Finite Group Databasebeta

← #13159100

Subgroups — #13159100

Order 1728.

848 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C3×#64097C211yesmaximal, characteristic, complemented
2#11723494C211yesmaximal, characteristic
3#11722179C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic, complemented
5#11722056C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic
8#10592589—31nomaximal
9#10590464—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634—21no—
12C2×#6289—21no—
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88804—21no—
21#64102—31no—
22#64097D311yescharacteristic, complemented
23C3×#888—34no—
24#63894—34no—
25C32×#107—31no—
26#63648—31no—
27#63641—31no—
28#63636—31no—
29#63636—34no—
30#63636—64no—
31#63634—31no—
32C3×#807D314yescomplemented
33#63576—34no—
34Dic3⋊(C3⋊C8)—34no—
35#63564—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C4⋊(C3⋊(C3⋊D3))—21no—
40C4⋊(C3⋊(C3⋊D3))Q812yescomplemented
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×#634—42no—
45C2×C32×Dic3—21no—
46#6289—42no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C32×C3⋊C8—21no—
49#5087—91no—
50#4890—94no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C4⋊(C3⋊D3)—34no—
53C2×C4⋊(C3⋊D3)—64no—
54C2×C4×C3⋊D3—61no—
55C2×#634—21no—
56C2×C3×Dic6—64no—
57C2×#624—64no—
58C2×C32×C8—31no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4—31no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63#3752—61no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65#3750—21no—
66C4×C3⋊Dic3C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72C3×Dic3⋊C4—64no—
73C3×C4×Dic3—34no—
74C2×C3×C3⋊C8C2×D314yes—
75#3722—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3⋊(C3⋊D3)—22no—
78C2×C3⋊(C3⋊D3)—41no—
79C2×C3⋊(C3⋊D3)—41no—
80C33×C4—41no—
81C33×C4C2×Q811yescharacteristic
82C33×C4#4111yescharacteristic
83C3×C3⋊Dic3—21no—
84C3×C3⋊Dic3—41no—
85C3×C3⋊Dic3D812yes—
86C32×Dic3—41no—
87C3×C4⋊Q8—91no—
88#893—94no—
89#888—34no—
90C4×D12—91no—
91C3×#107—34no—
92#821—94no—
93Dic3⋊C8—94no—
94#809—94no—
95#809—91no—
96#809—91no—
97#809—184no—
98#807—94no—
99#807—91no—
100#807C3⋊D311yescharacteristic, complemented
101C4⋊(C3⋊C8)—91no—
102C22×C3⋊D3—61no—
103C22×C3⋊D3—64no—
104C22×C3⋊D3—124no—
105C2×C32×C4—21no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—44no—
109C2×C32×C4—61no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3#6711yescharacteristic
114C2×C3⋊Dic3#6711yescharacteristic
115C4⋊(C3⋊D3)—32no—
116C4⋊(C3⋊D3)—38no—
117C4⋊(C3⋊D3)—68no—
118C4⋊(C3⋊D3)—61no—
119C4⋊(C3⋊D3)—64no—
120C4⋊(C3⋊D3)—124no—
121C4×C3⋊D3—122no—
122#634—42no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—64no—
127C3×Dic6—128no—
128#624—128no—
129C32×C8—61no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133C3×C3⋊C8—24no—
134#474—271noSylow
135C2×C33C22⋊Q811yescharacteristic
136C2×C33C2×D811yescharacteristic
137C2×C33#13711yescharacteristic
138C3⋊(C3⋊D3)—42no—
139C2×C3×Q8—181no—
140C2×D12—94no—
141C2×D12—91no—
142C2×D12—184no—
143C2×C4×D3—181no—
144C2×C4×D3—184no—
145C2×Dic6—64no—
146C2×C3×C8—34no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—34no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C4⋊C4D3214yes—
152C3×C4⋊C4C2×C3⋊D311yescharacteristic
153C3×C42—91no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—184no—
156C4⋊Dic3—34no—
157C4⋊Dic3C2×C3⋊D311yescharacteristic
158Dic3⋊C4—64no—
159C4×Dic3—34no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—184no—
163C2×C3⋊C8C2×C3⋊D311yescharacteristic
164C22×C32—24no—
165C22×C32D3×D414yes—
166C22×C32C2×#6711yescharacteristic
167C2×C3⋊D3—62no—
168C2×C3⋊D3—68no—
169C2×C3⋊D3—128no—
170C2×C3⋊D3—121no—
171C2×C3⋊D3—121no—
172C2×C3⋊D3—124no—
173C2×C3⋊D3—124no—
174C2×C3⋊D3—244no—
175C2×C3⋊D3—244no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—41no—
179C32×C4—44no—
180C32×C4—84no—
181C32×C4—121no—
182C32×C4#23514yes—
183C32×C4D3×Q814yes—
184C32×C4D3×Q811yescharacteristic
185C32×C4#23911yescharacteristic
186C3⋊Dic3—21no—
187C3⋊Dic3—41no—
188C3⋊Dic3#21312yes—
189C3×Dic3—38no—
190C3×Dic3—44no—
191C3×Dic3—64no—
192C3×Dic3—124no—
193C4⋊Q8—91no—
194C4×D4—271no—
195#107—31no—
196#107—271no—
197C4⋊C8—271no—
198#102—271no—
199#102—271no—
200C33#47411yescharacteristic, complemented, Sylow
201C22×D3—184no—
202C22×D3—181no—
203C22×D3—364no—
204C3×Q8—362no—
205C2×C3×C4—24no—
206C2×C3×C4—21no—
207C2×C3×C4—24no—
208C2×C3×C4—44no—
209C2×C3×C4—64no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—181no—
213C2×C3×C4C2×D3214yes—
214C2×C3×C4C22×C3⋊D311yescharacteristic
215C2×Dic3—21no—
216C2×Dic3—34no—
217C2×Dic3—34no—
218C2×Dic3—64no—
219D12—98no—
220D12—92no—
221D12—188no—
222D12—184no—
223D12—181no—
224D12—364no—
225C4×D3—368no—
226C4×D3—362no—
227Dic6—128no—
228C3×C8—64no—
229C3⋊C8—21no—
230C3⋊C8—184no—
231C3⋊C8—181no—
232C3⋊C8—364no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32—24no—
236C2×C32#89614yes—
237C2×C32D3×D814yes—
238C2×C32#91814yes—
239C2×C32C2×#21311yescharacteristic
240C2×C32#94211yescharacteristic
241C2×C32#94611yescharacteristic
242C3⋊D3—122no—
243C3⋊D3—128no—
244C3⋊D3—248no—
245C2×Q8—181no—
246C2×D4—271no—
247C22×C4—541no—
248C2×C8—31no—
249C2×C8—271no—
250C4⋊C4—31no—
251C4⋊C4—91no—
252C4⋊C4—181no—
253C4⋊C4D3×C3⋊D311yescharacteristic
254C22⋊C4—541no—
255C42—91no—
256C22×C3—24no—
257C22×C3D3×#6714yes—
258C22×C3D4×C3⋊D311yescharacteristic
259C2×D3—182no—
260C2×D3—188no—
261C2×D3—368no—
262C2×D3—361no—
263C2×D3—361no—
264C2×D3—364no—
265C2×D3—364no—
266C2×D3—724no—
267C2×D3—724no—
268C3×C4—24no—
269C3×C4—24no—
270C3×C4—41no—
271C3×C4—44no—
272C3×C4—84no—
273C3×C4—92no—
274C3×C4—124no—
275C3×C4—181no—
276C3×C4—361no—
277C3×C4#379714yes—
278C3×C4#379914yes—
279C3×C4#382811yescharacteristic
280C3×C4Q8×C3⋊D311yescharacteristic
281Dic3—38no—
282Dic3—41no—
283Dic3—64no—
284Dic3—124no—
285C32—24no—
286C32#489014yescomplemented
287C32#508711yescharacteristic, complemented
288C23—541no—
289Q8—362no—
290D4—272no—
291D4—541no—
292C2×C4—21no—
293C2×C4—61no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—181no—
297C2×C4—1082no—
298C2×C4C2×D3×C3⋊D311yescharacteristic
299C8—61no—
300C8—541no—
301C2×C3—24no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3#6388814yes—
305C2×C3D3×#21314yes—
306C2×C3#6394814yes—
307C2×C3#6410511yescharacteristic
308C2×C3D8×C3⋊D311yescharacteristic
309C2×C3#6412711yescharacteristic
310D3—368no—
311D3—362no—
312D3—728no—
313C22—542no—
314C22—1081no—
315C22—1081no—
316C22C3⋊D3×#6711yescharacteristic
317C4—41no—
318C4—92no—
319C4—121no—
320C4—181no—
321C4—361no—
322C4#8901911yescharacteristic
323C4#8902211yescharacteristic
324C3—24no—
325C3#1059046414yescomplemented
326C3#1059258911yescharacteristic, complemented
327C2—1082no—
328C2#1172348911yescharacteristic
329C2C3⋊D3×#21311yescharacteristic
330C2#1172358611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#474
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64097C2semidirect
4. C32×#807C2semidirect
5. #11722056C2semidirect
6. #11722051C2semidirect
22. #64097D3semidirect
32. C3×#807D3semidirect
40. C4⋊(C3⋊(C3⋊D3))Q8semidirect
100. #807C3⋊D3semidirect
200. C33#474semidirect
286. C32#4890semidirect
287. C32#5087semidirect
325. C3#10590464semidirect
326. C3#10592589semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
#117234942
#117221792
C32×#8072
#117220562
#117220512
#117220362
#105925893
#105904643