Finite Group Databasebeta

← #27183014

Subgroups — #27183014

Order 12928.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265861C211yesmaximal, characteristic, complemented
2#20265837C211yesmaximal, characteristic
3#20265533C211yesmaximal, characteristic, complemented
4#20265515C211yesmaximal, characteristic, complemented
5#20265492C211yesmaximal, characteristic, complemented
6#20265483C211yesmaximal, characteristic, complemented
7#20265409C211yesmaximal, characteristic, complemented
8#20265401C211yesmaximal, characteristic, complemented
9#20265337C211yesmaximal, characteristic, complemented
10#20265331C211yesmaximal, characteristic, complemented
11#20265269C211yesmaximal, characteristic, complemented
12C8×C22⋊D101C211yesmaximal, characteristic, complemented
13#20265174C211yesmaximal, characteristic, complemented
14#103×D101C211yesmaximal, characteristic, complemented
15#467×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C2×#13108064C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic
21#15692248—21no—
22#15692246C2211yescharacteristic
23#15692245—21no—
24#15692244C2211yescharacteristic
25#15692241—21no—
26#15692237—21no—
27#15692233—21no—
28#15692231C2211yescharacteristic
29#15692230C2211yescharacteristic
30#15692228—21no—
31Q8×Dic101C2211yescharacteristic
32#15692220C2211yescharacteristic
33#15692217—21no—
34C4×C22⋊D101C2211yescharacteristic
35#15692214C2211yescharacteristic
36#15692213—21no—
37C4⋊C4×D101C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692186C2211yescharacteristic
40#15692184C2211yescharacteristic
41#15692180C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692178C2211yescharacteristic
46#15692177C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692175C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×Q16×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic
54C22⋊Q8×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic
56#107×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#103×C101C2211yescharacteristic
59#13108074—22no—
60#13108074—41no—
61C2×Dic202—21no—
62C2×Dic202C2311yescharacteristic
63Q8×D101—22no—
64Q8×D101—41no—
65C2×D404—21no—
66C2×D404C2311yescharacteristic
67C2×C22⋊D101—21no—
68C2×C22⋊D101D411yescharacteristic
69C2×C4×D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101C2311yescharacteristic
73#13108064—22no—
74#13108061—41no—
75C2×#11717579C2311yescharacteristic
76C2×#11717579C2311yescharacteristic
77C8×D101—41no—
78C4⋊Dic101—21no—
79C4⋊Dic101—21no—
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053D411yescharacteristic
87C22⋊Dic101—21no—
88C22⋊Dic101D411yescharacteristic
89#13108051—21no—
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051D411yescharacteristic
94C4×Dic101—21no—
95C4×Dic101—21no—
96C4×Dic101—21no—
97C4×Dic101C2311yescharacteristic
98C2×Q8×C101C2311yescharacteristic
99C2×Q8×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101Q16×C101—41no—
102C2×C8×C101—41no—
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C4⋊C4×C101—21no—
106C4⋊C4×C101—21no—
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C22⋊C4×C101—21no—
110C22⋊C4×C101—21no—
111C23×C101C2×D411yescharacteristic
112C22×D101—21no—
113C22×D101C2×D411yescharacteristic
114Q8×C101—21no—
115Q8×C101—21no—
116Q8×C101—21no—
117Q8×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2411yescharacteristic
126C22⋊D101—22no—
127C22⋊D101—42no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133C2×Dic101C2×D411yescharacteristic
134D404—21no—
135D404—21no—
136D404—42no—
137C4×D101—21no—
138C4×D101—21no—
139C4×D101—42no—
140C4×D101—42no—
141C4×D101—42no—
142Dic202—21no—
143Dic202—42no—
144Dic202#4112yes—
145C8×C101—21no—
146C8×C101—21no—
147#11717579—21no—
148#11717579—21no—
149C22×C101—21no—
150C22×C101—21no—
151C22×C101—21no—
152C22×C101C22×D411yescharacteristic
153C2×D101—21no—
154C2×D101—21no—
155C2×D101—41no—
156C2×D101—41no—
157C2×D101—41no—
158C2×D101#13512yes—
159C4×C101—21no—
160C4×C101—41no—
161C4×C101—41no—
162C4×C101—41no—
163C4×C101—41no—
164C4×C101C2×#4111yescharacteristic
165C4×C101H(5,𝔽2)11yescharacteristic
166Dic101—22no—
167Dic101—21no—
168Dic101—21no—
169Dic101—21no—
170Dic101—41no—
171Dic101—41no—
172C2×C101—41no—
173C2×C101#54611yescharacteristic
174C2×C101C2×#13511yescharacteristic
175C2×C101#57811yescharacteristic
176D101—22no—
177D101—81no—
178#3300—1011nomaximal, Sylow
179C101#330011yescharacteristic, complemented, Sylow
180#552—1011no—
181#548—1011no—
182#488—1011no—
183#485—1011no—
184#480—1011no—
185#479—1011no—
186#467D10111yescharacteristic, complemented
187#465—1011no—
188#464—1011no—
189#452—1011no—
190#451—1011no—
191C4×Q16—1011no—
192C8×D4—1011no—
193#416—1011no—
194C2×#103—1011no—
195C2×#41—1011no—
196C22×Q8—1011no—
197C2×Q16—1011no—
198C2×Q16C2×D10111yescharacteristic
199C2×SD16—1011no—
200C22×C8—1011no—
201C22×C8C2×D10111yescharacteristic
202#126—2021no—
203#125—1011no—
204#125—2021no—
205#124—1011no—
206#124—2021no—
207#123—2021no—
208C22⋊Q8—1011no—
209C22⋊Q8—2021no—
210C22⋊Q8C2×D10111yescharacteristic
211C22⋊Q8C2×D10111yescharacteristic
212#121—1011no—
213#121—2021no—
214C4×Q8—1011no—
215C4×Q8—1011no—
216C4×D4—1011no—
217C4×D4—2021no—
218#117—1011no—
219#117—2021no—
220C2×C4⋊C4—1011no—
221#107—1011no—
222#107C2×D10111yescharacteristic
223#106—1011no—
224C4⋊C8—1011no—
225#103—1011no—
226#103—1011no—
227#103—1011no—
228#103—1011no—
229#103—1011no—
230#103C2×D10111yescharacteristic
231#103C2×D10111yescharacteristic
232#102—1011no—
233#102—1011no—
234#102—1011no—
235C22⋊C8—1011no—
236C22⋊C8—1011no—
237C4×C8—1011no—
238#41—4041no—
239#41—2022no—
240C2×Q8—1011no—
241C2×Q8—4041no—
242C2×Q8—2021no—
243C2×Q8—2022no—
244C2×Q8C22×D10111yescharacteristic
245C2×Q8C22×D10111yescharacteristic
246C2×D4—1011no—
247C2×D4—1011no—
248C2×D4—2021no—
249C2×D4—2021no—
250C22×C4—1011no—
251C22×C4—2021no—
252C22×C4—2021no—
253C22×C4—2021no—
254C22×C4C22×D10111yescharacteristic
255Q16—41no—
256Q16—2022no—
257SD16—4041no—
258C2×C8—41no—
259C2×C8—1011no—
260C2×C8—1011no—
261C2×C8—4041no—
262C2×C8C22×D10111yescharacteristic
263C2×C8C22×D10111yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—1011no—
267C4⋊C4—1011no—
268C4⋊C4—1011no—
269C4⋊C4—2021no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4C22×D10111yescharacteristic
276C4⋊C4C22×D10111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1011no—
280C22⋊C4—1011no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C42—1011no—
287C42—2021no—
288C42—2021no—
289C42—2021no—
290C23—1011no—
291C23—2021no—
292C23D4×D10111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—1012no—
298Q8—4042no—
299Q8—2021no—
300D4—4042no—
301D4—4042no—
302D4—2022no—
303D4—2021no—
304D4—2021no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1011no—
312C2×C4—1011no—
313C2×C4—1011no—
314C2×C4—4042no—
315C2×C4—4042no—
316C2×C4—4042no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4D4×D10111yescharacteristic
323C2×C4C23×D10111yescharacteristic
324C8—21no—
325C8—21no—
326C8—2021no—
327C8—2021no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—1012no—
332C22—4041no—
333C22—4041no—
334C22—4041no—
335C22—2021no—
336C22—2021no—
337C22C2×D4×D10111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—4041no—
344C4—4041no—
345C4—2021no—
346C4—2021no—
347C4—2022no—
348C4—2021no—
349C4#41×D10111yescharacteristic
350C4#1569230011yescharacteristic
351C2—41no—
352C2—8081no—
353C2—2022no—
354C2#2026581611yescharacteristic
355C2#135×D10111yescharacteristic
356C2#2026599011yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#467×C101C2
socleC22×C101C22×D4
prime radical#20265837C2
hypercenter#467D101
hypocenterC101#3300
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #20265861C2semidirect
3. #20265533C2semidirect
4. #20265515C2semidirect
5. #20265492C2semidirect
6. #20265483C2semidirect
7. #20265409C2semidirect
8. #20265401C2semidirect
9. #20265337C2semidirect
10. #20265331C2semidirect
11. #20265269C2semidirect
12. C8×C22⋊D101C2semidirect
13. #20265174C2semidirect
14. #103×D101C2semidirect
15. #467×C101C2semidirect
179. C101#3300semidirect
186. #467D101semidirect

Maximal subgroups

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

subgroupindex
#202658612
#202658372
#202655332
#202655152
#202654922
#202654832
#202654092
#202654012
#202653372
#202653312
#202652692
C8×C22⋊D1012
#202651742
#103×D1012
#467×C1012
#3300101