Finite Group Databasebeta

← #27182563

Subgroups — #27182563

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#20265796C211yesmaximal, characteristic
2#20265785C211yesmaximal, characteristic, complemented
3#20265486C211yesmaximal, characteristic, complemented
4#20265475C211yesmaximal, characteristic, complemented
5#20265473C211yesmaximal, characteristic, complemented
6#20265462C211yesmaximal, characteristic, complemented
7#20265434C211yesmaximal, characteristic, complemented
8#20265420C211yesmaximal, characteristic
9#20265415C211yesmaximal, characteristic, complemented
10#20265405C211yesmaximal, characteristic
11#20265396C211yesmaximal, characteristic, complemented
12#20265385C211yesmaximal, characteristic, complemented
13#20265380C211yesmaximal, characteristic, complemented
14#20265202C211yesmaximal, characteristic, complemented
15#464×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18C2×#13108057C2211yescharacteristic
19C2×C8×D101C2211yescharacteristic
20#15692255C2211yescharacteristic
21#15692253C2211yescharacteristic
22#15692250—21no—
23#15692249—21no—
24#15692248—21no—
25#15692247—21no—
26#15692246C2211yescharacteristic
27#15692245—21no—
28#15692241—21no—
29#15692239—21no—
30#15692235C2211yescharacteristic
31#15692232—21no—
32#15692231C2211yescharacteristic
33#15692231C2211yescharacteristic
34#15692226C2211yescharacteristic
35Q8×Dic101C2211yescharacteristic
36C4×Dic202C2211yescharacteristic
37D4×Dic101C2211yescharacteristic
38#15692212C2211yescharacteristic
39#15692189C2211yescharacteristic
40#15692189C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692186C2211yescharacteristic
43C4⋊#11717579C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692180C2211yescharacteristic
48#15692179C2211yescharacteristic
49#15692179C2211yescharacteristic
50#15692179C2211yescharacteristic
51#15692176C2211yescharacteristic
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
61C2×C22⋊D101—21no—
62C22×Dic101—21no—
63C2×C4×D101C2311yescharacteristic
64#13108064—41no—
65#13108062—41no—
66#13108057—41no—
67C2×#11717579C2311yescharacteristic
68C2×#11717579C2311yescharacteristic
69C8×D101—41no—
70C4⋊Dic101—21no—
71C4⋊Dic101—21no—
72C4⋊Dic101—21no—
73C4⋊Dic101—21no—
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#13108053D411yescharacteristic
84#13108053D411yescharacteristic
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051D411yescharacteristic
91#13108051D411yescharacteristic
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101C2311yescharacteristic
95C2×Q8×C101C2311yescharacteristic
96C2×Q8×C101C2311yescharacteristic
97C2×D4×C101C2311yescharacteristic
98Q16×C101—41no—
99SD16×C101—41no—
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—21no—
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—81no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—21no—
127C2×Dic101—81no—
128C2×Dic101C2×D411yescharacteristic
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131C4×D101—21no—
132C4×D101—21no—
133Dic202—21no—
134Dic202—21no—
135Dic202—81no—
136C8×C101—21no—
137C8×C101—21no—
138#11717579—21no—
139#11717579—21no—
140C22×C101—41no—
141C22×C101—41no—
142C22×C101—41no—
143C22×C101C22×D411yescharacteristic
144C2×D101—21no—
145C2×D101—21no—
146C2×D101—21no—
147C4×C101—21no—
148C4×C101—21no—
149C4×C101—41no—
150C4×C101—41no—
151C4×C101—41no—
152C4×C101H(5,𝔽2)11yescharacteristic
153C4×C1012−1+411yescharacteristic
154Dic101—21no—
155Dic101—21no—
156Dic101—21no—
157Dic101—41no—
158Dic101—41no—
159Dic101—41no—
160Dic101—41no—
161C2×C101—81no—
162C2×C101#53711yescharacteristic
163C2×C101#57711yescharacteristic
164C2×C101#57811yescharacteristic
165D101—41no—
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#464D10111yescharacteristic, complemented
180#462—1011no—
181#461—1011no—
182#424—1011no—
183C2×Q16—1011no—
184C2×Q16C2×D10111yescharacteristic
185C2×SD16—1011no—
186C2×SD16C2×D10111yescharacteristic
187C2×M16—1011no—
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—2021no—
198#124C2×D10111yescharacteristic
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×Q8C2×D10111yescharacteristic
208C4×D4—1011no—
209#117—1011no—
210#107—1011no—
211#107—1011no—
212#106—1011no—
213#106—1011no—
214C4⋊C8—1011no—
215C4⋊C8—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
226C2×Q8—1011no—
227C2×Q8—2021no—
228C2×Q8C22×D10111yescharacteristic
229C2×Q8C22×D10111yescharacteristic
230C2×D4—2021no—
231C2×D4C22×D10111yescharacteristic
232C22×C4—1011no—
233C22×C4—2021no—
234Q16—41no—
235Q16—4041no—
236SD16—41no—
237SD16—4041no—
238M16—4041no—
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—1011no—
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⋊C4—2021no—
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—21no—
271C42—1011no—
272C42—2021no—
273C42—2021no—
274C42C22×D10111yescharacteristic
275C23—21no—
276C23—1011no—
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—8081no—
282Q8—2021no—
283Q8—2021no—
284D4—21no—
285D4—21no—
286D4—8081no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—8081no—
291C2×C4—1011no—
292C2×C4—1011no—
293C2×C4—1011no—
294C2×C4—2021no—
295C2×C4—2021no—
296C2×C4—2021no—
297C2×C4—2021no—
298C2×C4—2021no—
299C2×C4—2021no—
300C2×C4D4×D10111yescharacteristic
301C2×C4D4×D10111yescharacteristic
302C2×C4C23×D10111yescharacteristic
303C8—21no—
304C8—21no—
305C8—2021no—
306C8—2021no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—2021no—
311C22—2021no—
312C22—2021no—
313C22C2×D4×D10111yescharacteristic
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—4041no—
320C4—4041no—
321C4—4041no—
322C4—4041no—
323C4—2021no—
324C4—2021no—
325C4—2021no—
326C4#1569230011yescharacteristic
327C4#1569230211yescharacteristic
328C2—81no—
329C2—4041no—
330C2#2026575611yescharacteristic
331C2#2026598611yescharacteristic
332C2#2026599111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#464×C101C2
socleC22×C101C22×D4
prime radical#15692226C22
hypercenter#464D101
hypocenterC101#3258
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. #20265785C2semidirect
3. #20265486C2semidirect
4. #20265475C2semidirect
5. #20265473C2semidirect
6. #20265462C2semidirect
7. #20265434C2semidirect
9. #20265415C2semidirect
11. #20265396C2semidirect
12. #20265385C2semidirect
13. #20265380C2semidirect
14. #20265202C2semidirect
15. #464×C101C2semidirect
167. C101#3258semidirect
179. #464D101semidirect

Maximal subgroups

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

subgroupindex
#202657962
#202657852
#202654862
#202654752
#202654732
#202654622
#202654342
#202654202
#202654152
#202654052
#202653962
#202653852
#202653802
#202652022
#464×C1012
#3258101