Finite Group Databasebeta

← #46837448

Subgroups — #46837448

Order 34688.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#16864643C211yesmaximal, characteristic
2C22×#16864641C211yesmaximal, characteristic
3C2×#22576158C211yesmaximal, characteristic, complemented
4C2×#22576083C211yesmaximal, characteristic, complemented
5#31773798C211yesmaximal, characteristic
6#31773651C211yesmaximal, characteristic, complemented
7C2×#103×C271C211yesmaximal, characteristic, complemented
8C22×Dic542C2211yescharacteristic
9C22×D1084C2211yescharacteristic
10C2×#16864643—22no—
11C2×#16864643—22no—
12C2×#16864643—22no—
13C2×#16864641—22no—
14C2×#16864641—22no—
15C2×#16864641—22no—
16C22×C271⋊C8C2211yescharacteristic
17#22576158—22no—
18#22576158—22no—
19C2×Dic271⋊C4—22no—
20C2×(C2×D271)⋊C4—22no—
21C2×C4×Dic271C2211yescharacteristic
22#22576083—22no—
23C22×Q8×C271C2211yescharacteristic
24C22×C8×C271C2211yescharacteristic
25C2×C4⋊C4×C271C2211yescharacteristic
26#103×C271—22no—
27C23×D271—21no—
28C2×Dic542—21no—
29C2×Dic542—21no—
30C2×Dic542—21no—
31C2×Dic542—21no—
32C2×Dic542—21no—
33C2×Dic542D412yes—
34C2×D1084—21no—
35C2×D1084—21no—
36C2×D1084—21no—
37C2×D1084—21no—
38C2×D1084—21no—
39C2×D1084D412yes—
40C22×Dic271—21no—
41C22×Dic271D412yes—
42#16864643—28no—
43#16864641—28no—
44C2×C271⋊C8—21no—
45C2×C271⋊C8—21no—
46C2×C271⋊C8D412yes—
47Dic271⋊C4—28no—
48(C2×D271)⋊C4—44no—
49C4×Dic271—21no—
50C4×Dic271—21no—
51C2×Q8×C271—21no—
52C2×Q8×C271—21no—
53C2×Q8×C271—21no—
54C2×Q8×C271—21no—
55C2×Q8×C271—21no—
56C2×Q8×C271D412yes—
57C22×C4×C271—21no—
58C22×C4×C271—21no—
59C22×C4×C271C2311yescharacteristic
60C2×C8×C271—21no—
61C2×C8×C271—21no—
62C2×C8×C271D412yes—
63C4⋊C4×C271—21no—
64C4⋊C4×C271—21no—
65C22×D271—22no—
66C22×D271—41no—
67C22×D271—41no—
68C22×D271—42no—
69C22×D271—42no—
70Dic542—22no—
71Dic542—22no—
72Dic542—41no—
73Dic542—41no—
74D1084—22no—
75D1084—22no—
76D1084—41no—
77D1084—41no—
78C2×Dic271—22no—
79C2×Dic271—22no—
80C2×Dic271—22no—
81C2×Dic271—41no—
82C2×Dic271—41no—
83C2×Dic271SD1614yescomplemented
84C271⋊C8—22no—
85C23×C271C2×D411yescharacteristic
86Q8×C271—22no—
87Q8×C271—22no—
88Q8×C271—41no—
89Q8×C271—41no—
90C2×C4×C271—22no—
91C2×C4×C271—41no—
92C2×C4×C271—41no—
93C2×C4×C271—41no—
94C2×C4×C271—42no—
95C2×C4×C271C2×D411yescharacteristic
96C2×C4×C271C2×D411yescharacteristic
97C2×C4×C271C2×D411yescharacteristic
98C2×C4×C271C2×D411yescharacteristic
99C2×C4×C271C2×D411yescharacteristic
100C2×C4×C271#4111yescharacteristic
101C8×C271—22no—
102C22×C271#12011yescharacteristic
103C22×C271#12111yescharacteristic
104C22×C271#12711yescharacteristic
105C22×C271C2×SD1612yes—
106C22×C271#13712yes—
107C2×D271—42no—
108C2×D271—44no—
109C2×D271—82no—
110C2×D271—81no—
111C2×D271—81no—
112C4×C271—42no—
113C4×C271—81no—
114C4×C271#12011yescharacteristic
115C4×C271#12011yescharacteristic
116C4×C271#12111yescharacteristic
117C4×C271#12111yescharacteristic
118Dic271—24no—
119Dic271—42no—
120C2×C271#39211yescharacteristic
121C2×C271#44812yes—
122C2×C271#46112yes—
123C2×C271#49211yescharacteristic
124C2×C271#49711yescharacteristic
125D271—82no—
126C271#201311yescharacteristic, complemented, Sylow
127#2013—2711nomaximal, Sylow
128C22×SD16—2711no—
129C22×SD16—2711no—
130C2×C4⋊Q8—2711no—
131C2×#103—2711no—
132C2×#103D27111yescharacteristic, complemented
133#390—2711no—
134#336—2711no—
135C22×Q8—2711no—
136C22×Q8C2×D27111yescharacteristic
137C22×D4—2711no—
138C2×SD16—5422no—
139C2×SD16—5422no—
140C2×SD16—5422no—
141C2×SD16—5422no—
142C2×SD16—5422no—
143C2×SD16—5422no—
144C22×C8—2711no—
145C22×C8C2×D27111yescharacteristic
146C4⋊Q8—5422no—
147C4⋊Q8—5422no—
148C2×C4⋊C4—5422no—
149C2×C4⋊C4C2×D27111yescharacteristic
150C2×C22⋊C4—5422no—
151C2×C42—2711no—
152#103—22no—
153#103—5422no—
154C24—5421no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—2712no—
161C2×Q8—5421no—
162C2×Q8—5421no—
163C2×Q8—5421no—
164C2×Q8—5421no—
165C2×Q8—5421no—
166C2×Q8D108412yes—
167C2×D4—2712no—
168C2×D4—5421no—
169C2×D4—5421no—
170C2×D4—5421no—
171C2×D4—5421no—
172C2×D4—5421no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—2712no—
176C22×C4—5421no—
177C22×C4C22×D27111yescharacteristic
178SD16—5428no—
179SD16—5428no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—2712no—
183C2×C8—5421no—
184C2×C8—5421no—
185C2×C8C22⋊D27112yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—5428no—
189C22⋊C4—10844no—
190C42—5421no—
191C42—5421no—
192C23—5422no—
193C23—10841no—
194C23—10841no—
195C23—10842no—
196C23—10842no—
197C23D4×D27111yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—5422no—
203Q8—5422no—
204Q8—10841no—
205Q8—10841no—
206D4—5422no—
207D4—5422no—
208D4—10841no—
209D4—10841no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—2714no—
216C2×C4—5422no—
217C2×C4—5422no—
218C2×C4—5422no—
219C2×C4—10841no—
220C2×C4—10841no—
221C2×C4D4×D27111yescharacteristic
222C2×C4D4×D27111yescharacteristic
223C2×C4D4×D27111yescharacteristic
224C2×C4C2×C22⋊D27111yescharacteristic
225C2×C4C2×D108411yescharacteristic
226C2×C4#1686465411yescharacteristic
227C8—22no—
228C8—5422no—
229C22—10842no—
230C22—10844no—
231C22—21681no—
232C22—21681no—
233C22—21682no—
234C22#2257613011yescharacteristic
235C22#2257613311yescharacteristic
236C22#2257615611yescharacteristic
237C22SD16×D27112yes—
238C22#2257619212yes—
239C4—42no—
240C4—81no—
241C4—5424no—
242C4—10842no—
243C4#2257612811yescharacteristic
244C4#2257613011yescharacteristic
245C4#2257613311yescharacteristic
246C4#2257613411yescharacteristic
247C2—21682no—
248C2#3177380611yescharacteristic
249C2#3177403712yes—
250C2#3177410112yes—
251C2#3177426811yescharacteristic
252C2#3177429011yescharacteristic
subgroupG/H
centerC23D4×D271
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
FittingC2×#103×C271C2
socleC23×C271C2×D4
prime radicalC22×D1084C22
hypercenterC2×#103D271
hypocenterC271#2013
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#22576158C2semidirect
4. C2×#22576083C2semidirect
6. #31773651C2semidirect
7. C2×#103×C271C2semidirect
83. C2×Dic271SD16semidirect
126. C271#2013semidirect
132. C2×#103D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646432
C22×#168646412
C2×#225761582
C2×#225760832
#317737982
#317736512
C2×#103×C2712
#2013271