Finite Group Databasebeta

← #46845655

Subgroups — #46845655

Order 34688.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22576208C211yesmaximal, characteristic
2C2×#22576191C211yesmaximal, characteristic, complemented
3#31774471C211yesmaximal, characteristic, complemented
4#31774058C214yesmaximal, complemented
5#31774051C214yesmaximal, complemented
6#31773906C211yesmaximal, characteristic, complemented
7#31773894C211yesmaximal, characteristic, complemented
8#31773863C211yesmaximal, characteristic, complemented
9C22×Q16×C271C211yesmaximal, characteristic, complemented
10#22576208—24no—
11#22576208—24no—
12C2×#16864655—21no—
13C2×#16864655C2212yes—
14C2×#16864654—22no—
15C2×#16864654C2211yescharacteristic
16C2×#16864654C2211yescharacteristic
17C2×Q8×D271—21no—
18C2×Q8×D271C2212yes—
19#22576191—24no—
20C2×#16864644C2212yes—
21C2×#16864641C2212yes—
22C2×#16864638C2211yescharacteristic
23#22576158—21no—
24#22576149—21no—
25#22576143—22no—
26#22576136C2214yes—
27#22576119C2211yescharacteristic
28#22576085C2212yes—
29#22576084C2214yes—
30#22576082C2212yes—
31(C2×D271)⋊C8C2214yes—
32C22×Q8×C271C2211yescharacteristic
33C22×Q8×C271C2211yescharacteristic
34C2×Q16×C271—24no—
35C2×Q16×C271C2214yes—
36C22×C8×C271C2211yescharacteristic
37#16864655—28no—
38#16864655—24no—
39#16864655—24no—
40#16864654—216no—
41#16864654—22no—
42#16864654—22no—
43#16864654—22no—
44#16864654D414yes—
45C2×Dic542—22no—
46C2×Dic542D411yescharacteristic
47C2×Dic542C2311yescharacteristic
48Q8×D271—28no—
49Q8×D271—24no—
50Q8×D271—24no—
51C2×D1084—22no—
52C2×D1084D411yescharacteristic
53C2×D1084C2311yescharacteristic
54C2×C22⋊D271—21no—
55C2×C22⋊D271D412yes—
56C2×C4×D271—22no—
57C2×C4×D271—22no—
58C2×C4×D271—21no—
59C2×C4×D271C2312yes—
60#16864644—24no—
61#16864641—24no—
62#16864638—22no—
63C2×C271⋊C8C2312yes—
64C4⋊Dic271C2312yes—
65Dic271⋊C4—24no—
66C22⋊Dic271—21no—
67(C2×D271)⋊C4—24no—
68C4×Dic271—21no—
69C2×Q8×C271—21no—
70C2×Q8×C271—22no—
71C2×Q8×C271—21no—
72C2×Q8×C271—22no—
73C2×Q8×C271—22no—
74C2×Q8×C271D414yes—
75C2×Q8×C271C2312yes—
76C2×Q8×C271C2312yes—
77C22×C4×C271—21no—
78C22×C4×C271—21no—
79C22×C4×C271C2311yescharacteristic
80Q16×C271—28no—
81C2×C8×C271—22no—
82C2×C8×C271C2312yes—
83C22×D271—21no—
84C22×D271C2×D412yes—
85Dic542—28no—
86Dic542—22no—
87Dic542—21no—
88Dic542C2×D412yes—
89D1084—28no—
90D1084—22no—
91D1084—21no—
92D1084C2×D412yes—
93C22⋊D271—24no—
94C22⋊D271—24no—
95C2×Dic271—21no—
96C2×Dic271—22no—
97C2×Dic271C2×D412yes—
98C4×D271—28no—
99C4×D271—28no—
100C4×D271—24no—
101C4×D271—22no—
102C4×D271—22no—
103C271⋊C8—22no—
104C23×C271C2×D411yescharacteristic
105Q8×C271—24no—
106Q8×C271—22no—
107Q8×C271—22no—
108Q8×C271—22no—
109Q8×C271—42no—
110Q8×C271C2×D414yes—
111C2×C4×C271—24no—
112C2×C4×C271—22no—
113C2×C4×C271—22no—
114C2×C4×C271—42no—
115C2×C4×C271C2×D411yescharacteristic
116C2×C4×C271C2×D412yes—
117C2×C4×C271C2×D412yes—
118C2×C4×C271C2411yescharacteristic
119C8×C271—22no—
120C22×C271—22no—
121C22×C271#12012yes—
122C22×C271C22×D411yescharacteristic
123C2×D271—22no—
124C2×D271—22no—
125C2×D271—24no—
126C2×D271—42no—
127C4×C271—24no—
128C4×C271—42no—
129C4×C271#12012yes—
130C4×C271C22×D411yescharacteristic
131C4×C271C22×D411yescharacteristic
132Dic271—22no—
133Dic271—22no—
134Dic271—42no—
135C2×C271—22no—
136C2×C271C2×#12011yescharacteristic
137C2×C271#57812yes—
138D271—42no—
139D271—42no—
140C271#301111yescharacteristic, complemented, Sylow
141#3011—2711nomaximal, Sylow
142C2×2−1+4—2711no—
143C2×#137—2711no—
144C22×Q16D27111yescharacteristic, complemented
145#536—2711no—
146#452—2714no—
147#451—2714no—
148#419—2711no—
149#416—2711no—
150#408—2711no—
1512−1+4—5424no—
1522−1+4—5424no—
153C2×#41—2711no—
154C2×#41—2712no—
155C2×#41—2711no—
156C2×#41—5422no—
157C2×#41—5421no—
158C22×Q8—2712no—
159C22×Q8—5421no—
160C22×Q8C2×D27111yescharacteristic
161C22×Q8C2×D27111yescharacteristic
162#137—5424no—
163C2×Q16—24no—
164C2×Q16—2712no—
165C2×Q16C2×D27114yes—
166C2×SD16—2712no—
167C2×M16—2711no—
168C22×C8C2×D27111yescharacteristic
169C4⋊Q8—5421no—
170#124—5421no—
171#123—5422no—
172C22⋊Q8—2714no—
173#117—2711no—
174#103—2714no—
175#103—2712no—
176#102—2712no—
177C22⋊C8—2714no—
178#41—2714no—
179#41—54216no—
180#41—5424no—
181#41—5428no—
182#41—5424no—
183#41—5422no—
184#41—5422no—
185#41—5422no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—2711no—
192C2×Q8—2711no—
193C2×Q8—5422no—
194C2×Q8—5424no—
195C2×Q8—5428no—
196C2×Q8—5424no—
197C2×Q8C22⋊D27114yes—
198C2×Q8C22×D27112yes—
199C2×Q8C22×D27112yes—
200C2×D4—2712no—
201C2×D4—2711no—
202C2×D4—2711no—
203C2×D4—5422no—
204C2×D4—5421no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—2712no—
208C22×C4—5422no—
209C22×C4—5422no—
210C22×C4—5421no—
211C22×C4C22×D27111yescharacteristic
212Q16—28no—
213Q16—5424no—
214SD16—5424no—
215M16—5422no—
216C2×C8—22no—
217C2×C8—2712no—
218C2×C8C22×D27112yes—
219C4⋊C4—2712no—
220C4⋊C4—5424no—
221C22⋊C4—5424no—
222C22⋊C4—5421no—
223C42—5421no—
224C23—2712no—
225C23—5421no—
226C23D4×D27111yescharacteristic
227Q8—24no—
228Q8—22no—
229Q8—22no—
230Q8—22no—
231Q8—42no—
232Q8—2712no—
233Q8—5428no—
234Q8—5422no—
235Q8—5421no—
236Q8C2×C22⋊D27114yes—
237D4—2712no—
238D4—5428no—
239D4—5424no—
240D4—5424no—
241D4—5422no—
242D4—5421no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—2712no—
248C2×C4—5428no—
249C2×C4—5428no—
250C2×C4—5421no—
251C2×C4—5424no—
252C2×C4—5422no—
253C2×C4—5422no—
254C2×C4—5422no—
255C2×C4D4×D27111yescharacteristic
256C2×C4D4×D27112yes—
257C2×C4C2×C22⋊D27112yes—
258C2×C4C23×D27111yescharacteristic
259C8—22no—
260C8—5422no—
261C22—22no—
262C22—5422no—
263C22—5422no—
264C22—5424no—
265C22—10842no—
266C22#2257612812yes—
267C22C2×D4×D27111yescharacteristic
268C4—24no—
269C4—42no—
270C4—5422no—
271C4—5422no—
272C4—10842no—
273C4#2257612812yes—
274C4C2×D4×D27111yescharacteristic
275C4C22×C22⋊D27111yescharacteristic
276C2—22no—
277C2—10842no—
278C2—10842no—
279C2C2×#2257612811yescharacteristic
280C2#3177471012yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC22×Q16×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576208C2
hypercenterC22×Q16D271
hypocenterC271#3011
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#22576191C2semidirect
3. #31774471C2semidirect
4. #31774058C2semidirect
5. #31774051C2semidirect
6. #31773906C2semidirect
7. #31773894C2semidirect
8. #31773863C2semidirect
9. C22×Q16×C271C2semidirect
140. C271#3011semidirect
144. C22×Q16D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225762082
C2×#225761912
#317744712
#317740582
#317740512
#317739062
#317738942
#317738632
C22×Q16×C2712
#3011271