Finite Group Databasebeta

← #13159658

Subgroups — #13159658

Order 1728.

792 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 160 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#11723535C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic
3#11722188C211yesmaximal, characteristic, complemented
4C3×#63641C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic
6#11722057C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic
8#10592458—31nomaximal
9#10590762—34nomaximal
10C2×C32×D12C2211yescharacteristic
11C2×#6293—21no—
12C2×D3×C3⋊Dic3—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88807—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21D4×C3⋊Dic3—31no—
22#64097—31no—
23#63918—34no—
24Dic3×D12—34no—
25C32×#102—31no—
26#63659—31no—
27#63645—31no—
28#63645—34no—
29#63645—64no—
30#63641D311yescharacteristic, complemented
31#63634—31no—
32C3×#821D314yes—
33C3×Dic3⋊C8—34no—
34#63575—34no—
35#63563—34no—
36C22×C32×D3—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C2×C3×C3⋊Dic3D411yescharacteristic
42C2×C32×Dic3—21no—
43C32×D12—21no—
44C32×D12Q812yescomplemented
45#6293—42no—
46D3×C3⋊Dic3—42no—
47C33×C8—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#4846—91no—
50#4759—94no—
51C2×C32×D4—31no—
52C22×C3⋊Dic3—61no—
53C2×#634—61no—
54C2×C3×D12C2×D314yes—
55C2×#627—64no—
56C2×D3×Dic3—64no—
57C2×C32×C8—24no—
58C2×C32×C8C2×D314yes—
59C2×C32×C8C2×D311yescharacteristic
60C32×C4⋊C4—31no—
61#3757—61no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—64no—
67C4⋊(C3⋊Dic3)—64no—
68#3750—61no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C4⋊Dic3C2×D314yes—
72C3×C4×Dic3—34no—
73C2×C3×C3⋊C8—34no—
74Dic3⋊Dic3—64no—
75(C2×D3)⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—22no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C33×C4C2×Q811yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3D812yes—
86C32×Dic3—41no—
87D4×Dic3—94no—
88#888—94no—
89C4×D12—91no—
90#869—91no—
91C3×C4⋊C8—91no—
92C3×#102—34no—
93#832—94no—
94#821—91no—
95#821C3⋊D311yescharacteristic
96#818—94no—
97#818—91no—
98#818—91no—
99#818—184no—
100Dic3⋊C8—34no—
101#807—94no—
102C23×C32—61no—
103C22×C3×D3—24no—
104C32×D4—32no—
105C32×D4—61no—
106C2×C32×C4—24no—
107C2×C32×C4—61no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—124no—
115C2×C3⋊Dic3—124no—
116C2×C3⋊Dic3—122no—
117C2×C3⋊Dic3D1211yescharacteristic
118C2×C3⋊Dic3D1211yescharacteristic
119#634—122no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—34no—
123C3×D12—24no—
124C3×D12Dic618yescomplemented
125#627—128no—
126D3×Dic3—128no—
127C32×C8—28no—
128C32×C8—24no—
129C32×C8—21no—
130C3⋊(C3⋊C8)—21no—
131C3×C3⋊C8—64no—
132#474—271noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33C2×D811yescharacteristic
135C2×C33#13711yescharacteristic
136C32×D3—42no—
137C2×C3×D4—34no—
138C22×Dic3—184no—
139C2×D12C2×C3⋊D311yescharacteristic
140C2×C4×D3—181no—
141C2×Dic6—181no—
142C2×Dic6—184no—
143C2×C3×C8—24no—
144C2×C3×C8—91no—
145C2×C3×C8D3214yes—
146C2×C3×C8C2×C3⋊D311yescharacteristic
147C3×C4⋊C4—34no—
148C3×C42—91no—
149#217—184no—
150(C2×D3)⋊C4—181no—
151C4⋊Dic3—94no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—184no—
156C4⋊Dic3—184no—
157C4⋊Dic3—181no—
158C4⋊Dic3C2×C3⋊D311yescharacteristic
159Dic3⋊C4—184no—
160C4×Dic3—34no—
161C2×C3⋊C8—34no—
162C22×C32—24no—
163C22×C32—62no—
164C22×C32—121no—
165C22×C32—121no—
166C22×C32C2×D1211yescharacteristic
167C22×C32D3×D414yes—
168C2×C3×D3—28no—
169C2×C3×D3—44no—
170C2×C3×D3—44no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—121no—
174C32×C4C2×Dic614yes—
175C32×C4#23714yes—
176C32×C4#23711yescharacteristic
177C32×C4D3×Q811yescharacteristic
178C3⋊Dic3—21no—
179C3⋊Dic3—121no—
180C3⋊Dic3—121no—
181C3⋊Dic3—124no—
182C3⋊Dic3—124no—
183C3⋊Dic3—244no—
184C3⋊Dic3—244no—
185C3⋊Dic3D2412yes—
186C3×Dic3—38no—
187C3×Dic3—44no—
188C3×Dic3—64no—
189C4⋊Q8—271no—
190C4×D4—271no—
191#107—271no—
192#107—271no—
193C4⋊C8—91no—
194#102—31no—
195#102—271no—
196C33#47411yescharacteristic, complemented, Sylow
197C23×C3—64no—
198C22×D3—21no—
199C3×D4—38no—
200C3×D4—64no—
201C2×C3×C4—24no—
202C2×C3×C4—64no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4C2×D3214yes—
206C2×C3×C4C22×C3⋊D311yescharacteristic
207C2×Dic3—21no—
208C2×Dic3—34no—
209C2×Dic3—34no—
210C2×Dic3—184no—
211C2×Dic3—184no—
212C2×Dic3—181no—
213C2×Dic3—181no—
214C2×Dic3—364no—
215C2×Dic3—364no—
216C2×Dic3—368no—
217D12—21no—
218D12#63412yescomplemented
219C4×D3—362no—
220Dic6—368no—
221Dic6—362no—
222C3×C8—28no—
223C3×C8—24no—
224C3×C8—21no—
225C3×C8—181no—
226C3⋊C8—64no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—122no—
231C2×C32#87814yes—
232C2×C32#89711yescharacteristic
233C2×C32C2×D2411yescharacteristic
234C2×C32#90911yescharacteristic
235C2×C32D3×D814yes—
236C2×C32#91514yes—
237C3×D3—48no—
238C2×Q8—541no—
239C2×D4—31no—
240C22×C4—541no—
241C2×C8—91no—
242C2×C8D3×C3⋊D311yescharacteristic
243C4⋊C4—31no—
244C4⋊C4—271no—
245C4⋊C4—271no—
246C4⋊C4—541no—
247C22⋊C4—541no—
248C42—91no—
249C22×C3—24no—
250C22×C3—68no—
251C22×C3—124no—
252C22×C3—124no—
253C22×C3D3×D1214yes—
254C22×C3D4×C3⋊D311yescharacteristic
255C2×D3—22no—
256C2×D3—41no—
257C2×D3—41no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—92no—
261C3×C4—124no—
262C3×C4—181no—
263C3×C4D3×Dic614yes—
264C3×C4#380514yes—
265C3×C4C2×#63411yescharacteristic
266C3×C4#383011yescharacteristic
267Dic3—38no—
268Dic3—41no—
269Dic3—64no—
270Dic3—364no—
271Dic3—364no—
272Dic3—361no—
273Dic3—361no—
274Dic3—724no—
275Dic3—724no—
276C32—24no—
277C32#475914yescomplemented
278C32#484611yescharacteristic, complemented
279C23—61no—
280Q8—1082no—
281D4—32no—
282D4—61no—
283C2×C4—61no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—541no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—21no—
291C8—181no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—128no—
296C2×C3D3×D2414yes—
297C2×C3#6380014yes—
298C2×C3#6389314yes—
299C2×C3#6408711yescharacteristic
300C2×C3D8×C3⋊D311yescharacteristic
301C2×C3#6412411yescharacteristic
302D3—42no—
303C22—62no—
304C22—121no—
305C22—121no—
306C22C3⋊D3×D1211yescharacteristic
307C4—92no—
308C4—121no—
309C4—181no—
310C4—1081no—
311C4—1081no—
312C4D3×#63411yescharacteristic
313C4#8902911yescharacteristic
314C3—24no—
315C3#1059076214yescomplemented
316C3#1059245811yescharacteristic, complemented
317C2—122no—
318C2C3⋊D3×D2411yescharacteristic
319C2#1172334011yescharacteristic
320C2#1172349111yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
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. #11723535C2semidirect
3. #11722188C2semidirect
4. C3×#63641C2semidirect
6. #11722057C2semidirect
30. #63641D3semidirect
44. C32×D12Q8semidirect
124. C3×D12Dic6semidirect
196. C33#474semidirect
218. D12#634semidirect
277. C32#4759semidirect
278. C32#4846semidirect
315. C3#10590762semidirect
316. C3#10592458semidirect

Maximal subgroups

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

subgroupindex
#117235352
D12×C3⋊Dic32
#117221882
C3×#636412
C32×#8212
#117220572
#117220322
#105924583
#105907623