Finite Group Databasebeta

← #13159685

Subgroups — #13159685

Order 1728.

792 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 168 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#11723534C211yesmaximal, characteristic, complemented
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3#11722188C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic
5C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
6#11722055C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic
8#10592545—31nomaximal
9#10590762—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×#6293—21no—
12C2×Dic3×C3⋊D3—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20#88803—21no—
21C4×C4⋊(C3⋊D3)—31no—
22#64078—31no—
23#63918—34no—
24Dic3×D12—34no—
25C32×C4⋊C8—31no—
26#63648—31no—
27#63648D311yescharacteristic
28#63645—31no—
29#63645—31no—
30#63645—34no—
31#63645—64no—
32C3×#821—34no—
33C3×Dic3⋊C8D314yescomplemented
34#63575—34no—
35#63563—34no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)Q812yescomplemented
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45#6293—42no—
46Dic3×C3⋊D3—42no—
47C33×C8—21no—
48C32×C3⋊C8—21no—
49#4846—94no—
50#4759—91no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C4×C3⋊D3—61no—
53C2×#634—61no—
54C2×C3×D12—34no—
55C2×#627—64no—
56C2×D3×Dic3—64no—
57C2×C32×C8—24no—
58C2×C32×C8—31no—
59C2×C32×C8C2×D314yes—
60C2×C32×C8C2×D311yescharacteristic
61C32×C42—31no—
62#3752—61no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)—64no—
69C4⋊(C3⋊Dic3)—64no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C3×C4⋊Dic3—34no—
72C3×C4×Dic3C2×D314yes—
73C2×C3×C3⋊C8C2×D314yes—
74Dic3⋊Dic3—64no—
75(C2×D3)⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—22no—
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C33×C4C2×Q811yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—41no—
85C32×Dic3—21no—
86C32×Dic3D812yes—
87D4×Dic3—91no—
88#888—91no—
89C4×D12—94no—
90#869—94no—
91C3×C4⋊C8—34no—
92C3×#102—91no—
93#832—91no—
94#821—34no—
95#821—94no—
96#818—94no—
97#818—94no—
98#818—91no—
99#818—184no—
100Dic3⋊C8C3⋊D311yescharacteristic, complemented
101#807—91no—
102C22×C3⋊D3—21no—
103C22×C3×D3—64no—
104C2×C32×C4—24no—
105C2×C32×C4—31no—
106C2×C32×C4—31no—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—124no—
115C2×C3⋊Dic3—124no—
116C4⋊(C3⋊D3)—21no—
117C4⋊(C3⋊D3)Dic612yescomplemented
118C4×C3⋊D3—122no—
119#634—122no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3D1214yes—
122C2×C3×Dic3D1214yes—
123C3×D12—38no—
124C3×D12—64no—
125#627—128no—
126D3×Dic3—128no—
127C32×C8—28no—
128C32×C8—24no—
129C32×C8—21no—
130C32×C8—61no—
131C3×C3⋊C8—24no—
132#474—271noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33C2×D811yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—42no—
137C2×C3×D4—91no—
138C22×Dic3—181no—
139C2×D12—34no—
140C2×C4×D3—184no—
141C2×Dic6—184no—
142C2×Dic6—181no—
143C2×C3×C8—24no—
144C2×C3×C8—34no—
145C2×C3×C8D3214yes—
146C2×C3×C8C2×C3⋊D311yescharacteristic
147C3×C4⋊C4—91no—
148C3×C42—34no—
149#217—181no—
150(C2×D3)⋊C4—184no—
151C4⋊Dic3—34no—
152C4⋊Dic3—94no—
153C4⋊Dic3—94no—
154C4⋊Dic3—91no—
155C4⋊Dic3—91no—
156C4⋊Dic3—184no—
157C4⋊Dic3—184no—
158C4⋊Dic3—184no—
159Dic3⋊C4—181no—
160C4×Dic3C2×C3⋊D311yescharacteristic
161C2×C3⋊C8C2×C3⋊D311yescharacteristic
162C22×C32—24no—
163C22×C32C2×D1214yes—
164C22×C32D3×D411yescharacteristic
165C2×C3⋊D3—22no—
166C2×C3⋊D3—41no—
167C2×C3⋊D3—41no—
168C2×C3×D3—68no—
169C2×C3×D3—124no—
170C2×C3×D3—124no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—32no—
174C32×C4—61no—
175C32×C4C2×Dic611yescharacteristic
176C32×C4#23714yes—
177C32×C4#23711yescharacteristic
178C32×C4D3×Q814yes—
179C3⋊Dic3—41no—
180C3⋊Dic3—121no—
181C3⋊Dic3—121no—
182C3⋊Dic3—124no—
183C3⋊Dic3—124no—
184C3⋊Dic3—244no—
185C3⋊Dic3—244no—
186C3×Dic3—24no—
187C3×Dic3—124no—
188C3×Dic3D2418yes—
189C4⋊Q8—271no—
190C4×D4—271no—
191#107—271no—
192#107—271no—
193C4⋊C8—31no—
194#102—91no—
195#102—271no—
196C33#47411yescharacteristic, complemented, Sylow
197C23×C3—181no—
198C22×D3—64no—
199C3×D4—92no—
200C3×D4—181no—
201C2×C3×C4—24no—
202C2×C3×C4—34no—
203C2×C3×C4—34no—
204C2×C3×C4—181no—
205C2×C3×C4C2×D3214yes—
206C2×C3×C4C22×C3⋊D311yescharacteristic
207C2×Dic3—64no—
208C2×Dic3—184no—
209C2×Dic3—184no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212C2×Dic3—364no—
213C2×Dic3—364no—
214C2×Dic3—362no—
215C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
216C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
217D12—38no—
218D12—64no—
219C4×D3—368no—
220Dic6—368no—
221Dic6—362no—
222C3×C8—28no—
223C3×C8—24no—
224C3×C8—21no—
225C3×C8—64no—
226C3⋊C8—21no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32#87811yescharacteristic
231C2×C32#89714yes—
232C2×C32C2×D2414yes—
233C2×C32#90914yes—
234C2×C32D3×D811yescharacteristic
235C2×C32#91511yescharacteristic
236C3⋊D3—42no—
237C3×D3—128no—
238C2×Q8—541no—
239C2×D4—91no—
240C22×C4—541no—
241C2×C8—31no—
242C2×C8D3×C3⋊D311yescharacteristic
243C4⋊C4—91no—
244C4⋊C4—271no—
245C4⋊C4—271no—
246C4⋊C4—541no—
247C22⋊C4—541no—
248C42—31no—
249C22×C3—24no—
250C22×C3—182no—
251C22×C3—361no—
252C22×C3—361no—
253C22×C3D3×D1214yes—
254C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
255C2×D3—68no—
256C2×D3—124no—
257C2×D3—124no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—38no—
261C3×C4—64no—
262C3×C4—361no—
263C3×C4D3×Dic614yes—
264C3×C4#380514yes—
265C3×C4#383011yescharacteristic
266C3×C4Q8×C3⋊D311yescharacteristic
267Dic3—21no—
268Dic3—124no—
269Dic3—364no—
270Dic3—364no—
271Dic3—361no—
272Dic3—361no—
273Dic3—724no—
274Dic3—724no—
275Dic3#374512yes—
276C32—24no—
277C32#475911yescharacteristic, complemented
278C32#484614yescomplemented
279C23—181no—
280Q8—1082no—
281D4—92no—
282D4—181no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—181no—
286C2×C4—541no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—21no—
291C8—61no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—362no—
296C2×C3D3×D2414yes—
297C2×C3#6380014yes—
298C2×C3#6389314yes—
299C2×C3#6410611yescharacteristic
300C2×C3C2×#374511yescharacteristic
301C2×C3#6411811yescharacteristic
302D3—128no—
303C22—182no—
304C22—361no—
305C22—361no—
306C22D3×C4⋊(C3⋊D3)11yescharacteristic
307C4—32no—
308C4—61no—
309C4—361no—
310C4—1081no—
311C4—1081no—
312C4C3⋊D3×Dic611yescharacteristic
313C4#8902911yescharacteristic
314C3—24no—
315C3#1059076214yescomplemented
316C3#1059254511yescharacteristic, complemented
317C2—362no—
318C2D3×#374511yescharacteristic
319C2#1172332611yescharacteristic
320C2#1172348311yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#474
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11723534C2semidirect
3. #11722188C2semidirect
5. C32×Dic3⋊C8C2semidirect
6. #11722055C2semidirect
33. C3×Dic3⋊C8D3semidirect
42. C3×C4⋊(C3⋊D3)Q8semidirect
100. Dic3⋊C8C3⋊D3semidirect
117. C4⋊(C3⋊D3)Dic6semidirect
196. C33#474semidirect
277. C32#4759semidirect
278. C32#4846semidirect
315. C3#10590762semidirect
316. C3#10592545semidirect

Maximal subgroups

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

subgroupindex
#117235342
Dic3×C4⋊(C3⋊D3)2
#117221882
C3×#636482
C32×Dic3⋊C82
#117220552
#117220342
#105925453
#105907623