Finite Group Databasebeta

← #13167018

Subgroups — #13167018

Order 1728.

492 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C4⋊#6296C211yesmaximal, characteristic, complemented
2C3×#63914C211yesmaximal, characteristic
3C4×C3⋊#622C211yesmaximal, characteristic, complemented
4#11722773C214yesmaximal, complemented
5#10593366—31nomaximal
6C2×C32×D12—21no—
7C2×#6296—21no—
8C2×C3×#625—21no—
9C2×C3×#625C2212yes—
10C2×C3⋊#622C2212yes—
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13#88815C2212yes—
14C3×#940—61no—
15#63919—31no—
16#63918—61no—
17#63914D311yescharacteristic
18C4×#622—31no—
19#63739—34no—
20C22×C32×D3—22no—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3—22no—
23C2×C3×C3⋊Dic3D411yescharacteristic
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C32×D12—42no—
26#6296—42no—
27C3×#625—24no—
28C3×#625—42no—
29C3⋊#622—22no—
30#5070—91nomaximal
31C2×C32×D4—61no—
32C2×C3×#67—62no—
33C2×C3×#67—62no—
34C2×C3×D12—21no—
35C2×C3×D12—21no—
36C2×C3×D12—41no—
37C2×#627—62no—
38C2×#627—61no—
39C2×#625—21no—
40C2×#625C2×D312yes—
41C2×#622—32no—
42C4⋊(C3⋊Dic3)—21no—
43C4×C3⋊Dic3C2×D311yescharacteristic
44C3×C4⋊Dic3—61no—
45C3×C4⋊Dic3—61no—
46C3×C4⋊Dic3—121no—
47C3×C4×Dic3—61no—
48C3×C4×Dic3—61no—
49#3724—32no—
50Dic3⋊Dic3—62no—
51C22×C33C2×D411yescharacteristic
52C2×C32×D3—42no—
53C2×C32×D3—42no—
54C2×C32×D3—42no—
55C33×C4SD1612yes—
56C3×C3⋊Dic3—42no—
57C3×C3⋊Dic3D812yes—
58C3×C3⋊Dic3#4112yes—
59C3×#127—91no—
60#940—61no—
61#888—181no—
62#869—91no—
63#832—94no—
64C4×C3⋊C8—91no—
65C23×C32—62no—
66C22×C3×D3—22no—
67C22×C3×D3—22no—
68C22×C3×D3—42no—
69C32×D4—122no—
70C2×C32×C4—21no—
71C2×C32×C4—21no—
72C2×C32×C4—41no—
73C2×C32×C4—41no—
74C2×C32×C4#6711yescharacteristic
75C2×C3⋊Dic3—22no—
76C2×C3⋊Dic3#6711yescharacteristic
77C2×C3⋊Dic3C22×D311yescharacteristic
78C3×#67—124no—
79C3×#67—124no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—122no—
87C3×D12—42no—
88C3×D12—44no—
89C3×D12—42no—
90#627—124no—
91#627—122no—
92#625—24no—
93#625—42no—
94#622—62no—
95#486—271noSylow
96C2×C33#12411yescharacteristic
97C2×C33C2×D811yescharacteristic
98C2×C33C2×SD1611yescharacteristic
99C32×D3—82no—
100C2×C3×D4—61no—
101C2×C3×D4—61no—
102C2×C3×D4—92no—
103C2×C3×D4—121no—
104C2×C3×D4—181no—
105C2×#67—62no—
106C2×#67—62no—
107C2×D12—21no—
108C2×Dic6—181no—
109C2×Dic6—182no—
110C3×C4⋊C4—181no—
111C3×C42—91no—
112C4⋊Dic3—61no—
113C4⋊Dic3—61no—
114C4⋊Dic3—92no—
115C4⋊Dic3—121no—
116Dic3⋊C4—182no—
117C4×Dic3—61no—
118C4×Dic3—61no—
119C2×C3⋊C8—92no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32—122no—
125C22×C32—122no—
126C22×C32—122no—
127C22×C32C2×#6711yescharacteristic
128C2×C3×D3—42no—
129C2×C3×D3—42no—
130C2×C3×D3—42no—
131C2×C3×D3—44no—
132C2×C3×D3—42no—
133C2×C3×D3—42no—
134C2×C3×D3—42no—
135C2×C3×D3—82no—
136C2×C3×D3—82no—
137C32×C4—22no—
138C32×C4—22no—
139C32×C4—42no—
140C32×C4—42no—
141C32×C4#21412yes—
142C3⋊Dic3—42no—
143C3⋊Dic3#21312yes—
144C3⋊Dic3#23712yes—
145C3×Dic3—62no—
146C3×Dic3—62no—
147C3×Dic3—62no—
148C3×Dic3—62no—
149C3×Dic3—122no—
150C3×Dic3—122no—
151C3×Dic3—242no—
152C4⋊Q8—271no—
153#127—91no—
154#102—274no—
155C4×C8—271no—
156C33#48611yescharacteristic, complemented, Sylow
157C23×C3—62no—
158C23×C3—62no—
159C23×C3—122no—
160C22×D3—22no—
161C3×D4—122no—
162C3×D4—124no—
163C3×D4—122no—
164C3×D4—184no—
165C3×D4—362no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—91no—
171C2×C3×C4—91no—
172C2×C3×C4—182no—
173C2×C3×C4GO+(4,𝔽2)11yescharacteristic
174#67—124no—
175#67—124no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—61no—
180C2×Dic3—62no—
181C2×Dic3—62no—
182C2×Dic3—122no—
183D12—42no—
184Dic6—364no—
185Dic6—362no—
186C3⋊C8—182no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—41no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—242no—
200C2×C32C2×#21311yescharacteristic
201C2×C32C2×#21411yescharacteristic
202C2×C32#93611yescharacteristic
203C3×D3—82no—
204C3×D3—84no—
205C3×D3—82no—
206C2×Q8—541no—
207C2×D4—61no—
208C2×D4—92no—
209C2×D4—181no—
210C2×C8—272no—
211C4⋊C4—181no—
212C4⋊C4—272no—
213C42—91no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—122no—
219C22×C3—122no—
220C22×C3—122no—
221C22×C3—124no—
222C22×C3—122no—
223C22×C3—122no—
224C22×C3—122no—
225C22×C3—242no—
226C22×C3—242no—
227C22×C3C2×GO+(4,𝔽2)11yescharacteristic
228C2×D3—42no—
229C2×D3—42no—
230C2×D3—42no—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—42no—
234C3×C4—42no—
235C3×C4—92no—
236C3×C4—92no—
237C3×C4—362no—
238C3×C4#377512yes—
239Dic3—62no—
240Dic3—62no—
241Dic3—62no—
242Dic3—62no—
243Dic3—122no—
244Dic3—122no—
245Dic3—242no—
246C32—21no—
247C32—21no—
248C32—41no—
249C32—41no—
250C32#507011yescharacteristic, complemented
251C23—62no—
252Q8—1082no—
253D4—122no—
254D4—184no—
255D4—362no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—182no—
259C2×C4D32⋊D311yescharacteristic
260C8—542no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—41no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—242no—
274C2×C3—244no—
275C2×C3—242no—
276C2×C3#6422111yescharacteristic
277C2×C3#6422511yescharacteristic
278C2×C3C2×#377511yescharacteristic
279D3—82no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22C2×D32⋊D311yescharacteristic
284C4—92no—
285C4—92no—
286C4—362no—
287C4#8893712yes—
288C3—21no—
289C3—21no—
290C3—41no—
291C3—41no—
292C3#1059336611yescharacteristic, complemented
293C2—242no—
294C2#1172430111yescharacteristic
295C2#1172430611yescharacteristic
296C2C2×#8893711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypercenterC2×C4D32⋊D3
hypocenterC33#486
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C4⋊#6296C2semidirect
3. C4×C3⋊#622C2semidirect
4. #11722773C2semidirect
156. C33#486semidirect
250. C32#5070semidirect
292. C3#10593366semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×#639142
C4×C3⋊#6222
#117227732
#105933663
#50709