Finite Group Databasebeta

← #13161139

Subgroups — #13161139

Order 1728.

744 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C32×#946C211yesmaximal, characteristic
2#11723424C211yesmaximal, characteristic
3#11722205C211yesmaximal, characteristic, complemented
4C3×#63662C211yesmaximal, characteristic, complemented
5#11722056C211yesmaximal, characteristic, complemented
6#11722054C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
8#10592744—31nomaximal
9#10591559—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×(C2×D3)⋊C4—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21C32×C22⋊Q8—31no—
22#64135—31no—
23C3×#946D314yes—
24#63841—34no—
25#63662—31no—
26#63662—34no—
27#63662—64no—
28#63662D311yescharacteristic, complemented
29#63657—31no—
30#63635—31no—
31#63634—31no—
32C3×#835—34no—
33#63576—34no—
34#63573—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C32×D3D411yescharacteristic
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C32×C4×D3—21no—
46C32×C4×D3—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#5045—94no—
50#4815—91no—
51C2×C32×Q8—24no—
52C2×C32×Q8C2×D314yes—
53C2×C32×Q8C2×D311yescharacteristic
54C22×C32×C4—31no—
55C2×C3×C4×D3C2×D314yes—
56C32×C4⋊C4—31no—
57C32×C4⋊C4—61no—
58C32×C22⋊C4—61no—
59#3757—61no—
60C4⋊(C3⋊Dic3)—31no—
61C4⋊(C3⋊Dic3)—34no—
62C4⋊(C3⋊Dic3)—64no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4×C3⋊Dic3—61no—
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×(C2×D3)⋊C4—24no—
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72C3×Dic3⋊C4—24no—
73C2×C3×C3⋊C8—34no—
74(C2×D3)⋊Dic3—64no—
75Dic32—64no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—21no—
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C33×C4—41no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—41no—
85C32×Dic3—21no—
86C32×Dic3—41no—
87C3×C22⋊Q8—34no—
88#946C3⋊D311yescharacteristic
89#926—94no—
90#892—91no—
91C3×#103—91no—
92#835—34no—
93#835—94no—
94#835—91no—
95#835—184no—
96#830—94no—
97(C2×D3)⋊C8—91no—
98#817—91no—
99#808—94no—
100#807—94no—
101#807—91no—
102C23×C32—31no—
103C22×C3×D3#6714yes—
104C32×Q8—28no—
105C32×Q8—21no—
106C32×Q8—24no—
107C32×Q8—21no—
108C32×Q8—24no—
109C32×Q8—44no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—44no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117C2×C32×C4—61no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—124no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3#6714yes—
127C3×C4×D3—24no—
128C3×C4×D3—24no—
129C3⋊(C3⋊C8)—21no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133C3×C3⋊C8—64no—
134#485—271noSylow
135C2×C33#12311yescharacteristic
136C2×C33#13511yescharacteristic
137C2×C33#13711yescharacteristic
138C32×D3—41no—
139C2×C3×Q8—24no—
140C2×C3×Q8D3214yes—
141C2×C3×Q8C2×C3⋊D311yescharacteristic
142C22×C3×C4—34no—
143C2×C4×D3C2×C3⋊D311yescharacteristic
144C2×C3×C8—91no—
145C3×C4⋊C4—34no—
146C3×C4⋊C4—64no—
147C3×C4⋊C4—91no—
148C3×C22⋊C4—64no—
149#217—184no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—181no—
152C4⋊Dic3—34no—
153C4⋊Dic3—94no—
154C4⋊Dic3—91no—
155C4⋊Dic3—184no—
156C4⋊Dic3C2×C3⋊D311yescharacteristic
157Dic3⋊C4—21no—
158C4×Dic3—184no—
159C4×Dic3—181no—
160C2×C3⋊C8—34no—
161C2×C3⋊C8—94no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—184no—
164C22×C32—24no—
165C22×C32—61no—
166C22×C32—61no—
167C22×C32—61no—
168C22×C32D3×D411yescharacteristic
169C22×C32C2×#6714yes—
170C2×C3×D3—24no—
171C2×C3×D3—24no—
172C2×C3×D3—24no—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—48no—
176C32×C4—41no—
177C32×C4—44no—
178C32×C4—61no—
179C32×C4—121no—
180C32×C4#23511yescharacteristic
181C32×C4#23714yes—
182C32×C4#23714yes—
183C32×C4#23711yescharacteristic
184C3⋊Dic3—41no—
185C3⋊Dic3—121no—
186C3⋊Dic3—124no—
187C3⋊Dic3—244no—
188C3×Dic3—24no—
189C3×Dic3—44no—
190C3×Dic3—124no—
191C22⋊Q8—31no—
192#117—271no—
193#107—271no—
194#106—271no—
195#103—91no—
196#103—271no—
197C22⋊C8—271no—
198C33#48511yescharacteristic, complemented, Sylow
199C23×C3—34no—
200C22×D3#63811yescharacteristic
201C3×Q8—28no—
202C3×Q8—24no—
203C3×Q8—24no—
204C3×Q8—21no—
205C3×Q8—21no—
206C3×Q8—44no—
207C2×C3×C4—24no—
208C2×C3×C4—21no—
209C2×C3×C4—24no—
210C2×C3×C4—34no—
211C2×C3×C4—44no—
212C2×C3×C4—64no—
213C2×C3×C4—64no—
214C2×C3×C4—64no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—64no—
220C2×Dic3—184no—
221C2×Dic3—181no—
222C2×Dic3—364no—
223C2×Dic3#63811yescharacteristic
224C4×D3—21no—
225C4×D3—21no—
226C3×C8—181no—
227C3⋊C8—64no—
228C3⋊C8—184no—
229C3⋊C8—181no—
230C3⋊C8—364no—
231C2×C32—24no—
232C2×C32—24no—
233C2×C32—24no—
234C2×C32—121no—
235C2×C32#88311yescharacteristic
236C2×C32#91611yescharacteristic
237C2×C32#91811yescharacteristic
238C2×C32#93514yes—
239C2×C32#94214yes—
240C2×C32#95014yes—
241C3×D3—44no—
242C2×Q8D3×C3⋊D311yescharacteristic
243C22×C4—31no—
244C2×C8—91no—
245C2×C8—271no—
246C4⋊C4—31no—
247C4⋊C4—61no—
248C4⋊C4—91no—
249C4⋊C4—271no—
250C22⋊C4—61no—
251C22⋊C4—541no—
252C42—541no—
253C22×C3—24no—
254C22×C3—64no—
255C22×C3—64no—
256C22×C3—64no—
257C22×C3D3×#6714yes—
258C22×C3C2×#63811yescharacteristic
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—21no—
262C3×C4—24no—
263C3×C4—24no—
264C3×C4—48no—
265C3×C4—41no—
266C3×C4—44no—
267C3×C4—64no—
268C3×C4—124no—
269C3×C4—361no—
270C3×C4#380414yes—
271C3×C4#380514yes—
272C3×C4#383011yescharacteristic
273C3×C4#383011yescharacteristic
274Dic3—21no—
275Dic3—41no—
276Dic3—124no—
277Dic3—361no—
278Dic3—364no—
279Dic3—724no—
280C32—24no—
281C32#481511yescharacteristic, complemented
282C32#504514yescomplemented
283C23—31no—
284Q8—21no—
285Q8—21no—
286C2×C4—21no—
287C2×C4—31no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—181no—
292C2×C4—541no—
293C2×C4C2×D3×C3⋊D311yescharacteristic
294C8—181no—
295C8—541no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—124no—
300C2×C3#6394714yes—
301C2×C3#6394814yes—
302C2×C3#6396914yes—
303C2×C3#6414411yescharacteristic
304C2×C3#6415111yescharacteristic
305C2×C3#6415911yescharacteristic
306D3—41no—
307C22—61no—
308C22—61no—
309C22—61no—
310C22D3×#63811yescharacteristic
311C4—41no—
312C4—61no—
313C4—121no—
314C4—361no—
315C4—1081no—
316C4#8902811yescharacteristic
317C4#8902911yescharacteristic
318C3—24no—
319C3#1059155914yescomplemented
320C3#1059274411yescharacteristic, complemented
321C2—121no—
322C2#1172358011yescharacteristic
323C2#1172358111yescharacteristic
324C2#1172361411yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. #11722205C2semidirect
4. C3×#63662C2semidirect
5. #11722056C2semidirect
6. #11722054C2semidirect
28. #63662D3semidirect
198. C33#485semidirect
281. C32#4815semidirect
282. C32#5045semidirect
319. C3#10591559semidirect
320. C3#10592744semidirect

Maximal subgroups

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

subgroupindex
C32×#9462
#117234242
#117222052
C3×#636622
#117220562
#117220542
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927443
#105915593