Finite Group Databasebeta

← #27182575

Subgroups — #27182575

Order 12928.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265788C211yesmaximal, characteristic, complemented
2#20265778C211yesmaximal, characteristic
3#20265486C211yesmaximal, characteristic, complemented
4#20265482C211yesmaximal, characteristic, complemented
5#20265478C211yesmaximal, characteristic, complemented
6#20265477C211yesmaximal, characteristic, complemented
7#20265426C211yesmaximal, characteristic, complemented
8#20265420C211yesmaximal, characteristic, complemented
9#20265409C211yesmaximal, characteristic, complemented
10#20265403C211yesmaximal, characteristic, complemented
11#20265379C211yesmaximal, characteristic, complemented
12#20265375C211yesmaximal, characteristic, complemented
13#20265375C211yesmaximal, characteristic, complemented
14#20265202C211yesmaximal, characteristic, complemented
15#461×C101C211yesmaximal, characteristic, complemented
16C2×#13108061C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18C2×#13108057C2211yescharacteristic
19C2×C8×D101C2211yescharacteristic
20#15692251C2211yescharacteristic
21#15692250—21no—
22#15692250—21no—
23#15692247C2211yescharacteristic
24#15692246C2211yescharacteristic
25#15692245—21no—
26#15692245—21no—
27#15692238—21no—
28#15692237—21no—
29#15692231C2211yescharacteristic
30#15692231C2211yescharacteristic
31#15692230C2211yescharacteristic
32#15692229—21no—
33#15692228—21no—
34#15692226C2211yescharacteristic
35Q8×Dic101C2211yescharacteristic
36Q8×Dic101C2211yescharacteristic
37C4×D404C2211yescharacteristic
38#15692212C2211yescharacteristic
39#15692189C2211yescharacteristic
40#15692188C2211yescharacteristic
41#15692187C2211yescharacteristic
42#15692186C2211yescharacteristic
43C4⋊#11717579C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692179C2211yescharacteristic
48#15692177C2211yescharacteristic
49#15692177C2211yescharacteristic
50#15692176C2211yescharacteristic
51#15692176C2211yescharacteristic
52C2×SD16×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C4⋊Q8×C101C2211yescharacteristic
55C4×D4×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#103×C101C2211yescharacteristic
59C2×D404—21no—
60C2×D404C2311yescharacteristic
61C2×C22⋊D101—21no—
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C2×C4×D101—21no—
65C2×C4×D101C2311yescharacteristic
66#13108061—41no—
67#13108061—41no—
68#13108057—41no—
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C8×D101—41no—
72C4⋊Dic101—21no—
73C4⋊Dic101—21no—
74C4⋊Dic101—21no—
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053D411yescharacteristic
84#13108053D411yescharacteristic
85C22⋊Dic101—21no—
86#13108051—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
98C22×C4×C101—21no—
99SD16×C101—41no—
100SD16×C101—41no—
101C2×C8×C101C2311yescharacteristic
102C2×C8×C101C2311yescharacteristic
103C4⋊C4×C101—21no—
104C4⋊C4×C101—21no—
105C4⋊C4×C101C2311yescharacteristic
106C22⋊C4×C101—21no—
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C22×D101—21no—
110C22×D101C2×D411yescharacteristic
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×C101—21no—
114Q8×C101—21no—
115D4×C101—21no—
116D4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—81no—
121C2×C4×C101C2×D411yescharacteristic
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2411yescharacteristic
124C22⋊D101—81no—
125C22⋊D101—81no—
126C22⋊D101—81no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101C2×D411yescharacteristic
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133D404—21no—
134D404—21no—
135D404—81no—
136C4×D101—21no—
137C4×D101—21no—
138C4×D101—81no—
139C8×C101—21no—
140C8×C101—21no—
141#11717579—21no—
142#11717579—21no—
143C22×C101—41no—
144C22×C101—41no—
145C22×C101—41no—
146C22×C101C22×D411yescharacteristic
147C2×D101—21no—
148C2×D101—21no—
149C2×D101—21no—
150C2×D101—41no—
151C2×D101—41no—
152C2×D101—41no—
153C4×C101—21no—
154C4×C101—21no—
155C4×C101—41no—
156C4×C101—41no—
157C4×C101—41no—
158C4×C101H(5,𝔽2)11yescharacteristic
159C4×C1012−1+411yescharacteristic
160Dic101—21no—
161Dic101—21no—
162Dic101—21no—
163Dic101—41no—
164Dic101—41no—
165Dic101—41no—
166C2×C101—81no—
167C2×C101#53711yescharacteristic
168C2×C101#57711yescharacteristic
169C2×C101#57711yescharacteristic
170D101—41no—
171D101—81no—
172#3260—1011nomaximal, Sylow
173C101#326011yescharacteristic, complemented, Sylow
174#541—1011no—
175#540—1011no—
176#479—1011no—
177#479—1011no—
178#478—1011no—
179#478—1011no—
180#468—1011no—
181#468—1011no—
182#465—1011no—
183#465—1011no—
184#461—1011no—
185#461D10111yescharacteristic, complemented
186#460—1011no—
187#460—1011no—
188#424—1011no—
189C2×SD16—1011no—
190C2×SD16—1011no—
191C2×SD16C2×D10111yescharacteristic
192C2×SD16C2×D10111yescharacteristic
193C2×M16—1011no—
194C22×C8—1011no—
195C4⋊Q8C2×D10111yescharacteristic
196#127—1011no—
197#126—2021no—
198#126—2021no—
199#125—1011no—
200#125—1011no—
201#125—2021no—
202#125—2021no—
203#123—2021no—
204#123—2021no—
205C22⋊Q8—1011no—
206C22⋊Q8—1011no—
207#121—1011no—
208#121—1011no—
209#121—2021no—
210#121—2021no—
211C4×Q8—1011no—
212C4×Q8—1011no—
213C4×D4—1011no—
214C4×D4C2×D10111yescharacteristic
215#117—1011no—
216#107—1011no—
217#107—1011no—
218#106—1011no—
219#106—1011no—
220C4⋊C8—1011no—
221C4⋊C8—1011no—
222C4⋊C8—1011no—
223C4⋊C8C2×D10111yescharacteristic
224#103—1011no—
225#103—1011no—
226#103C2×D10111yescharacteristic
227#103C2×D10111yescharacteristic
228#102—1011no—
229#102—1011no—
230#102—1011no—
231#102—1011no—
232C2×Q8C22×D10111yescharacteristic
233C2×Q8C22×D10111yescharacteristic
234C2×D4—1011no—
235C2×D4—2021no—
236C2×D4—2021no—
237C2×D4—2021no—
238C2×D4—2021no—
239C2×D4C22×D10111yescharacteristic
240C22×C4—21no—
241C22×C4—1011no—
242C22×C4—2021no—
243SD16—41no—
244SD16—41no—
245SD16—4041no—
246SD16—4041no—
247M16—4041no—
248C2×C8—1011no—
249C2×C8—1011no—
250C2×C8—4041no—
251C2×C8C22×D10111yescharacteristic
252C2×C8C22×D10111yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—1011no—
256C4⋊C4—1011no—
257C4⋊C4—1011no—
258C4⋊C4—1011no—
259C4⋊C4—1011no—
260C4⋊C4—2021no—
261C4⋊C4—2021no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4C22×D10111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—1011no—
271C22⋊C4—1011no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C22⋊C4—2021no—
277C42—1011no—
278C42—2021no—
279C42—2021no—
280C42C22×D10111yescharacteristic
281C23—21no—
282C23—1011no—
283C23—2021no—
284Q8—21no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288D4—21no—
289D4—21no—
290D4—8081no—
291D4—8081no—
292D4—8081no—
293D4—8081no—
294D4—2021no—
295D4—2021no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—8081no—
301C2×C4—1011no—
302C2×C4—1011no—
303C2×C4—1011no—
304C2×C4—2021no—
305C2×C4—2021no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4D4×D10111yescharacteristic
310C2×C4D4×D10111yescharacteristic
311C2×C4C23×D10111yescharacteristic
312C8—21no—
313C8—21no—
314C8—2021no—
315C8—2021no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—4041no—
320C22—4041no—
321C22—4041no—
322C22—2021no—
323C22—2021no—
324C22—2021no—
325C22C2×D4×D10111yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—4041no—
332C4—4041no—
333C4—4041no—
334C4—2021no—
335C4—2021no—
336C4—2021no—
337C4#1569230011yescharacteristic
338C4#1569230211yescharacteristic
339C2—81no—
340C2—8081no—
341C2—4041no—
342C2#2026575611yescharacteristic
343C2#2026598311yescharacteristic
344C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#461×C101C2
socleC22×C101C22×D4
prime radical#20265778C2
hypercenter#461D101
hypocenterC101#3260
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #20265788C2semidirect
3. #20265486C2semidirect
4. #20265482C2semidirect
5. #20265478C2semidirect
6. #20265477C2semidirect
7. #20265426C2semidirect
8. #20265420C2semidirect
9. #20265409C2semidirect
10. #20265403C2semidirect
11. #20265379C2semidirect
12. #20265375C2semidirect
13. #20265375C2semidirect
14. #20265202C2semidirect
15. #461×C101C2semidirect
173. C101#3260semidirect
185. #461D101semidirect

Maximal subgroups

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

subgroupindex
#202657882
#202657782
#202654862
#202654822
#202654782
#202654772
#202654262
#202654202
#202654092
#202654032
#202653792
#202653752
#202652022
#461×C1012
#3260101