Finite Group Databasebeta

← #27174420

Subgroups — #27174420

Order 12928.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×#15692220C211yesmaximal, characteristic, complemented
2C2×D4×Dic101C211yesmaximal, characteristic
3#20265117C211yesmaximal, characteristic, complemented
4#20265116C211yesmaximal, characteristic, complemented
5#20265105C212yesmaximal, complemented
6#20265091C211yesmaximal, characteristic
7#20265089C212yesmaximal, complemented
8C22⋊C4×Dic101C212yesmaximal, complemented
9C22⋊C4×Dic101C212yesmaximal, complemented
10#20265022C211yesmaximal, characteristic, complemented
11#394×C101C211yesmaximal, characteristic
12C22×Dic202C2211yescharacteristic
13C23×Dic101C2212yes—
14#15692220—24no—
15D4×Dic101—24no—
16C2×C4⋊Dic101C2211yescharacteristic
17C2×#13108053C2211yescharacteristic
18C2×#13108053C2211yescharacteristic
19C2×C22⋊Dic101C2212yes—
20C2×C22⋊Dic101C2212yes—
21C2×C22⋊Dic101C2212yes—
22C2×C4×Dic101C2212yes—
23C2×C4×Dic101C2211yescharacteristic
24C2×C4×Dic101C2211yescharacteristic
25C2×C4×Dic101C2211yescharacteristic
26#15692162C2212yes—
27#15692162C2211yescharacteristic
28#15692162C2212yes—
29#15692162C2212yes—
30#15692162C2212yes—
31#15692162C2212yes—
32C22×D4×C101C2211yescharacteristic
33C2×C4⋊C4×C101C2211yescharacteristic
34C2×C22⋊C4×C101C2212yes—
35C2×C22⋊C4×C101C2212yes—
36#95×C101C2211yescharacteristic
37C2×Dic202—22no—
38C2×Dic202—22no—
39C2×Dic202—22no—
40C22×Dic101—24no—
41C22×Dic101—24no—
42C22×Dic101—24no—
43C22×Dic101C2312yes—
44C22×Dic101C2312yes—
45C22×Dic101C2311yescharacteristic
46C22×Dic101C2311yescharacteristic
47C22×Dic101C2311yescharacteristic
48C22×Dic101C2311yescharacteristic
49C4⋊Dic101—22no—
50#13108053—22no—
51#13108053—22no—
52C22⋊Dic101—24no—
53C22⋊Dic101—24no—
54C22⋊Dic101—24no—
55C4×Dic101—22no—
56C4×Dic101—22no—
57C4×Dic101—24no—
58C4×Dic101D414yes—
59C24×C101C2312yes—
60C2×D4×C101—24no—
61C2×D4×C101—22no—
62C22×C4×C101C2312yes—
63C22×C4×C101C2311yescharacteristic
64C22×C4×C101C2311yescharacteristic
65C22×C4×C101C2311yescharacteristic
66C4⋊C4×C101—22no—
67C22⋊C4×C101—24no—
68C22⋊C4×C101—24no—
69C23×C101—22no—
70C23×C101—22no—
71C23×C101—22no—
72C23×C101—22no—
73C23×C101—22no—
74C23×C101—22no—
75C23×C101#4114yes—
76C23×C101C2411yescharacteristic
77D4×C101—28no—
78C2×C4×C101—21no—
79C2×C4×C101—22no—
80C2×C4×C101—21no—
81C2×C4×C101—21no—
82C2×C4×C101—22no—
83C2×C4×C101—22no—
84C2×C4×C101—22no—
85C2×C4×C101—22no—
86C2×C4×C101C2×D412yes—
87C2×C4×C101#4112yes—
88C2×Dic101—21no—
89C2×Dic101—21no—
90C2×Dic101—21no—
91C2×Dic101—21no—
92C2×Dic101—21no—
93C2×Dic101—22no—
94C2×Dic101—22no—
95C2×Dic101—22no—
96C2×Dic101—22no—
97C2×Dic101—22no—
98C2×Dic101—22no—
99C2×Dic101—22no—
100C2×Dic101—48no—
101C2×Dic101C2×D412yes—
102C2×Dic101C2×D412yes—
103C2×Dic101#4112yes—
104C2×Dic101#4114yes—
105Dic202—28no—
106C22×C101—24no—
107C22×C101—24no—
108C22×C101—24no—
109C22×C101—42no—
110C22×C101—42no—
111C22×C101—42no—
112C22×C101—42no—
113C22×C101C22×D411yescharacteristic
114C22×C101C2×#4111yescharacteristic
115C22×C101C2×#4112yes—
116C22×C101C2×#4112yes—
117C22×C101C2×#4111yescharacteristic
118C4×C101—22no—
119C4×C101—22no—
120C4×C101—41no—
121C4×C101—42no—
122Dic101—22no—
123Dic101—22no—
124Dic101—41no—
125Dic101—42no—
126Dic101—42no—
127Dic101#12414yes—
128C2×C101—44no—
129C2×C101#52511yescharacteristic
130C2×C101#52511yescharacteristic
131C2×C101C2×#12411yescharacteristic
132C2×C101#52912yes—
133C2×C101#52912yes—
134#2398—1011nomaximal, Sylow
135C101#239811yescharacteristic, complemented, Sylow
136C2×C4×Q8—1011no—
137C2×C4×D4—1011no—
138#400—1011no—
139#400—1011no—
140#397—1012no—
141#394—1011no—
142#394D10111yescharacteristic
143#393—1012no—
144C4×C22⋊C4—1012no—
145C4×C22⋊C4—1012no—
146#376—1011no—
147C22×Q8—1011no—
148C22×D4C2×D10111yescharacteristic
149C23×C4—1012no—
150C4×Q8—2024no—
151C4×D4—2024no—
152C2×C4⋊C4—1011no—
153C2×C4⋊C4—1011no—
154C2×C4⋊C4—1011no—
155C2×C4⋊C4C2×D10111yescharacteristic
156C2×C22⋊C4—1012no—
157C2×C22⋊C4—1012no—
158C2×C22⋊C4—1012no—
159C2×C22⋊C4C2×D10112yes—
160C2×C22⋊C4C2×D10112yes—
161C2×C42—1011no—
162C2×C42—1011no—
163C2×C42—1011no—
164C2×C42—1012no—
165#95—1012no—
166#95—1011no—
167#95—1012no—
168#95—1012no—
169#95—1012no—
170#95—1012no—
171#95C2×D10111yescharacteristic
172C24C22×D10112yes—
173C2×Q8—2022no—
174C2×Q8—2022no—
175C2×Q8—2022no—
176C2×D4—24no—
177C2×D4—22no—
178C22×C4—1011no—
179C22×C4—1011no—
180C22×C4—1011no—
181C22×C4—1012no—
182C22×C4—1012no—
183C22×C4—1011no—
184C22×C4—2024no—
185C22×C4—2024no—
186C22×C4—2024no—
187C22×C4C22×D10112yes—
188C22×C4C22×D10111yescharacteristic
189C22×C4C22×D10111yescharacteristic
190C22×C4C22×D10111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—2022no—
193C4⋊C4—2022no—
194C4⋊C4—2022no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—2024no—
198C22⋊C4—2024no—
199C22⋊C4—2024no—
200C42—1014no—
201C42—2024no—
202C42—2022no—
203C42—2022no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23#1310807314yes—
211C23C23×D10111yescharacteristic
212Q8—2028no—
213D4—28no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—1012no—
223C2×C4—1014no—
224C2×C4—1012no—
225C2×C4—1012no—
226C2×C4—4048no—
227C2×C4—2021no—
228C2×C4—2022no—
229C2×C4—2022no—
230C2×C4—2022no—
231C2×C4—2022no—
232C2×C4—2022no—
233C2×C4—2022no—
234C2×C4—2022no—
235C2×C4—2021no—
236C2×C4—2021no—
237C2×C4—2021no—
238C2×C4—2021no—
239C2×C4D4×D10112yes—
240C2×C4#1310807312yes—
241C22—24no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D10111yescharacteristic
249C22#41×D10111yescharacteristic
250C22#41×D10112yes—
251C22C2×#1310807312yes—
252C22C2×#1310807311yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—1014no—
258C4—4041no—
259C4—4042no—
260C4—4042no—
261C4—2022no—
262C4—2022no—
263C2—44no—
264C2#2026568311yescharacteristic
265C2#2026568711yescharacteristic
266C2#124×D10111yescharacteristic
267C2#2026570412yes—
268C2#2026570812yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#394×C101C2
prime radicalC22×D4×C101C22
hypercenter#394D101
hypocenterC101#2398
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#15692220C2semidirect
3. #20265117C2semidirect
4. #20265116C2semidirect
5. #20265105C2semidirect
7. #20265089C2semidirect
8. C22⋊C4×Dic101C2semidirect
9. C22⋊C4×Dic101C2semidirect
10. #20265022C2semidirect
135. C101#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#156922202
C2×D4×Dic1012
#202651172
#202651162
#202651052
#202650912
#202650892
C22⋊C4×Dic1012
#202650222
#394×C1012
#2398101