Finite Group Databasebeta

← #27172115

Subgroups — #27172115

Order 12928.

324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692232C211yesmaximal, characteristic
2#20265035C211yesmaximal, characteristic
3C22⋊C4×Dic101C211yesmaximal, characteristic
4#20264944C211yesmaximal, characteristic
5#20264943C211yesmaximal, characteristic
6#20264943C211yesmaximal, characteristic
7#342×C101C211yesmaximal, characteristic
8C22×Dic202C411yescharacteristic, complemented
9C23×Dic101C2211yescharacteristic
10#15692232—21no—
11#15692232—21no—
12#15692232—21no—
13#15692232—21no—
14C2×C4⋊Dic101C411yescharacteristic, complemented
15C2×#13108053—21no—
16C2×#13108053C411yescharacteristic, complemented
17C2×#13108053C411yescharacteristic, complemented
18C2×C22⋊Dic101C2211yescharacteristic
19C2×C22⋊Dic101C2211yescharacteristic
20C2×C22⋊Dic101C2211yescharacteristic
21C2×C4×Dic101—21no—
22#15692162—21no—
23#15692162—21no—
24C2×C22⋊C4×C101C2211yescharacteristic
25C2×C22⋊C4×C101C2211yescharacteristic
26C2×C22⋊C4×C101C2211yescharacteristic
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202—42no—
30C22×Dic101—21no—
31C22×Dic101—21no—
32C22×Dic101—42no—
33C22×Dic101C2×C411yescharacteristic
34C22×Dic101C2×C411yescharacteristic
35C22×Dic101C2×C411yescharacteristic
36C22×Dic101C2×C411yescharacteristic
37C22×Dic101D411yescharacteristic
38C22×Dic101D411yescharacteristic
39C22×Dic101D411yescharacteristic
40C22×Dic101D411yescharacteristic
41C4⋊Dic101—21no—
42C4⋊Dic101—21no—
43#13108053—21no—
44#13108053—21no—
45#13108053—21no—
46#13108053—21no—
47#13108053—42no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C22⋊Dic101—21no—
51C22⋊Dic101—21no—
52C22⋊Dic101—21no—
53C22⋊Dic101Q812yescomplemented
54C4×Dic101—42no—
55C24×C101C2311yescharacteristic
56C22×C4×C101—21no—
57C22×C4×C101—21no—
58C22×C4×C101C2×C411yescharacteristic
59C22×C4×C101C2×C411yescharacteristic
60C22⋊C4×C101—21no—
61C22⋊C4×C101—21no—
62C22⋊C4×C101—21no—
63C22⋊C4×C101—21no—
64C22⋊C4×C101—21no—
65C22⋊C4×C101Q812yescomplemented
66C23×C101—41no—
67C23×C101—41no—
68C23×C101C22×C411yescharacteristic
69C23×C101C2×D411yescharacteristic
70C23×C101C2×D411yescharacteristic
71C23×C101C2×Q811yescharacteristic
72C23×C101C2×Q811yescharacteristic
73C23×C101#4111yescharacteristic
74C23×C101#4111yescharacteristic
75C2×C4×C101—21no—
76C2×C4×C101—21no—
77C2×C4×C101—22no—
78C2×C4×C101—41no—
79C2×C4×C101—41no—
80C2×C4×C101—41no—
81C2×C4×C101—41no—
82C2×C4×C101—41no—
83C2×C4×C101—41no—
84C2×C4×C101—41no—
85C2×Dic101—22no—
86C2×Dic101—22no—
87C2×Dic101—22no—
88C2×Dic101—21no—
89C2×Dic101—21no—
90C2×Dic101—22no—
91C2×Dic101—41no—
92C2×Dic101—41no—
93C2×Dic101—41no—
94C2×Dic101—41no—
95C2×Dic101—41no—
96C2×Dic101—41no—
97C2×Dic101—41no—
98C2×Dic101—41no—
99C2×Dic101—41no—
100C2×Dic101—41no—
101C2×Dic101C22⋊C411yescharacteristic
102C2×Dic101C22⋊C411yescharacteristic
103C2×Dic101C22⋊C411yescharacteristic
104C2×Dic101C22⋊C411yescharacteristic
105Dic202—42no—
106Dic202—42no—
107C22×C101—21no—
108C22×C101—21no—
109C22×C101—21no—
110C22×C101—21no—
111C22×C101—21no—
112C22×C101—21no—
113C22×C101—41no—
114C22×C101—41no—
115C22×C101—41no—
116C22×C101—41no—
117C22×C101C2×C22⋊C411yescharacteristic
118C22×C101C4×Q811yescharacteristic
119C22×C101C4×Q811yescharacteristic
120C22×C101C22⋊Q811yescharacteristic
121C22×C101C22⋊Q811yescharacteristic
122C22×C101#12411yescharacteristic
123C22×C101C4⋊Q811yescharacteristic
124C4×C101—42no—
125C4×C101—41no—
126C4×C101—41no—
127C4×C101—81no—
128Dic101—21no—
129Dic101—21no—
130Dic101—21no—
131Dic101—42no—
132Dic101—41no—
133Dic101—41no—
134Dic101—81no—
135Dic101#9912yes—
136C2×C101—21no—
137C2×C101—21no—
138C2×C101—21no—
139C2×C101—21no—
140C2×C101—21no—
141C2×C101—21no—
142C2×C101#39111yescharacteristic
143C2×C101C2×#9911yescharacteristic
144C2×C101#41011yescharacteristic
145#1996—1011nomaximal, Sylow
146C101#199611yescharacteristic, complemented, Sylow
147C2×C22⋊Q8—1011no—
148#380—1011no—
149C4×C22⋊C4—1011no—
150#342—1011no—
151#342—1011no—
152#342—1011no—
153#342D10111yescharacteristic
154C22×Q8—1011no—
155C23×C4—1011no—
156C22⋊Q8—2021no—
157C22⋊Q8—2021no—
158C22⋊Q8—2021no—
159C22⋊Q8—2021no—
160C2×C4⋊C4—1011no—
161C2×C4⋊C4—1011no—
162C2×C4⋊C4—1011no—
163C2×C4⋊C4—2021no—
164C2×C22⋊C4—1011no—
165C2×C22⋊C4—1011no—
166C2×C22⋊C4—1011no—
167C2×C22⋊C4C2×D10111yescharacteristic
168C2×C22⋊C4C2×D10111yescharacteristic
169C2×C22⋊C4C2×D10111yescharacteristic
170C2×C42—2021no—
171#95—2021no—
172#95—2021no—
173C24C22×D10111yescharacteristic
174C2×Q8—4042no—
175C2×Q8—2021no—
176C2×Q8—2021no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—1011no—
180C22×C4—1011no—
181C22×C4—1011no—
182C22×C4—1011no—
183C22×C4—1011no—
184C22×C4—1011no—
185C22×C4—1011no—
186C22×C4—1011no—
187C22×C4—4042no—
188C22×C4—2021no—
189C22×C4—2021no—
190C22×C4C4×D10111yescharacteristic
191C22×C4C4×D10111yescharacteristic
192C4⋊C4—4042no—
193C4⋊C4—2021no—
194C4⋊C4—2021no—
195C4⋊C4—2021no—
196C4⋊C4—2021no—
197C4⋊C4—2021no—
198C4⋊C4—2021no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—1012no—
205C22⋊C4—2021no—
206C22⋊C4—2021no—
207C22⋊C4—2021no—
208C22⋊C4—2021no—
209C22⋊C4—2021no—
210C22⋊C4Dic20212yescomplemented
211C42—4042no—
212C23—41no—
213C23—41no—
214C23C2×C4×D10111yescharacteristic
215C23D4×D10111yescharacteristic
216C23D4×D10111yescharacteristic
217C23Q8×D10111yescharacteristic
218C23C2×Dic20211yescharacteristic
219C23#1310807311yescharacteristic
220C23#1310807411yescharacteristic
221Q8—4042no—
222Q8—4042no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—1011no—
234C2×C4—1011no—
235C2×C4—1011no—
236C2×C4—1011no—
237C2×C4—4041no—
238C2×C4—4041no—
239C2×C4—4041no—
240C2×C4—4041no—
241C2×C4—4041no—
242C2×C4—4041no—
243C2×C4—4041no—
244C2×C4—4041no—
245C2×C4—4041no—
246C2×C4—4041no—
247C2×C4—2022no—
248C2×C4—2022no—
249C2×C4—2022no—
250C2×C4—2021no—
251C2×C4—2021no—
252C2×C4—2022no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22C22⋊C4×D10111yescharacteristic
264C22#1569222011yescharacteristic
265C22C4×Dic20211yescharacteristic
266C22#1569223211yescharacteristic
267C22#1569223311yescharacteristic
268C22#1569224111yescharacteristic
269C22#1569225411yescharacteristic
270C4—42no—
271C4—41no—
272C4—41no—
273C4—81no—
274C4—8081no—
275C4—1012no—
276C4—4042no—
277C4—4041no—
278C4—4041no—
279C4—2021no—
280C4—2021no—
281C4—2021no—
282C2—21no—
283C2—21no—
284C2—21no—
285C2—21no—
286C2—21no—
287C2—21no—
288C2#2026508111yescharacteristic
289C2#99×D10111yescharacteristic
290C2#2026514911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#342×C101C2
socleC22×C101C2×C22⋊C4
prime radical, cosocleC24×C101C23
hypercenter#342D101
hypocenterC101#1996
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
8. C22×Dic202C4semidirect
14. C2×C4⋊Dic101C4semidirect
16. C2×#13108053C4semidirect
17. C2×#13108053C4semidirect
53. C22⋊Dic101Q8semidirect
65. C22⋊C4×C101Q8semidirect
146. C101#1996semidirect
210. C22⋊C4Dic202semidirect

Maximal subgroups

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

subgroupindex
C2×#156922322
#202650352
C22⋊C4×Dic1012
#202649442
#202649432
#342×C1012
#1996101