Finite Group Databasebeta

← #27182552

Subgroups — #27182552

Order 12928.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265798C211yesmaximal, characteristic
2#20265481C211yesmaximal, characteristic, complemented
3#20265474C211yesmaximal, characteristic, complemented
4#20265472C211yesmaximal, characteristic, complemented
5#20265462C211yesmaximal, characteristic, complemented
6#20265433C211yesmaximal, characteristic, complemented
7#20265425C211yesmaximal, characteristic
8#20265416C211yesmaximal, characteristic, complemented
9#20265408C211yesmaximal, characteristic
10#20265399C211yesmaximal, characteristic, complemented
11#20265399C211yesmaximal, characteristic, complemented
12#20265384C211yesmaximal, characteristic, complemented
13#20265379C211yesmaximal, characteristic, complemented
14#20265204C211yesmaximal, characteristic, complemented
15#541×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20C2×#13108058C2211yescharacteristic
21C22×#11717579C2211yescharacteristic
22#15692255C2211yescharacteristic
23#15692250—21no—
24#15692243C2211yescharacteristic
25#15692241—21no—
26#15692237—21no—
27#15692234C2211yescharacteristic
28#15692233—21no—
29#15692229C2211yescharacteristic
30C4×Dic202C2211yescharacteristic
31C4×D404C2211yescharacteristic
32#15692188C2211yescharacteristic
33#15692188C2211yescharacteristic
34#15692186C2211yescharacteristic
35#15692186C2211yescharacteristic
36C4⋊#11717579C2211yescharacteristic
37C4⋊#11717579C2211yescharacteristic
38C4⋊#11717579C2211yescharacteristic
39C4⋊#11717579C2211yescharacteristic
40#15692179C2211yescharacteristic
41#15692179C2211yescharacteristic
42#15692179C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692178C2211yescharacteristic
45#15692178C2211yescharacteristic
46#15692175C2211yescharacteristic
47#15692175C2211yescharacteristic
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
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C4×D101—21no—
64#13108064—41no—
65#13108064—41no—
66#13108061—41no—
67#13108061—41no—
68#13108058—41no—
69C2×#11717579—41no—
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C4⋊Dic101—21no—
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77#13108053—21no—
78#13108053—21no—
79C22⋊Dic101—21no—
80#13108051—21no—
81#13108051—21no—
82#13108051—21no—
83#13108051—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—81no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130D404—21no—
131D404—21no—
132C4×D101—81no—
133Dic202—21no—
134Dic202—21no—
135Dic202—81no—
136#11717579—21no—
137#11717579—21no—
138#11717579—21no—
139#11717579—21no—
140C22×C101—21no—
141C22×C101—21no—
142C22×C101—21no—
143C22×C101C22×D411yescharacteristic
144C2×D101—41no—
145C2×D101—41no—
146C2×D101—41no—
147C4×C101—21no—
148C4×C101—21no—
149C4×C101—21no—
150C4×C101—21no—
151C4×C101—21no—
152C4×C101—41no—
153C4×C101—41no—
154C4×C101—41no—
155C4×C101—41no—
156C4×C101H(5,𝔽2)11yescharacteristic
157C4×C1012−1+411yescharacteristic
158Dic101—41no—
159Dic101—41no—
160Dic101—41no—
161C2×C101—41no—
162C2×C101#53711yescharacteristic
163C2×C101#57711yescharacteristic
164C2×C101#57811yescharacteristic
165D101—81no—
166#3258—1011nomaximal, Sylow
167C101#325811yescharacteristic, complemented, Sylow
168#542—1011no—
169#541D10111yescharacteristic, complemented
170#479—1011no—
171#478—1011no—
172#477—1011no—
173#475—1011no—
174#470—1011no—
175#468—1011no—
176#467—1011no—
177#465—1011no—
178#464—1011no—
179#464—1011no—
180#462—1011no—
181#461—1011no—
182#424—1011no—
183C2×Q16—1011no—
184C2×Q16—1011no—
185C2×SD16—1011no—
186C2×SD16—1011no—
187C2×M16—1011no—
188C22×C8—1011no—
189C4⋊Q8—1011no—
190C4⋊Q8C2×D10111yescharacteristic
191#126—21no—
192#126—21no—
193#126—2021no—
194#125—21no—
195#125—21no—
196#125C2×D10111yescharacteristic
197#124—1011no—
198#124—2021no—
199#123—2021no—
200C22⋊Q8—1011no—
201C22⋊Q8—2021no—
202C22⋊Q8C2×D10111yescharacteristic
203C22⋊Q8C2×D10111yescharacteristic
204#121—1011no—
205C4×Q8—1011no—
206C4×Q8C2×D10111yescharacteristic
207C4×Q8C2×D10111yescharacteristic
208C4×D4—1011no—
209#117C2×D10111yescharacteristic
210#107—1011no—
211#107—1011no—
212#106—1011no—
213#106—1011no—
214C4⋊C8—1011no—
215C4⋊C8—1011no—
216C4⋊C8—1011no—
217C4⋊C8—1011no—
218#103—1011no—
219#103—1011no—
220#103—1011no—
221#103—1011no—
222#103—1011no—
223#103—1011no—
224#102—1011no—
225#102—1011no—
226C2×Q8—1011no—
227C2×Q8—2021no—
228C2×Q8C22×D10111yescharacteristic
229C2×Q8C22×D10111yescharacteristic
230C2×D4—1011no—
231C2×D4—2021no—
232C22×C4—2021no—
233C22×C4C22×D10111yescharacteristic
234Q16—4041no—
235Q16—4041no—
236SD16—4041no—
237SD16—4041no—
238M16—4041no—
239C2×C8—1011no—
240C2×C8—1011no—
241C2×C8—1011no—
242C2×C8—1011no—
243C2×C8—4041no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—1011no—
253C4⋊C4—1011no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4C22⋊D10111yescharacteristic
258C4⋊C4C22⋊D10111yescharacteristic
259C4⋊C4C22×D10111yescharacteristic
260C4⋊C4C22×D10111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—2021no—
264C22⋊C4—2021no—
265C22⋊C4—2021no—
266C22⋊C4—2021no—
267C22⋊C4—2021no—
268C22⋊C4C22⋊D10111yescharacteristic
269C22⋊C4C22⋊D10111yescharacteristic
270C42—21no—
271C42—21no—
272C42—2021no—
273C42C22×D10111yescharacteristic
274C42C22×D10111yescharacteristic
275C23—2021no—
276C23C2×C22⋊D10111yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—8081no—
282Q8—2021no—
283Q8—2021no—
284D4—8081no—
285D4—2021no—
286D4—2021no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—8081no—
294C2×C4—2021no—
295C2×C4—2021no—
296C2×C4—2021no—
297C2×C4C2×C22⋊D10111yescharacteristic
298C2×C4C2×C22⋊D10111yescharacteristic
299C2×C4C2×C22⋊D10111yescharacteristic
300C2×C4C2×C22⋊D10111yescharacteristic
301C2×C4C2×C22⋊D10111yescharacteristic
302C2×C4C23×D10111yescharacteristic
303C8—2021no—
304C8—2021no—
305C8—2021no—
306C8—2021no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—4041no—
311C22—4041no—
312C22—4041no—
313C22C22×C22⋊D10111yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—4041no—
324C4—4041no—
325C4—4041no—
326C4#1569230111yescharacteristic
327C4#1569230211yescharacteristic
328C2—41no—
329C2—8081no—
330C2#2026575511yescharacteristic
331C2#2026598511yescharacteristic
332C2#2026599211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#541×C101C2
socleC22×C101C22×D4
prime radical#15692229C22
hypercenter#541D101
hypocenterC101#3258
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. #20265481C2semidirect
3. #20265474C2semidirect
4. #20265472C2semidirect
5. #20265462C2semidirect
6. #20265433C2semidirect
8. #20265416C2semidirect
10. #20265399C2semidirect
11. #20265399C2semidirect
12. #20265384C2semidirect
13. #20265379C2semidirect
14. #20265204C2semidirect
15. #541×C101C2semidirect
167. C101#3258semidirect
169. #541D101semidirect

Maximal subgroups

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

subgroupindex
#202657982
#202654812
#202654742
#202654722
#202654622
#202654332
#202654252
#202654162
#202654082
#202653992
#202653842
#202653792
#202652042
#541×C1012
#3258101