Finite Group Databasebeta

← #27182560

Subgroups — #27182560

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#20265789C211yesmaximal, characteristic, complemented
3#20265485C211yesmaximal, characteristic, complemented
4#20265478C211yesmaximal, characteristic, complemented
5#20265473C211yesmaximal, characteristic, complemented
6#20265463C211yesmaximal, characteristic, complemented
7#20265435C211yesmaximal, characteristic, complemented
8#20265423C211yesmaximal, characteristic
9#20265418C211yesmaximal, characteristic, complemented
10#20265406C211yesmaximal, characteristic
11#20265400C211yesmaximal, characteristic, complemented
12#20265400C211yesmaximal, characteristic, complemented
13#20265388C211yesmaximal, characteristic, complemented
14#20265383C211yesmaximal, characteristic, complemented
15#424×C101C211yesmaximal, characteristic, complemented
16C2×Dic404C2211yescharacteristic
17C2×Dic404C2211yescharacteristic
18C2×#13108063C2211yescharacteristic
19C2×#13108063C2211yescharacteristic
20#15692255C2211yescharacteristic
21#15692255C2211yescharacteristic
22#15692250—21no—
23#15692248—21no—
24#15692248—21no—
25#15692247C2211yescharacteristic
26#15692245—21no—
27#15692245—21no—
28#15692243C2211yescharacteristic
29#15692241—21no—
30#15692237—21no—
31#15692234C2211yescharacteristic
32#15692234C2211yescharacteristic
33#15692234C2211yescharacteristic
34#15692233—21no—
35#15692229C2211yescharacteristic
36C4×Dic202C2211yescharacteristic
37C4×Dic202C2211yescharacteristic
38C4×Dic202C2211yescharacteristic
39C4×D404C2211yescharacteristic
40#15692189C2211yescharacteristic
41#15692189C2211yescharacteristic
42#15692187C2211yescharacteristic
43#15692187C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692180C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692180C2211yescharacteristic
48#15692180C2211yescharacteristic
49#15692180C2211yescharacteristic
50#15692177C2211yescharacteristic
51#15692177C2211yescharacteristic
52C2×M16×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic
54#117×C101C2211yescharacteristic
55C4⋊C8×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57C4⋊C8×C101C2211yescharacteristic
58C4⋊C8×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C2×Dic202C2311yescharacteristic
62C2×Dic202C2311yescharacteristic
63C2×D404C2311yescharacteristic
64C2×C22⋊D101—21no—
65C2×C4×D101—21no—
66Dic404—41no—
67Dic404—41no—
68#13108063—41no—
69#13108063—41no—
70C4⋊Dic101—21no—
71C4⋊Dic101—21no—
72C4⋊Dic101—21no—
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051—21no—
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101—21no—
95C22×C4×C101C2311yescharacteristic
96M16×C101—41no—
97C2×C8×C101—41no—
98C2×C8×C101C2311yescharacteristic
99C2×C8×C101C2311yescharacteristic
100C2×C8×C101C2311yescharacteristic
101C2×C8×C101C2311yescharacteristic
102C4⋊C4×C101D411yescharacteristic
103C4⋊C4×C101D411yescharacteristic
104C22⋊C4×C101D411yescharacteristic
105C22⋊C4×C101D411yescharacteristic
106C42×C101C2311yescharacteristic
107C42×C101C2311yescharacteristic
108C23×C101C2×D411yescharacteristic
109C22×D101—21no—
110C2×C4×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101C2×D411yescharacteristic
113C2×C4×C101C2×D411yescharacteristic
114C2×C4×C101C2×D411yescharacteristic
115C2×C4×C101C2×D411yescharacteristic
116C2×C4×C101C2×D411yescharacteristic
117C2×C4×C101C2411yescharacteristic
118C22⋊D101—81no—
119C2×Dic101—21no—
120C2×Dic101—21no—
121C2×Dic101—21no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126D404—21no—
127D404—21no—
128C4×D101—81no—
129Dic202—21no—
130Dic202—21no—
131Dic202—21no—
132Dic202—21no—
133Dic202—21no—
134Dic202—21no—
135Dic202—81no—
136C8×C101—21no—
137C8×C101—21no—
138C8×C101—21no—
139C8×C101—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×C101H(5,𝔽2)11yescharacteristic
153C4×C1012−1+411yescharacteristic
154Dic101—41no—
155Dic101—41no—
156Dic101—41no—
157Dic101—41no—
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#541—1011no—
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#424D10111yescharacteristic, complemented
183C2×Q16—1011no—
184C2×Q16—1011no—
185C2×SD16—1011no—
186C2×SD16—1011no—
187C2×M16C2×D10111yescharacteristic
188C22×C8C2×D10111yescharacteristic
189C4⋊Q8—1011no—
190C4⋊Q8—1011no—
191#126—2021no—
192#126—2021no—
193#126—2021no—
194#125—1011no—
195#125—2021no—
196#125—2021no—
197#124—1011no—
198#124—2021no—
199#123—2021no—
200C22⋊Q8—1011no—
201C22⋊Q8—1011no—
202C22⋊Q8—1011no—
203C22⋊Q8—2021no—
204#121—1011no—
205C4×Q8—1011no—
206C4×Q8—1011no—
207C4×Q8—1011no—
208C4×D4—1011no—
209#117C2×D10111yescharacteristic
210#107—1011no—
211#107—1011no—
212#106—1011no—
213#106—1011no—
214C4⋊C8C2×D10111yescharacteristic
215C4⋊C8C2×D10111yescharacteristic
216C4⋊C8C2×D10111yescharacteristic
217C4⋊C8C2×D10111yescharacteristic
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—1011no—
228C2×Q8—1011no—
229C2×Q8—2021no—
230C2×D4—1011no—
231C2×D4—2021no—
232C22×C4—2021no—
233C22×C4C22×D10111yescharacteristic
234Q16—4041no—
235Q16—4041no—
236SD16—4041no—
237SD16—4041no—
238M16—41no—
239C2×C8—41no—
240C2×C8C22×D10111yescharacteristic
241C2×C8C22×D10111yescharacteristic
242C2×C8C22×D10111yescharacteristic
243C2×C8C22×D10111yescharacteristic
244C4⋊C4—1011no—
245C4⋊C4—1011no—
246C4⋊C4—1011no—
247C4⋊C4—1011no—
248C4⋊C4—2021no—
249C4⋊C4—2021no—
250C4⋊C4—2021no—
251C4⋊C4—2021no—
252C4⋊C4—2021no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
259C4⋊C4D40411yescharacteristic
260C4⋊C4D40411yescharacteristic
261C22⋊C4—2021no—
262C22⋊C4—2021no—
263C22⋊C4—2021no—
264C22⋊C4—2021no—
265C22⋊C4—2021no—
266C22⋊C4—2021no—
267C22⋊C4—2021no—
268C22⋊C4D40411yescharacteristic
269C22⋊C4D40411yescharacteristic
270C42—2021no—
271C42—2021no—
272C42—2021no—
273C42C22×D10111yescharacteristic
274C42C22×D10111yescharacteristic
275C23—2021no—
276C23C2×D40411yescharacteristic
277Q8—8081no—
278Q8—2021no—
279Q8—2021no—
280Q8—2021no—
281Q8—2021no—
282Q8—2021no—
283Q8—2021no—
284D4—8081no—
285D4—2021no—
286D4—2021no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—8081no—
290C2×C4—2021no—
291C2×C4—2021no—
292C2×C4—2021no—
293C2×C4—2021no—
294C2×C4—2021no—
295C2×C4—2021no—
296C2×C4—2021no—
297C2×C4C2×D40411yescharacteristic
298C2×C4C2×D40411yescharacteristic
299C2×C4C2×D40411yescharacteristic
300C2×C4C2×D40411yescharacteristic
301C2×C4C2×D40411yescharacteristic
302C2×C4C23×D10111yescharacteristic
303C8—21no—
304C8—21no—
305C8—21no—
306C8—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—4041no—
311C22—4041no—
312C22—4041no—
313C22C22×D40411yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—4041no—
320C4—4041no—
321C4—4041no—
322C4—4041no—
323C4—4041no—
324C4—4041no—
325C4—4041no—
326C4#1569230011yescharacteristic
327C4#1569230311yescharacteristic
328C2—41no—
329C2—8081no—
330C2#2026575811yescharacteristic
331C2#2026598911yescharacteristic
332C2#2026599411yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#424×C101C2
socleC22×C101C22×D4
prime radical#15692229C22
hypercenter#424D101
hypocenterC101#3258
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #20265789C2semidirect
3. #20265485C2semidirect
4. #20265478C2semidirect
5. #20265473C2semidirect
6. #20265463C2semidirect
7. #20265435C2semidirect
9. #20265418C2semidirect
11. #20265400C2semidirect
12. #20265400C2semidirect
13. #20265388C2semidirect
14. #20265383C2semidirect
15. #424×C101C2semidirect
167. C101#3258semidirect
182. #424D101semidirect

Maximal subgroups

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

subgroupindex
#202657982
#202657892
#202654852
#202654782
#202654732
#202654632
#202654352
#202654232
#202654182
#202654062
#202654002
#202653882
#202653832
#424×C1012
#3258101