Finite Group Databasebeta

← #46837504

Subgroups — #46837504

Order 34688.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#22576159C211yesmaximal, characteristic
2C2×#22576085C211yesmaximal, characteristic
3C2×#22576083C211yesmaximal, characteristic
4#31773802C211yesmaximal, characteristic
5#31773659C211yesmaximal, characteristic
6#31773651C211yesmaximal, characteristic
7#340×C271C211yesmaximal, characteristic
8C22×Dic542—21no—
9C22×Dic542C2211yescharacteristic
10C22×C271⋊C8C2211yescharacteristic
11#22576159—22no—
12#22576159—22no—
13C2×C4⋊Dic271C2211yescharacteristic
14C2×Dic271⋊C4—21no—
15C2×C4×Dic271C2211yescharacteristic
16#22576085—22no—
17#22576083—22no—
18C271⋊#95—21no—
19C271⋊#95—21no—
20C22×C8×C271C2211yescharacteristic
21C2×C4⋊C4×C271C2211yescharacteristic
22C2×C4⋊C4×C271C2211yescharacteristic
23C2×Dic542—24no—
24C2×Dic542—21no—
25C2×Dic542—21no—
26C2×Dic542—21no—
27C2×Dic542—21no—
28C2×Dic542—21no—
29C2×Dic542—42no—
30C2×Dic542—42no—
31C2×Dic542Q812yescomplemented
32C22×Dic271—21no—
33C22×Dic271—21no—
34C22×Dic271D411yescharacteristic
35C22×Dic271D411yescharacteristic
36C2×C271⋊C8—21no—
37C2×C271⋊C8—21no—
38C2×C271⋊C8—21no—
39C4⋊Dic271—21no—
40C4⋊Dic271D412yes—
41Dic271⋊C4—24no—
42C4×Dic271—21no—
43C4×Dic271—21no—
44C22×C4×C271—21no—
45C22×C4×C271—21no—
46C22×C4×C271C2311yescharacteristic
47C2×C8×C271—21no—
48C2×C8×C271—21no—
49C2×C8×C271—21no—
50C4⋊C4×C271—21no—
51C4⋊C4×C271—21no—
52C4⋊C4×C271—21no—
53C4⋊C4×C271D412yes—
54Dic542—22no—
55Dic542—22no—
56Dic542—48no—
57Dic542—41no—
58Dic542—41no—
59C2×Dic271—21no—
60C2×Dic271—21no—
61C2×Dic271—21no—
62C2×Dic271—21no—
63C2×Dic271—22no—
64C2×Dic271—22no—
65C2×Dic271—41no—
66C2×Dic271—41no—
67C2×Dic271—41no—
68C2×Dic271—41no—
69C2×Dic271SD1612yes—
70C2×Dic271Q1612yes—
71C271⋊C8—41no—
72C23×C271C2×D411yescharacteristic
73C2×C4×C271—22no—
74C2×C4×C271—41no—
75C2×C4×C271—41no—
76C2×C4×C271—41no—
77C2×C4×C271—41no—
78C2×C4×C271—41no—
79C2×C4×C271C2×D411yescharacteristic
80C2×C4×C271C2×D411yescharacteristic
81C2×C4×C271C2×Q811yescharacteristic
82C2×C4×C271#4111yescharacteristic
83C2×C4×C271#4111yescharacteristic
84C2×C4×C271#4111yescharacteristic
85C8×C271—41no—
86C22×C271#12011yescharacteristic
87C22×C271C22⋊Q811yescharacteristic
88C22×C271#12411yescharacteristic
89C22×C271C2×SD1611yescharacteristic
90C22×C271C2×Q1611yescharacteristic
91C22×C271AGL(1,ℤ/8)11yescharacteristic
92C22×C271#13711yescharacteristic
93C4×C271—42no—
94C4×C271—81no—
95C4×C271C22⋊Q811yescharacteristic
96C4×C271C22⋊Q811yescharacteristic
97C4×C271#12411yescharacteristic
98C4×C271#12411yescharacteristic
99Dic271—22no—
100Dic271—22no—
101Dic271—42no—
102Dic271—42no—
103C2×C271#39311yescharacteristic
104C2×C271#45011yescharacteristic
105C2×C271#45111yescharacteristic
106C2×C271#47511yescharacteristic
107C2×C271#47711yescharacteristic
108C2×C271#48711yescharacteristic
109C2×C271#48911yescharacteristic
110C271#202411yescharacteristic, complemented, Sylow
111#2024—2711nomaximal, Sylow
112C2×C4⋊Q8—2711no—
113C2×#103—2711no—
114C2×#103—2711no—
115#391—2711no—
116#340—2711no—
117#340D27111yescharacteristic
118#336—2711no—
119C22×Q8—2711no—
120C22×Q8—5421no—
121C22×C8—2711no—
122C22×C8C2×D27111yescharacteristic
123C4⋊Q8—5422no—
124C4⋊Q8—5422no—
125C2×C4⋊C4—2711no—
126C2×C4⋊C4—5421no—
127C2×C4⋊C4C2×D27111yescharacteristic
128C2×C4⋊C4C2×D27111yescharacteristic
129C2×C42—2711no—
130#103—5422no—
131#103—5422no—
132#95—5421no—
133#95—5421no—
134C2×Q8—2712no—
135C2×Q8—5424no—
136C2×Q8—5421no—
137C2×Q8—5421no—
138C2×Q8—5421no—
139C2×Q8—5421no—
140C2×Q8—5421no—
141C2×Q8—10842no—
142C2×Q8—10842no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—2711no—
146C22×C4—2711no—
147C22×C4—5421no—
148C22×C4—5421no—
149C22×C4C22×D27111yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—5421no—
154C2×C8—5421no—
155C2×C8—5421no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—2712no—
160C4⋊C4—5424no—
161C4⋊C4—5421no—
162C4⋊C4D108412yes—
163C42—5421no—
164C42—5421no—
165C23D4×D27111yescharacteristic
166Q8—5422no—
167Q8—5422no—
168Q8—10848no—
169Q8—10841no—
170Q8—10841no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—2712no—
178C2×C4—2712no—
179C2×C4—5421no—
180C2×C4—5421no—
181C2×C4—5421no—
182C2×C4—5421no—
183C2×C4—5422no—
184C2×C4—5422no—
185C2×C4—10841no—
186C2×C4—10841no—
187C2×C4—10841no—
188C2×C4—10841no—
189C2×C4D4×D27111yescharacteristic
190C2×C4C2×D108411yescharacteristic
191C2×C4Q8×D27111yescharacteristic
192C2×C4#1686465311yescharacteristic
193C2×C4#1686465411yescharacteristic
194C2×C4#1686465411yescharacteristic
195C8—41no—
196C8—10841no—
197C22#2257613011yescharacteristic
198C22#2257613811yescharacteristic
199C22#2257614611yescharacteristic
200C22SD16×D27111yescharacteristic
201C22Q16×D27111yescharacteristic
202C22#2257618411yescharacteristic
203C22#2257618911yescharacteristic
204C4—42no—
205C4—81no—
206C4—5422no—
207C4—5422no—
208C4—10842no—
209C4—10842no—
210C4#2257613811yescharacteristic
211C4#2257614011yescharacteristic
212C4#2257614611yescharacteristic
213C4#2257614811yescharacteristic
214C2#3177381011yescharacteristic
215C2#3177404911yescharacteristic
216C2#3177405411yescharacteristic
217C2#3177418111yescharacteristic
218C2#3177419111yescharacteristic
219C2#3177424611yescharacteristic
220C2#3177425511yescharacteristic
subgroupG/H
centerC23D4×D271
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#340×C271C2
socle, prime radicalC23×C271C2×D4
hypercenter#340D271
hypocenterC271#2024
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic542Q8semidirect
110. C271#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#225761592
C2×#225760852
C2×#225760832
#317738022
#317736592
#317736512
#340×C2712
#2024271