Finite Group Databasebeta

← #13166302

Subgroups — #13166302

Order 1728.

380 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 70 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#11723390C211yesmaximal, characteristic
2C4⋊(C3⋊#622)C211yesmaximal, characteristic
3C3×#63777C211yesmaximal, characteristic
4Dic3⋊#622C211yesmaximal, characteristic
5Dic3⋊#622C211yesmaximal, characteristic
6#11722724C211yesmaximal, characteristic
7Dic3×#622C211yesmaximal, characteristic
8#10593756—31nomaximal
9C2×#6293C411yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88857C411yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×Dic3⋊C4C411yescharacteristic
17#88807C411yescharacteristic
18Dic3×C3⋊Dic3C2211yescharacteristic
19Dic3×C3⋊Dic3C2211yescharacteristic
20#64096—31no—
21#63837—62no—
22#63777—31no—
23#63777D311yescharacteristic
24C4⋊#622—31no—
25C4⋊#622—31no—
26C4⋊#622—31no—
27C4×#622—31no—
28C2×C33×C4C2×C411yescharacteristic
29C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3C2×C411yescharacteristic
34C2×C32×Dic3C2×C411yescharacteristic
35#6293—22no—
36C3⋊#622—21no—
37C3⋊#622—21no—
38C3×#622—21no—
39C3×#622Q812yescomplemented
40#4701—91nomaximal
41C2×#634—31no—
42C2×#627—62no—
43C2×#622—31no—
44C2×#622—31no—
45C2×#622C2×D311yescharacteristic
46C2×#622C2×D311yescharacteristic
47C32×C4⋊C4—31no—
48#3750—31no—
49#3750—31no—
50#3750—62no—
51#3750—122no—
52C4×C3⋊Dic3—31no—
53C4×C3⋊Dic3—31no—
54C4×C3⋊Dic3C2×D311yescharacteristic
55C3×Dic3⋊C4—22no—
56C3×C4×Dic3—62no—
57Dic3⋊Dic3—62no—
58Dic32—62no—
59Dic32—62no—
60C22×C33C22×C411yescharacteristic
61C33×C4—21no—
62C3⋊(C3⋊Dic3)—21no—
63C3⋊(C3⋊Dic3)—21no—
64C3×C3⋊Dic3—21no—
65C3×C3⋊Dic3C2×Q811yescharacteristic
66C3×C3⋊Dic3#4111yescharacteristic
67C32×Dic3—21no—
68C32×Dic3M1612yes—
69#887—182no—
70C4×Dic6—91no—
71C3×#97—91no—
72#815—91no—
73Dic3⋊C8—91no—
74Dic3⋊C8—91no—
75C8×Dic3—91no—
76C4⋊(C3⋊C8)—91no—
77C2×C32×C4—22no—
78C2×C32×C4—31no—
79C2×C32×C4—31no—
80C2×C32×C4—42no—
81C2×C32×C4C4×D311yescharacteristic
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—31no—
84C2×C3⋊Dic3—62no—
85C2×C3⋊Dic3—62no—
86C2×C3⋊Dic3—122no—
87C2×C3⋊Dic3—122no—
88C2×C3⋊Dic3C4×D311yescharacteristic
89C2×C3⋊Dic3C22×D311yescharacteristic
90#634—62no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95#627—68no—
96#622—21no—
97#622—61no—
98#622—61no—
99#622Dic612yescomplemented
100#446—271noSylow
101C2×C33C4×Q811yescharacteristic
102C2×C33C2×M1611yescharacteristic
103C2×C33#13111yescharacteristic
104C2×Dic6—91no—
105C2×Dic6—182no—
106C2×C3×C8—91no—
107C2×C3×C8—91no—
108C3×C4⋊C4—62no—
109C3×C42—91no—
110C4⋊Dic3—91no—
111Dic3⋊C4—91no—
112Dic3⋊C4—182no—
113Dic3⋊C4—182no—
114Dic3⋊C4—362no—
115Dic3⋊C4#15711yescharacteristic
116C4×Dic3—62no—
117C4×Dic3—91no—
118C4×Dic3—91no—
119C4×Dic3—182no—
120C4×Dic3—182no—
121C2×C3⋊C8—91no—
122C2×C3⋊C8—91no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32C2×C4×D311yescharacteristic
126C32×C4—21no—
127C32×C4—32no—
128C32×C4—42no—
129C32×C4—61no—
130C32×C4—82no—
131C3⋊Dic3—21no—
132C3⋊Dic3—61no—
133C3⋊Dic3—64no—
134C3⋊Dic3—61no—
135C3⋊Dic3—124no—
136C3⋊Dic3—122no—
137C3⋊Dic3—242no—
138C3⋊Dic3C2×Dic611yescharacteristic
139C3⋊Dic3#23511yescharacteristic
140C3×Dic3—24no—
141C3×Dic3—42no—
142C3×Dic3—62no—
143C3×Dic3—62no—
144C3×Dic3—64no—
145C4×Q8—271no—
146C4⋊C8—271no—
147C4⋊C8—271no—
148C4⋊C8—271no—
149#97—91no—
150#97—271no—
151C4×C8—271no—
152C33#44611yescharacteristic, complemented, Sylow
153C2×C3×C4—22no—
154C2×C3×C4—42no—
155C2×C3×C4—62no—
156C2×C3×C4—62no—
157C2×C3×C4—91no—
158C2×C3×C4—91no—
159C2×C3×C4C2×#15711yescharacteristic
160C2×Dic3—62no—
161C2×Dic3—62no—
162C2×Dic3—91no—
163C2×Dic3—91no—
164C2×Dic3—182no—
165C2×Dic3—182no—
166C2×Dic3—362no—
167C2×Dic3—362no—
168C2×Dic3C2×#15711yescharacteristic
169C2×Dic3C2×#15711yescharacteristic
170Dic6—188no—
171Dic6—182no—
172C3×C8—92no—
173C3×C8—181no—
174C3⋊C8—181no—
175C3⋊C8—181no—
176C2×C32—22no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—42no—
180C2×C32—42no—
181C2×C32—42no—
182C2×C32C4×Dic611yescharacteristic
183C2×C32D3×M1611yescharacteristic
184C2×C32#90711yescharacteristic
185C2×Q8—271no—
186C2×C8—91no—
187C2×C8—91no—
188C2×C8—271no—
189C2×C8—271no—
190C4⋊C4—31no—
191C4⋊C4—271no—
192C4⋊C4—271no—
193C42—91no—
194C42—271no—
195C42—271no—
196C22×C3—22no—
197C22×C3—42no—
198C22×C3C22×#15711yescharacteristic
199C3×C4—21no—
200C3×C4—42no—
201C3×C4—64no—
202C3×C4—82no—
203C3×C4—91no—
204C3×C4—91no—
205C3×C4—122no—
206C3×C4—181no—
207Dic3—21no—
208Dic3—62no—
209Dic3—62no—
210Dic3—64no—
211Dic3—184no—
212Dic3—181no—
213Dic3—181no—
214Dic3—362no—
215Dic3—364no—
216Dic3—722no—
217Dic3#378812yes—
218C32—22no—
219C32—42no—
220C32#470111yescharacteristic, complemented
221Q8—542no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—91no—
225C2×C4—91no—
226C2×C4—271no—
227C2×C4—271no—
228C2×C4D3×#15711yescharacteristic
229C8—92no—
230C8—181no—
231C8—541no—
232C8—541no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—22no—
236C2×C3—42no—
237C2×C3—42no—
238C2×C3—42no—
239C2×C3C2×#378811yescharacteristic
240C2×C3#6428711yescharacteristic
241C2×C3Q8×#15711yescharacteristic
242C22C2×D3×#15711yescharacteristic
243C4—21no—
244C4—32no—
245C4—61no—
246C4—91no—
247C4—91no—
248C4—181no—
249C4—541no—
250C4—541no—
251C3—22no—
252C3—42no—
253C3#1059375611yescharacteristic, complemented
254C2Dic6×#15711yescharacteristic
255C2D3×#378811yescharacteristic
256C2#1172426711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socle, prime radicalC22×C33C22×C4
FittingC2×C33×C4C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#446
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
39. C3×#622Q8semidirect
99. #622Dic6semidirect
152. C33#446semidirect
220. C32#4701semidirect
253. C3#10593756semidirect

Maximal subgroups

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

subgroupindex
#117233902
C4⋊(C3⋊#622)2
C3×#637772
Dic3⋊#6222
#117227242
Dic3×#6222
#105937563
#47019