Finite Group Databasebeta

← #27182559

Subgroups — #27182559

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#20265794C211yesmaximal, characteristic
2#20265787C211yesmaximal, characteristic, complemented
3#20265483C211yesmaximal, characteristic, complemented
4#20265479C211yesmaximal, characteristic, complemented
5#20265471C211yesmaximal, characteristic, complemented
6#20265460C211yesmaximal, characteristic, complemented
7#20265435C211yesmaximal, characteristic, complemented
8#20265416C211yesmaximal, characteristic, complemented
9#20265404C211yesmaximal, characteristic
10#20265398C211yesmaximal, characteristic, complemented
11#20265397C211yesmaximal, characteristic, complemented
12#20265386C211yesmaximal, characteristic, complemented
13#20265380C211yesmaximal, characteristic, complemented
14#20265205C211yesmaximal, characteristic, complemented
15#468×C101C211yesmaximal, characteristic
16C2×Dic404C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19C22×#11717579C2211yescharacteristic
20#15692254C2211yescharacteristic
21#15692253C2211yescharacteristic
22#15692248—21no—
23#15692248—21no—
24#15692248—21no—
25#15692246—21no—
26#15692246C2211yescharacteristic
27#15692245—21no—
28#15692242—21no—
29#15692242C2211yescharacteristic
30#15692236—21no—
31#15692234C2211yescharacteristic
32#15692234C2211yescharacteristic
33#15692232—21no—
34Q8×Dic101C2211yescharacteristic
35#15692220C2211yescharacteristic
36#15692220C2211yescharacteristic
37D4×Dic101C2211yescharacteristic
38#15692214C2211yescharacteristic
39#15692188C2211yescharacteristic
40#15692187C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692184C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692180C2211yescharacteristic
48#15692179C2211yescharacteristic
49#15692178C2211yescharacteristic
50#15692178C2211yescharacteristic
51#15692176C2211yescharacteristic
52C2×SD16×C101C2211yescharacteristic
53C2×M16×C101C2211yescharacteristic
54C22⋊Q8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#107×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C2×Dic202C2311yescharacteristic
62C22×Dic101—21no—
63Dic404—41no—
64#13108064—41no—
65#13108062—41no—
66C2×#11717579—41no—
67C2×#11717579C2311yescharacteristic
68C2×#11717579C2311yescharacteristic
69C4⋊Dic101—21no—
70C4⋊Dic101—21no—
71C4⋊Dic101D411yescharacteristic
72C4⋊Dic101D411yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101D411yescharacteristic
89C22⋊Dic101D411yescharacteristic
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101—21no—
93C4×Dic101C2311yescharacteristic
94C4×Dic101C2311yescharacteristic
95C2×Q8×C101C2311yescharacteristic
96C2×D4×C101—21no—
97C2×D4×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99SD16×C101—41no—
100M16×C101—41no—
101C2×C8×C101C2311yescharacteristic
102C2×C8×C101C2311yescharacteristic
103C4⋊C4×C101—21no—
104C4⋊C4×C101C2311yescharacteristic
105C4⋊C4×C101C2311yescharacteristic
106C22⋊C4×C101—21no—
107C22⋊C4×C101—21no—
108C23×C101—21no—
109C23×C101C2×D411yescharacteristic
110Q8×C101—21no—
111Q8×C101—21no—
112D4×C101—21no—
113D4×C101—21no—
114D4×C101—81no—
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×C101C2411yescharacteristic
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—81no—
127C2×Dic101C2×D411yescharacteristic
128C2×Dic101C2×D411yescharacteristic
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131Dic202—21no—
132Dic202—21no—
133Dic202—21no—
134Dic202—21no—
135Dic202—81no—
136C8×C101—21no—
137C8×C101—21no—
138#11717579—21no—
139#11717579—21no—
140C22×C101—21no—
141C22×C101—21no—
142C22×C101—21no—
143C22×C101—41no—
144C22×C101—41no—
145C22×C101—41no—
146C22×C101C22×D411yescharacteristic
147C4×C101—21no—
148C4×C101—41no—
149C4×C101—41no—
150C4×C101—41no—
151C4×C101H(5,𝔽2)11yescharacteristic
152C4×C1012−1+411yescharacteristic
153Dic101—21no—
154Dic101—21no—
155Dic101—21no—
156Dic101—21no—
157Dic101—41no—
158Dic101—41no—
159Dic101—41no—
160Dic101—41no—
161C2×C101—41no—
162C2×C101—81no—
163C2×C101#53711yescharacteristic
164C2×C101#57711yescharacteristic
165C2×C101#57811yescharacteristic
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#468D10111yescharacteristic
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×SD16C2×D10111yescharacteristic
187C2×M16C2×D10111yescharacteristic
188C22×C8—1011no—
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—2021no—
203C22⋊Q8C2×D10111yescharacteristic
204#121C2×D10111yescharacteristic
205C4×Q8—1011no—
206C4×Q8—1011no—
207C4×Q8—1011no—
208C4×D4—1011no—
209#117—1011no—
210#107—1011no—
211#107C2×D10111yescharacteristic
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#103C2×D10111yescharacteristic
224#102—1011no—
225#102C2×D10111yescharacteristic
226C2×Q8—1011no—
227C2×Q8—1011no—
228C2×Q8—2021no—
229C2×Q8C22×D10111yescharacteristic
230C2×D4—21no—
231C2×D4C22×D10111yescharacteristic
232C22×C4—2021no—
233C22×C4C22×D10111yescharacteristic
234Q16—4041no—
235Q16—4041no—
236SD16—41no—
237SD16—4041no—
238M16—41no—
239C2×C8—1011no—
240C2×C8—1011no—
241C2×C8—4041no—
242C2×C8C22×D10111yescharacteristic
243C2×C8C22×D10111yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—1011no—
246C4⋊C4—1011no—
247C4⋊C4—1011no—
248C4⋊C4—1011no—
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⋊C4C22×D10111yescharacteristic
260C4⋊C4C22×D10111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—1011no—
264C22⋊C4—1011no—
265C22⋊C4—2021no—
266C22⋊C4—2021no—
267C22⋊C4—2021no—
268C22⋊C4—2021no—
269C22⋊C4—2021no—
270C42—1011no—
271C42—1011no—
272C42—2021no—
273C42—2021no—
274C42—2021no—
275C23—21no—
276C23D4×D10111yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—8081no—
280Q8—2021no—
281Q8—2021no—
282Q8—2021no—
283Q8—2021no—
284D4—21no—
285D4—21no—
286D4—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—8081no—
293C2×C4—1011no—
294C2×C4—1011no—
295C2×C4—1011no—
296C2×C4—1011no—
297C2×C4—2021no—
298C2×C4—2021no—
299C2×C4—2021no—
300C2×C4—2021no—
301C2×C4D4×D10111yescharacteristic
302C2×C4C23×D10111yescharacteristic
303C8—21no—
304C8—21no—
305C8—2021no—
306C8—2021no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C2×D4×D10111yescharacteristic
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—4041no—
319C4—4041no—
320C4—4041no—
321C4—4041no—
322C4—2021no—
323C4—2021no—
324C4—2021no—
325C4—2021no—
326C4#1569230011yescharacteristic
327C4#1569230211yescharacteristic
328C2—41no—
329C2—81no—
330C2#2026575411yescharacteristic
331C2#2026598711yescharacteristic
332C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#468×C101C2
socleC22×C101C22×D4
prime radical#121×C101C22
hypercenter#468D101
hypocenterC101#3258
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #20265787C2semidirect
3. #20265483C2semidirect
4. #20265479C2semidirect
5. #20265471C2semidirect
6. #20265460C2semidirect
7. #20265435C2semidirect
8. #20265416C2semidirect
10. #20265398C2semidirect
11. #20265397C2semidirect
12. #20265386C2semidirect
13. #20265380C2semidirect
14. #20265205C2semidirect
167. C101#3258semidirect

Maximal subgroups

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

subgroupindex
#202657942
#202657872
#202654832
#202654792
#202654712
#202654602
#202654352
#202654162
#202654042
#202653982
#202653972
#202653862
#202653802
#202652052
#468×C1012
#3258101