Finite Group Databasebeta

← #13167364

Subgroups — #13167364

Order 1728.

404 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 51 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#11724333C212yesmaximal
2#11723649C211yesmaximal, characteristic, complemented
3C8×#1113C211yesmaximal, characteristic
4C3×#63769C211yesmaximal, characteristic, complemented
5#11722784C212yesmaximal, complemented
6#10593288—31nomaximal
7C4×#1113C2211yescharacteristic
8#89040C2212yes—
9C3×C4⋊#157C2212yes—
10#88940—22no—
11#88939—22no—
12C3×#3746—21no—
13C3×C8×C3⋊D3—21no—
14C3×C8×C3⋊D3C2211yescharacteristic
15#88812—21no—
16#88808—21no—
17#88808C2211yescharacteristic
18D32⋊Q8—32no—
19#63802—31no—
20D3×Dic12—61no—
21D32×C8—31no—
22D32×C8—61no—
23#63769D311yescharacteristic, complemented
24#63730—32no—
25C2×#1113D411yescharacteristic
26C2×C3×#157—22no—
27C3×C4×C3⋊D3—21no—
28C3×C4×C3⋊D3C2311yescharacteristic
29C3×#634—22no—
30#6296—22no—
31#6294—21no—
32#6294—22no—
33#6294D411yescharacteristic
34C33×C8D411yescharacteristic
35C3×C3⋊(C3⋊C8)—21no—
36C3×C3⋊(C3⋊C8)D411yescharacteristic
37#4887—91nomaximal
38C4×D32—31no—
39C4×D32—61no—
40#3797—32no—
41D3×Dic6—62no—
42C4⋊#157C2×D312yes—
43#3773—62no—
44D32⋊C4—62no—
45#3746—21no—
46C8×C3⋊D3—21no—
47C8×C3⋊D3C2×D311yescharacteristic
48C3×Dic12—61no—
49C3×Dic12—61no—
50C3×Dic12—121no—
51C3×D3×C8—61no—
52C3×D3×C8—61no—
53C3×D3×C8—61no—
54C3×D3×C8—61no—
55C3×D3×C8—121no—
56#3719—62no—
57#3718—61no—
58#3710—31no—
59#3710—61no—
60D3×C3⋊C8—61no—
61D3×C3⋊C8—61no—
62D3×C3⋊C8—61no—
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3C2×D411yescharacteristic
65#1113—21no—
66#1113Q1612yescomplemented
67C3×#157—42no—
68C33×C4C2×D411yescharacteristic
69C3×C3⋊Dic3—21no—
70C3×C3⋊Dic3—42no—
71C3×C3⋊Dic3#4111yescharacteristic
72D3×Q16—181no—
73C2×Dic12—91no—
74C2×D3×C8—91no—
75C2×D3×C8—181no—
76#897—92no—
77C3×#107—91no—
78#810—92no—
79C2×D32—31no—
80C2×D32—61no—
81C2×#157—22no—
82C4×C3⋊D3—21no—
83C4×C3⋊D3C22×D311yescharacteristic
84#634—22no—
85C3×C4×D3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—121no—
90C3×Dic6—62no—
91C3×Dic6—62no—
92C3×Dic6—122no—
93#627—62no—
94#627—122no—
95#624—31no—
96#624—61no—
97#624—62no—
98D3×Dic3—61no—
99D3×Dic3—61no—
100D3×Dic3—61no—
101D3×Dic3—122no—
102C32×C8—21no—
103C32×C8—21no—
104C32×C8—41no—
105C32×C8—41no—
106C32×C8D1211yescharacteristic
107C3⋊(C3⋊C8)—21no—
108C3⋊(C3⋊C8)D1211yescharacteristic
109C3×C3⋊C8—61no—
110C3×C3⋊C8—61no—
111C3×C3⋊C8—61no—
112C3×C3⋊C8—61no—
113C3×C3⋊C8—121no—
114#467—271noSylow
115C2×C33#12111yescharacteristic
116C3×C3⋊D3—22no—
117C3×C3⋊D3C2×Q1611yescharacteristic
118C3×C3⋊D3#13511yescharacteristic
119D3×Q8—182no—
120C2×C4×D3—91no—
121C2×C4×D3—181no—
122C2×Dic6—92no—
123C3×Q16—181no—
124C2×C3×C8—91no—
125C2×C3×C8—181no—
126C3×C4⋊C4—92no—
127#216—182no—
128(C2×D3)⋊C4—182no—
129C4⋊Dic3—182no—
130C2×C3⋊C8—91no—
131C2×C3⋊C8—181no—
132Dic12—61no—
133Dic12—61no—
134Dic12—121no—
135Dic12—181no—
136D3×C8—61no—
137D3×C8—61no—
138D3×C8—61no—
139D3×C8—61no—
140D3×C8—121no—
141D3×C8—181no—
142D3×C8—181no—
143D3×C8—181no—
144C2×C3⋊D3—21no—
145C2×C3⋊D3D3×D411yescharacteristic
146C2×C3×D3—61no—
147C2×C3×D3—61no—
148C2×C3×D3—61no—
149C2×C3×D3—61no—
150C2×C3×D3—121no—
151D32—32no—
152D32—62no—
153D32—62no—
154D32—61no—
155#157—42no—
156C32×C4—21no—
157C32×C4—21no—
158C32×C4—41no—
159C32×C4—41no—
160C32×C4C2×D1211yescharacteristic
161C3⋊Dic3—21no—
162C3⋊Dic3—42no—
163C3⋊Dic3#23911yescharacteristic
164C3×Dic3—61no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—121no—
169C3×Dic3—122no—
170C3×Dic3—122no—
171C3×Dic3—242no—
172C2×Q16—271no—
173C22×C8—271no—
174C22⋊Q8—272no—
175#107—91no—
176#103—272no—
177C33#46711yescharacteristic, complemented, Sylow
178C22×D3—91no—
179C22×D3—181no—
180C3×Q8—182no—
181C2×C3×C4—91no—
182C2×C3×C4—181no—
183C2×C3×C4—182no—
184C2×Dic3—91no—
185C2×Dic3—181no—
186C2×Dic3—182no—
187C4×D3—61no—
188C4×D3—61no—
189C4×D3—61no—
190C4×D3—61no—
191C4×D3—121no—
192C4×D3—181no—
193C4×D3—181no—
194C4×D3—181no—
195C4×D3—362no—
196Dic6—62no—
197Dic6—62no—
198Dic6—122no—
199Dic6—182no—
200Dic6—362no—
201C3×C8—21no—
202C3×C8—21no—
203C3×C8—41no—
204C3×C8—41no—
205C3×C8—91no—
206C3×C8—181no—
207C3×C8GO+(4,𝔽2)11yescharacteristic
208C3⋊C8—61no—
209C3⋊C8—61no—
210C3⋊C8—61no—
211C3⋊C8—61no—
212C3⋊C8—121no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32#89511yescharacteristic
218C3⋊D3—22no—
219C3⋊D3#91211yescharacteristic
220C3⋊D3D3×Q1611yescharacteristic
221C3×D3—61no—
222C3×D3—61no—
223C3×D3—61no—
224C3×D3—61no—
225C3×D3—62no—
226C3×D3—62no—
227C3×D3—122no—
228C2×Q8—272no—
229C22×C4—271no—
230Q16—181no—
231Q16—541no—
232C2×C8—91no—
233C2×C8—181no—
234C2×C8—271no—
235C2×C8—541no—
236C4⋊C4—92no—
237C4⋊C4—542no—
238C22⋊C4—542no—
239C22×C3—91no—
240C22×C3—181no—
241C2×D3—61no—
242C2×D3—61no—
243C2×D3—61no—
244C2×D3—61no—
245C2×D3—92no—
246C2×D3—121no—
247C2×D3—182no—
248C2×D3—181no—
249C2×D3—182no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—91no—
255C3×C4—181no—
256C3×C4—362no—
257C3×C4—362no—
258C3×C4C2×GO+(4,𝔽2)11yescharacteristic
259Dic3—61no—
260Dic3—61no—
261Dic3—61no—
262Dic3—61no—
263Dic3—121no—
264Dic3—122no—
265Dic3—122no—
266Dic3—242no—
267C32—21no—
268C32—21no—
269C32—41no—
270C32—41no—
271C32#488711yescharacteristic, complemented
272C23—271no—
273Q8—182no—
274Q8—542no—
275C2×C4—91no—
276C2×C4—181no—
277C2×C4—182no—
278C2×C4—271no—
279C2×C4—541no—
280C2×C4—542no—
281C8—91no—
282C8—181no—
283C8#157⋊D311yescharacteristic
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3—91no—
289C2×C3—91no—
290C2×C3—182no—
291C2×C3#6423011yescharacteristic
292D3—61no—
293D3—61no—
294D3—61no—
295D3—61no—
296D3—62no—
297D3—62no—
298D3—122no—
299C22—91no—
300C22—181no—
301C22—272no—
302C22—541no—
303C4—91no—
304C4—181no—
305C4—362no—
306C4—362no—
307C4C2×#157⋊D311yescharacteristic
308C3—21no—
309C3—21no—
310C3—41no—
311C3—41no—
312C3#1059328811yescharacteristic, complemented
313C2—91no—
314C2—91no—
315C2—182no—
316C2#1172434311yescharacteristic
subgroupG/H
centerC2#11724343
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×#157⋊D3
FittingC33×C8D4
socleC2×C33#121
prime radicalC2×#1113D4
hypercenterC8#157⋊D3
hypocenterC33#467
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11723649C2semidirect
4. C3×#63769C2semidirect
5. #11722784C2semidirect
23. #63769D3semidirect
66. #1113Q16semidirect
177. C33#467semidirect
271. C32#4887semidirect
312. C3#10593288semidirect

Maximal subgroups

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

subgroupindex
#117243332
#117236492
C8×#11132
C3×#637692
#117227842
#105932883
#48879