharacteristic
23C4×D404C2211yescharacteristic
24C4×D404C2211yescharacteristic
25#15692215—21no—
26#15692215—21no—
27#15692213—21no—
28C2×C4⋊Dic101C2211yescharacteristic
29C4⋊C4×D101—21no—
30#15692188C2211yescharacteristic
31#15692188C2211yescharacteristic
32#15692186C2211yescharacteristic
33#15692186C2211yescharacteristic
34C4⋊#11717579C2211yescharacteristic
35C4⋊#11717579C2211yescharacteristic
36#15692179C2211yescharacteristic
37#15692179C2211yescharacteristic
38#15692179C2211yescharacteristic
39#15692179C2211yescharacteristic
40#15692178C2211yescharacteristic
41#15692178C2211yescharacteristic
42#15692175C2211yescharacteristic
43#15692175C2211yescharacteristic
44C22⋊#11717579C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46#15692166C2211yescharacteristic
47C4×#11717579C2211yescharacteristic
48C4⋊Q8×C101C2211yescharacteristic
49#126×C101—21no—
50#126×C101—21no—
51#125×C101—21no—
52#125×C101—21no—
53#125×C101C2211yescharacteristic
54C22⋊Q8×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic
56C4×Q8×C101C2211yescharacteristic
57C4×Q8×C101C2211yescharacteristic
58#117×C101C2211yescharacteristic
59#13108074—22no—
60#13108074—22no—
61C2×Dic202C2311yescharacteristic
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64C22×Dic101—21no—
65C2×C4×D101—21no—
66C2×C4×D101—21no—
67C2×C4×D101—21no—
68#13108064—22no—
69#13108064—22no—
70#13108061—22no—
71#13108061—22no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C2×#11717579C2311yescharacteristic
75C2×#11717579C2311yescharacteristic
76C4⋊Dic101—21no—
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79#13108053—21no—
80#13108053—21no—
81#13108051—21no—
82#13108051—21no—
83C4×Dic101—21no—
84C4×Dic101—21no—
85C2×Q8×C101C2311yescharacteristic
86C2×Q8×C101C2311yescharacteristic
87C22×C4×C101C2311yescharacteristic
88C4⋊C4×C101—21no—
89C4⋊C4×C101—21no—
90C4⋊C4×C101—21no—
91C4⋊C4×C101—21no—
92C4⋊C4×C101—21no—
93C4⋊C4×C101—21no—
94C4⋊C4×C101—21no—
95C4⋊C4×C101—21no—
96C4⋊C4×C101D411yescharacteristic
97C4⋊C4×C101D411yescharacteristic
98C4⋊C4×C101C2311yescharacteristic
99C4⋊C4×C101C2311yescharacteristic
100C22⋊C4×C101—21no—
101C22⋊C4×C101—21no—
102C22⋊C4×C101D411yescharacteristic
103C22⋊C4×C101D411yescharacteristic
104C42×C101—21no—
105C42×C101—21no—
106C42×C101C2311yescharacteristic
107C42×C101C2311yescharacteristic
108C23×C101C2×D411yescharacteristic
109C22×D101—21no—
110Q8×C101—21no—
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×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×C101C2×D411yescharacteristic
121C2×C4×C101C2×D411yescharacteristic
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2411yescharacteristic
126C22⋊D101—24no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—81no—
131D404—21no—
132D404#4112yes—
133C4×D101—24no—
134C4×D101—81no—
135C4×D101—81no—
136Dic202—21no—
137Dic202#4112yes—
138#11717579—21no—
139#11717579—21no—
140#11717579—21no—
141#11717579—21no—
142C22×C101—21no—
143C22×C101—21no—
144C22×C101—21no—
145C22×C101C22×D411yescharacteristic
146C2×D101—22no—
147C2×D101—41no—
148C2×D101—41no—
149C4×C101—21no—
150C4×C101—21no—
151C4×C101—21no—
152C4×C101—21no—
153C4×C101—21no—
154C4×C101—41no—
155C4×C101—41no—
156C4×C101—41no—
157C4×C101—41no—
158C4×C101C2×#4111yescharacteristic
159C4×C101C2×#4111yescharacteristic
160Dic101—22no—
161Dic101—41no—
162Dic101—41no—
163C2×C101—41no—
164C2×C101#52511yescharacteristic
165C2×C101#57711yescharacteristic
166C2×C101#57811yescharacteristic
167D101—42no—
168#3120—1011nomaximal, Sylow
169C101#312011yescharacteristic, complemented, Sylow
170#541D10111yescharacteristic, complemented
171#520—1011no—
172#484—1011no—
173#483—1011no—
174#479—1011no—
175#478—1011no—
176#462—1011no—
177#461—1011no—
178#452—1011no—
179#452—1011no—
180#441—1011no—
181#440—1011no—
182C4×Q16—1011no—
183C4×SD16—1011no—
184#433—1011no—
185C2×#41—1011no—
186C2×Q16—1011no—
187C2×Q16—1011no—
188C2×SD16—1011no—
189C2×SD16—1011no—
190C4⋊Q8C2×D10111yescharacteristic
191#126—21no—
192#126—21no—
193#125—21no—
194#125—21no—
195#125C2×D10111yescharacteristic
196C22⋊Q8C2×D10111yescharacteristic
197C22⋊Q8C2×D10111yescharacteristic
198C4×Q8—1011no—
199C4×Q8—1011no—
200C4×Q8C2×D10111yescharacteristic
201C4×Q8C2×D10111yescharacteristic
202C4×D4—1011no—
203C4×D4—1011no—
204C4×D4—2021no—
205C4×D4—2021no—
206#117—2021no—
207#117C2×D10111yescharacteristic
208C2×C4⋊C4—1011no—
209C2×C4⋊C4—2021no—
210#107—1011no—
211#107—1011no—
212#106—1011no—
213#106—1011no—
214C4⋊C8—1011no—
215C4⋊C8—1011no—
216#103—1011no—
217#103—1011no—
218#103—1011no—
219#103—1011no—
220#103—1011no—
221#103—1011no—
222#102—1011no—
223#102—1011no—
224C22⋊C8—1011no—
225C22⋊C8—1011no—
226#97—1011no—
227C4×C8—1011no—
228#41—2022no—
229#41—2022no—
230C2×Q8—1011no—
231C2×Q8C22×D10111yescharacteristic
232C2×Q8C22×D10111yescharacteristic
233C2×D4—1011no—
234C2×D4—2021no—
235C22×C4—2021no—
236C22×C4—2021no—
237C22×C4—2021no—
238C22×C4—2021no—
239C22×C4C22×D10111yescharacteristic
240Q16—2022no—
241Q16—2022no—
242SD16—2022no—
243SD16—2022no—
244C2×C8—1011no—
245C2×C8—1011no—
246C2×C8—1011no—
247C2×C8—1011no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—1011no—
257C4⋊C4—1011no—
258C4⋊C4—2021no—
259C4⋊C4—2021no—
260C4⋊C4—2021no—
261C4⋊C4C22⋊D10111yescharacteristic
262C4⋊C4C22⋊D10111yescharacteristic
263C4⋊C4C22×D10111yescharacteristic
264C4⋊C4C22×D10111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—2021no—
268C22⋊C4—2021no—
269C22⋊C4C22⋊D10111yescharacteristic
270C22⋊C4C22⋊D10111yescharacteristic
271C42—21no—
272C42—21no—
273C42—2021no—
274C42—2021no—
275C42C22×D10111yescharacteristic
276C42C22×D10111yescharacteristic
277C23—2021no—
278C23C2×C22⋊D10111yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—1012no—
284Q8—2021no—
285D4—1012no—
286D4—2024no—
287D4—2021no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—8081no—
295C2×C4—8081no—
296C2×C4—8081no—
297C2×C4—2024no—
298C2×C4—2021no—
299C2×C4—2021no—
300C2×C4—2021no—
301C2×C4C2×C22⋊D10111yescharacteristic
302C2×C4C2×C22⋊D10111yescharacteristic
303C2×C4C2×C22⋊D10111yescharacteristic
304C2×C4C2×C22⋊D10111yescharacteristic
305C2×C4C2×C22⋊D10111yescharacteristic
306C2×C4C23×D10111yescharacteristic
307C8—2021no—
308C8—2021no—
309C8—2021no—
310C8—2021no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—4041no—
315C22—4041no—
316C22—2022no—
317C22C22×C22⋊D10111yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—4041no—
328C4—4041no—
329C4—2022no—
330C4#41×D10111yescharacteristic
331C4#41×D10111yescharacteristic
332C2—41no—
333C2—4042no—
334C2#2026568411yescharacteristic
335C2#2026598511yescharacteristic
336C2#2026599211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#541×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108074C22
hypercenter#541D101
hypocenterC101#3120
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #20265510C2semidirect
3. #20265505C2semidirect
4. #20265481C2semidirect
5. #20265474C2semidirect
6. #20265384C2semidirect
7. #20265379C2semidirect
10. #20265282C2semidirect
11. #20265275C2semidirect
12. C4×#13108064C2semidirect
13. C4×#13108061C2semidirect
14. #20265238C2semidirect
15. #541×C101C2semidirect
169. C101#3120semidirect
170. #541D101semidirect

Maximal subgroups

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

subgroupindex
#202656572
#202655102
#202655052
#202654812
#202654742
#202653842
#202653792
#202653372
#202652822
#202652752
C4×#131080642
C4×#131080612
#202652382
#541×C1012
#3120101