Finite Group Databasebeta

← #27183021

Subgroups — #27183021

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#20265867C211yesmaximal, characteristic, complemented
2#20265836C211yesmaximal, characteristic
3#20265532C211yesmaximal, characteristic, complemented
4#20265519C211yesmaximal, characteristic, complemented
5#20265491C211yesmaximal, characteristic, complemented
6#20265485C211yesmaximal, characteristic, complemented
7#20265418C211yesmaximal, characteristic, complemented
8#20265406C211yesmaximal, characteristic, complemented
9#20265400C211yesmaximal, characteristic, complemented
10#20265341C211yesmaximal, characteristic, complemented
11#20265335C211yesmaximal, characteristic, complemented
12C4×Dic404C211yesmaximal, characteristic, complemented
13#20265174C211yesmaximal, characteristic, complemented
14C2×#15692180C211yesmaximal, characteristic, complemented
15C8×D4×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×Dic202C2211yescharacteristic
18C2×Dic404C2211yescharacteristic
19C2×Dic404C2211yescharacteristic
20C2×#13108063C2211yescharacteristic
21#15692248—21no—
22#15692247C2211yescharacteristic
23#15692245—21no—
24#15692243C2211yescharacteristic
25#15692241—21no—
26#15692236—21no—
27#15692234C2211yescharacteristic
28#15692234C2211yescharacteristic
29#15692234C2211yescharacteristic
30#15692233—21no—
31#15692229C2211yescharacteristic
32#15692227—21no—
33C4×Dic202C2211yescharacteristic
34C4×Dic202C2211yescharacteristic
35#15692215—21no—
36#15692214C2211yescharacteristic
37#15692211—21no—
38C2×C4⋊Dic101C2211yescharacteristic
39#15692189C2211yescharacteristic
40#15692189C2211yescharacteristic
41#15692187C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692180C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692180C2211yescharacteristic
48#15692180C2211yescharacteristic
49#15692177C2211yescharacteristic
50#15692177C2211yescharacteristic
51#15692177C2211yescharacteristic
52C22×C8×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic
55C4⋊C8×C101C2211yescharacteristic
56C22⋊C8×C101C2211yescharacteristic
57C22⋊C8×C101C2211yescharacteristic
58C4×C8×C101C2211yescharacteristic
59#13108074—22no—
60#13108074—41no—
61C2×Dic202—22no—
62C2×Dic202—21no—
63C2×Dic202—41no—
64C2×Dic202C2311yescharacteristic
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67C2×D404C2311yescharacteristic
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C22×Dic101—21no—
71C22×Dic101—21no—
72C2×C4×D101—21no—
73Dic404—22no—
74Dic404—41no—
75#13108063—41no—
76C4⋊Dic101—21no—
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053—21no—
87#13108053—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101—21no—
90C22⋊Dic101—21no—
91C22⋊Dic101—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95C4×Dic101—21no—
96C4×Dic101—21no—
97C4×Dic101—21no—
98C2×D4×C101D411yescharacteristic
99C22×C4×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101C2×C8×C101—41no—
102C2×C8×C101—41no—
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101D411yescharacteristic
108C22⋊C4×C101D411yescharacteristic
109C22⋊C4×C101D411yescharacteristic
110C42×C101C2311yescharacteristic
111C23×C101C2×D411yescharacteristic
112C23×C101C2×D411yescharacteristic
113C22×D101—21no—
114D4×C101—22no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101C2×D411yescharacteristic
120C2×C4×C101C2×D411yescharacteristic
121C2×C4×C101C2×D411yescharacteristic
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2411yescharacteristic
124C22⋊D101—42no—
125C22⋊D101—42no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—42no—
134C2×Dic101—42no—
135D404—21no—
136D404—21no—
137C4×D101—42no—
138Dic202—21no—
139Dic202—21no—
140Dic202—21no—
141Dic202—21no—
142Dic202—21no—
143Dic202—42no—
144Dic202#4112yes—
145C8×C101—21no—
146C8×C101—21no—
147C8×C101—21no—
148C8×C101—21no—
149C22×C101—21no—
150C22×C101—21no—
151C22×C101—21no—
152C22×C101—21no—
153C22×C101—21no—
154C22×C101#13512yes—
155C22×C101C22×D411yescharacteristic
156C2×D101—41no—
157C2×D101—41no—
158C2×D101—41no—
159C4×C101—21no—
160C4×C101—21no—
161C4×C101—21no—
162C4×C101—21no—
163C4×C101C2×#4111yescharacteristic
164C4×C101H(5,𝔽2)11yescharacteristic
165Dic101—22no—
166Dic101—41no—
167Dic101—41no—
168Dic101—41no—
169Dic101—41no—
170Dic101—41no—
171Dic101—41no—
172C2×C101—22no—
173C2×C101—41no—
174C2×C101#54611yescharacteristic
175C2×C101C2×#13511yescharacteristic
176C2×C101#57811yescharacteristic
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#467—1011no—
187#465—1011no—
188#464—1011no—
189#452—1011no—
190#451—1011no—
191C4×Q16—1011no—
192C8×D4D10111yescharacteristic, complemented
193#416—1011no—
194C2×#103—1011no—
195C2×#41—1011no—
196C22×Q8—1011no—
197C2×Q16—1011no—
198C2×Q16—1011no—
199C2×SD16—1011no—
200C22×C8C2×D10111yescharacteristic
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—1011no—
210C22⋊Q8—1011no—
211C22⋊Q8—2021no—
212#121—1011no—
213#121—2021no—
214C4×Q8—1011no—
215C4×Q8—1011no—
216C4×D4—2021no—
217C4×D4C2×D10111yescharacteristic
218#117—1011no—
219#117—2021no—
220C2×C4⋊C4—1011no—
221#107—1011no—
222#107—1011no—
223#106—1011no—
224C4⋊C8C2×D10111yescharacteristic
225#103—1011no—
226#103—1011no—
227#103—1011no—
228#103—1011no—
229#103—1011no—
230#103—1011no—
231#103—1011no—
232#102—1011no—
233#102—1011no—
234#102—1011no—
235C22⋊C8C2×D10111yescharacteristic
236C22⋊C8C2×D10111yescharacteristic
237C4×C8C2×D10111yescharacteristic
238#41—4041no—
239#41—2022no—
240C2×Q8—1011no—
241C2×Q8—1011no—
242C2×Q8—1011no—
243C2×Q8—4041no—
244C2×Q8—2022no—
245C2×Q8—2021no—
246C2×D4—1011no—
247C2×D4—2021no—
248C2×D4—2021no—
249C2×D4D40411yescharacteristic
250C22×C4—2021no—
251C22×C4—2021no—
252C22×C4—2021no—
253C22×C4C22×D10111yescharacteristic
254C22×C4C22×D10111yescharacteristic
255Q16—4041no—
256Q16—2022no—
257SD16—4041no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D10111yescharacteristic
261C2×C8C22×D10111yescharacteristic
262C2×C8C22×D10111yescharacteristic
263C2×C8C22×D10111yescharacteristic
264C4⋊C4—1011no—
265C4⋊C4—1011no—
266C4⋊C4—1011no—
267C4⋊C4—1011no—
268C4⋊C4—2021no—
269C4⋊C4—2021no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4—2021no—
276C4⋊C4D40411yescharacteristic
277C22⋊C4—2021no—
278C22⋊C4—2021no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4D40411yescharacteristic
285C22⋊C4D40411yescharacteristic
286C42—2021no—
287C42—2021no—
288C42—2021no—
289C42C22×D10111yescharacteristic
290C23—2021no—
291C23C2×D40411yescharacteristic
292C23C2×D40411yescharacteristic
293Q8—1012no—
294Q8—4042no—
295Q8—2021no—
296Q8—2021no—
297Q8—2021no—
298Q8—2021no—
299Q8—2021no—
300D4—22no—
301D4—4042no—
302D4—4042no—
303D4—2021no—
304D4—2021no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—4042no—
310C2×C4—4042no—
311C2×C4—4042no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4—2021no—
316C2×C4—2021no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4C2×D40411yescharacteristic
320C2×C4C2×D40411yescharacteristic
321C2×C4C2×D40411yescharacteristic
322C2×C4C2×D40411yescharacteristic
323C2×C4C23×D10111yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—4041no—
334C22—4041no—
335C22—4041no—
336C22#1569227712yes—
337C22C22×D40411yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—4041no—
343C4—4041no—
344C4—4041no—
345C4—4041no—
346C4—4041no—
347C4—4041no—
348C4—2022no—
349C4#41×D10111yescharacteristic
350C4#1569230011yescharacteristic
351C2—22no—
352C2—41no—
353C2—8081no—
354C2#2026582211yescharacteristic
355C2C2×#1569227711yescharacteristic
356C2#2026599411yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC8×D4×C101C2
socleC22×C101C22×D4
prime radical#20265836C2
hypercenterC8×D4D101
hypocenterC101#3300
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20265867C2semidirect
3. #20265532C2semidirect
4. #20265519C2semidirect
5. #20265491C2semidirect
6. #20265485C2semidirect
7. #20265418C2semidirect
8. #20265406C2semidirect
9. #20265400C2semidirect
10. #20265341C2semidirect
11. #20265335C2semidirect
12. C4×Dic404C2semidirect
13. #20265174C2semidirect
14. C2×#15692180C2semidirect
15. C8×D4×C101C2semidirect
179. C101#3300semidirect
192. C8×D4D101semidirect

Maximal subgroups

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

subgroupindex
#202658672
#202658362
#202655322
#202655192
#202654912
#202654852
#202654182
#202654062
#202654002
#202653412
#202653352
C4×Dic4042
#202651742
C2×#156921802
C8×D4×C1012
#3300101