Finite Group Databasebeta

← #46835672

Subgroups — #46835672

Order 34688.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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
1#31774639C211yesmaximal, characteristic
2#31774291C211yesmaximal, characteristic, complemented
3#31774257C211yesmaximal, characteristic, complemented
4#31774256C211yesmaximal, characteristic, complemented
5#31773636C211yesmaximal, characteristic, complemented
6#31773636C211yesmaximal, characteristic, complemented
7#329×C271C211yesmaximal, characteristic, complemented
8C2×Dic1084—21no—
9C2×#16864643—21no—
10#22576160C2211yescharacteristic, complemented
11#22576149—21no—
12#22576148C2211yescharacteristic
13#22576148C2211yescharacteristic
14#22576146—21no—
15#22576146—21no—
16#22576130—21no—
17#22576130—21no—
18#22576128—21no—
19#22576085—21no—
20#22576084—21no—
21#22576082—21no—
22#22576081—21no—
23#22576071C2211yescharacteristic, complemented
24#124×C271C2211yescharacteristic
25#97×C271C2211yescharacteristic, complemented
26#97×C271C2211yescharacteristic, complemented
27C23×D271—21no—
28C2×Dic542—21no—
29C2×Dic542D411yescharacteristic
30C2×Dic542D411yescharacteristic
31C2×D1084—21no—
32C2×D1084D411yescharacteristic
33C2×D1084D411yescharacteristic
34C2×C22⋊D271—21no—
35C2×C22⋊D271—21no—
36Dic1084—42no—
37#16864643—42no—
38C2×C271⋊C8—21no—
39C4⋊Dic271—21no—
40C4⋊Dic271—21no—
41C22⋊Dic271—21no—
42(C2×D271)⋊C4—21no—
43(C2×D271)⋊C4—21no—
44(C2×D271)⋊C4—21no—
45(C2×D271)⋊C4—21no—
46(C2×D271)⋊C4—21no—
47(C2×D271)⋊C4—21no—
48C4×Dic271—21no—
49C2×Q8×C271D411yescharacteristic
50C2×D4×C271D411yescharacteristic
51C2×C8×C271—21no—
52C2×C8×C271—21no—
53C22⋊C4×C271—21no—
54C22⋊C4×C271—21no—
55C42×C271C2311yescharacteristic
56C22×D271—21no—
57C22×D271—21no—
58C22×D271—81no—
59C22×D271—81no—
60C22×D271—81no—
61Dic542—41no—
62Dic542—41no—
63Dic542—41no—
64Dic542—41no—
65D1084—41no—
66D1084—41no—
67D1084—81no—
68C22⋊D271—81no—
69C22⋊D271—81no—
70C2×Dic271—21no—
71C2×Dic271—21no—
72C2×Dic271—41no—
73C271⋊C8—41no—
74C23×C271—21no—
75Q8×C271—41no—
76D4×C271—41no—
77C2×C4×C271—21no—
78C2×C4×C271C2×D411yescharacteristic
79C2×C4×C271C2×D411yescharacteristic
80C2×C4×C271C2×D411yescharacteristic
81C8×C271—22no—
82C8×C271—41no—
83C22×C271—41no—
84C22×C271—41no—
85C22×C271—41no—
86C22×C271#12011yescharacteristic
87C2×D271—41no—
88C2×D271—41no—
89C2×D271—41no—
90C2×D271—41no—
91C2×D271—41no—
92C2×D271—41no—
93C2×D271—84no—
94C4×C271—21no—
95C4×C271—21no—
96C4×C271—21no—
97C4×C271—41no—
98Dic271—41no—
99Dic271—41no—
100Dic271—81no—
101C2×C271—81no—
102C2×C271#45311yescharacteristic
103C2×C271#45511yescharacteristic
104C2×C271#45511yescharacteristic
105D271—81no—
106D271—81no—
107C271#165911yescharacteristic, complemented, Sylow
108#1659—2711nomaximal, Sylow
109H(3,𝔽4)—2711no—
110#497—2711no—
111#489—2711no—
112#489—2711no—
113#329—2711no—
114#329—2711no—
115#329D27111yescharacteristic, complemented
116C2×Q16—5421no—
117C2×SD16—5421no—
118C4⋊Q8—2711no—
119#124—2711no—
120#124—2711no—
121#124—5421no—
122#124—5421no—
123#124—5421no—
124#124C2×D27111yescharacteristic
125#120—5421no—
126#120—5421no—
127#120—5421no—
128#103—5421no—
129#103—5421no—
130#102—5421no—
131#102—5421no—
132#97—2711no—
133#97C2×D27111yescharacteristic, complemented
134#97C2×D27111yescharacteristic, complemented
135C24—5421no—
136C2×Q8—2711no—
137C2×Q8—2711no—
138C2×Q8—5421no—
139C2×Q8D108411yescharacteristic
140C2×D4—2711no—
141C2×D4—2711no—
142C2×D4—5421no—
143C2×D4—5421no—
144C2×D4—5421no—
145C2×D4D108411yescharacteristic
146Q16—10842no—
147SD16—10842no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—5421no—
151C4⋊C4—5421no—
152C4⋊C4—5421no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—5421no—
156C22⋊C4—5421no—
157C22⋊C4—5421no—
158C22⋊C4—5421no—
159C22⋊C4—5421no—
160C22⋊C4—5421no—
161C22⋊C4—5421no—
162C42—5421no—
163C42C22×D27111yescharacteristic
164C23—21no—
165C23—5421no—
166C23—5421no—
167C23—21681no—
168C23—21681no—
169C23—21681no—
170Q8—41no—
171Q8—10841no—
172Q8—10841no—
173Q8—10841no—
174Q8—10841no—
175D4—41no—
176D4—10841no—
177D4—10841no—
178D4—21681no—
179D4—21681no—
180D4—21681no—
181C2×C4—21no—
182C2×C4—5421no—
183C2×C4—5421no—
184C2×C4—10841no—
185C2×C4D4×D27111yescharacteristic
186C2×C4D4×D27111yescharacteristic
187C2×C4C2×D108411yescharacteristic
188C8—22no—
189C8—41no—
190C8—10841no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—10841no—
195C22—10841no—
196C22—10841no—
197C22—10841no—
198C22—10841no—
199C22—10841no—
200C22—21684no—
201C22#2257613011yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—10841no—
207C4—10841no—
208C4—21681no—
209C2—81no—
210C2—21681no—
211C2—21681no—
212C2#3177406611yescharacteristic
213C2#3177407611yescharacteristic
214C2#3177407711yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#329×C271C2
socleC22×C271#120
prime radical#31774639C2
hypercenter#329D271
hypocenterC271#1659
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #31774291C2semidirect
3. #31774257C2semidirect
4. #31774256C2semidirect
5. #31773636C2semidirect
6. #31773636C2semidirect
7. #329×C271C2semidirect
10. #22576160C22semidirect
23. #22576071C22semidirect
25. #97×C271C22semidirect
26. #97×C271C22semidirect
107. C271#1659semidirect
115. #329D271semidirect
133. #97C2×D271semidirect
134. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746392
#317742912
#317742572
#317742562
#317736362
#329×C2712
#1659271