Finite Group Databasebeta

← #13159091

Subgroups — #13159091

Order 1728.

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

normal subgroups only

This group has 720 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×#897C211yesmaximal, characteristic
2#11723424C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic, complemented
4C3×#63634C211yesmaximal, characteristic, complemented
5#11722038C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
8#10592744—31nomaximal
9#10590448—34nomaximal
10C2×C32×C4×D3C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21C32×C22⋊Q8—31no—
22#64135—31no—
23C3×#897D314yes—
24#63841—34no—
25#63662—31no—
26#63662—31no—
27#63657—31no—
28#63635—31no—
29#63635—34no—
30#63635—64no—
31#63634D311yescharacteristic, complemented
32C3×#807—34no—
33#63566—34no—
34#63565—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C32×D3D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3D411yescharacteristic
43C32×C4×D3—21no—
44C32×C4×D3—21no—
45C32×Dic6—21no—
46C32×Dic6—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#5045—94no—
50#4888—91no—
51C2×C32×Q8—31no—
52C22×C32×C4—31no—
53C2×C3×C4×D3C2×D314yes—
54C2×C3×Dic6C2×D314yes—
55C32×C4⋊C4—24no—
56C32×C4⋊C4—61no—
57C32×C4⋊C4C2×D314yes—
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C22⋊C4—61no—
60#3757—61no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—64no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4×C3⋊Dic3—61no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×(C2×D3)⋊C4—24no—
71C3×C4⋊Dic3—24no—
72C3×C4⋊Dic3—34no—
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#926—94no—
89#897C3⋊D311yescharacteristic
90#892—91no—
91C3×#107—91no—
92#835—94no—
93#835—94no—
94#830—94no—
95(C2×D3)⋊C8—91no—
96#816—91no—
97#810—91no—
98#808—94no—
99#808—91no—
100#808—184no—
101#807—34no—
102C23×C32—31no—
103C22×C3×D3#6714yes—
104C32×Q8—61no—
105C32×Q8—61no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—31no—
110C2×C32×C4—44no—
111C2×C32×C4—61no—
112C2×C32×C4—61no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—124no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3#6714yes—
123C3×C4×D3—24no—
124C3×C4×D3—24no—
125C3×Dic6—24no—
126C3×Dic6—24no—
127C3⋊(C3⋊C8)—21no—
128C3⋊(C3⋊C8)—68no—
129C3⋊(C3⋊C8)—64no—
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8—64no—
132#485—271noSylow
133C2×C33#12311yescharacteristic
134C2×C33#13511yescharacteristic
135C2×C33#13711yescharacteristic
136C32×D3—41no—
137C2×C3×Q8—34no—
138C22×C3×C4—34no—
139C2×C4×D3C2×C3⋊D311yescharacteristic
140C2×Dic6C2×C3⋊D311yescharacteristic
141C2×C3×C8—91no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—64no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4D3214yes—
146C3×C4⋊C4C2×C3⋊D311yescharacteristic
147C3×C22⋊C4—64no—
148#217—184no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—181no—
151C4⋊Dic3—21no—
152C4⋊Dic3—34no—
153C4⋊Dic3—94no—
154C4⋊Dic3—91no—
155C4⋊Dic3—184no—
156C4×Dic3—184no—
157C4×Dic3—181no—
158C2×C3⋊C8—34no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—184no—
162C22×C32—24no—
163C22×C32—61no—
164C22×C32—61no—
165C22×C32—61no—
166C22×C32D3×D411yescharacteristic
167C22×C32C2×#6714yes—
168C2×C3×D3—24no—
169C2×C3×D3—24no—
170C2×C3×D3—24no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—41no—
174C32×C4—44no—
175C32×C4—61no—
176C32×C4—84no—
177C32×C4—121no—
178C32×C4#23511yescharacteristic
179C32×C4#23714yes—
180C32×C4#23714yes—
181C32×C4#23911yescharacteristic
182C3⋊Dic3—41no—
183C3⋊Dic3—121no—
184C3⋊Dic3—124no—
185C3⋊Dic3—244no—
186C3×Dic3—24no—
187C3×Dic3—44no—
188C3×Dic3—124no—
189C22⋊Q8—31no—
190#117—271no—
191#107—91no—
192#106—271no—
193#103—271no—
194#103—271no—
195C22⋊C8—271no—
196C33#48511yescharacteristic, complemented, Sylow
197C23×C3—34no—
198C22×D3#63811yescharacteristic
199C3×Q8—64no—
200C3×Q8—64no—
201C2×C3×C4—24no—
202C2×C3×C4—21no—
203C2×C3×C4—24no—
204C2×C3×C4—34no—
205C2×C3×C4—44no—
206C2×C3×C4—64no—
207C2×C3×C4—64no—
208C2×C3×C4—64no—
209C2×C3×C4—181no—
210C2×C3×C4C2×D3214yes—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—64no—
214C2×Dic3—184no—
215C2×Dic3—181no—
216C2×Dic3—364no—
217C2×Dic3#63811yescharacteristic
218C4×D3—21no—
219C4×D3—21no—
220Dic6—21no—
221Dic6—21no—
222C3×C8—181no—
223C3⋊C8—64no—
224C3⋊C8—188no—
225C3⋊C8—184no—
226C3⋊C8—181no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—121no—
231C2×C32#89411yescharacteristic
232C2×C32#91211yescharacteristic
233C2×C32#91811yescharacteristic
234C2×C32#93514yes—
235C2×C32#94214yes—
236C2×C32#95014yes—
237C3×D3—44no—
238C2×Q8—31no—
239C22×C4—31no—
240C2×C8—91no—
241C2×C8—271no—
242C4⋊C4—61no—
243C4⋊C4—91no—
244C4⋊C4—271no—
245C4⋊C4D3×C3⋊D311yescharacteristic
246C22⋊C4—61no—
247C22⋊C4—541no—
248C42—541no—
249C22×C3—24no—
250C22×C3—64no—
251C22×C3—64no—
252C22×C3—64no—
253C22×C3D3×#6714yes—
254C22×C3C2×#63811yescharacteristic
255C2×D3—21no—
256C2×D3—21no—
257C2×D3—21no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—41no—
261C3×C4—44no—
262C3×C4—64no—
263C3×C4—84no—
264C3×C4—124no—
265C3×C4—361no—
266C3×C4#379514yes—
267C3×C4#379614yes—
268C3×C4#383011yescharacteristic
269C3×C4#383011yescharacteristic
270Dic3—21no—
271Dic3—41no—
272Dic3—124no—
273Dic3—361no—
274Dic3—364no—
275Dic3—724no—
276C32—24no—
277C32#488811yescharacteristic, complemented
278C32#504514yescomplemented
279C23—31no—
280Q8—61no—
281Q8—61no—
282C2×C4—21no—
283C2×C4—31no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—181no—
288C2×C4—541no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—181no—
291C8—541no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—124no—
296C2×C3#6388414yes—
297C2×C3#6393314yes—
298C2×C3#6394814yes—
299C2×C3#6414411yescharacteristic
300C2×C3#6415111yescharacteristic
301C2×C3#6415911yescharacteristic
302D3—41no—
303C22—61no—
304C22—61no—
305C22—61no—
306C22D3×#63811yescharacteristic
307C4—41no—
308C4—61no—
309C4—121no—
310C4—361no—
311C4—1081no—
312C4#8901411yescharacteristic
313C4#8901511yescharacteristic
314C3—24no—
315C3#1059044814yescomplemented
316C3#1059274411yescharacteristic, complemented
317C2—121no—
318C2#1172347311yescharacteristic
319C2#1172355611yescharacteristic
320C2#1172358111yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#485
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
3. #11722178C2semidirect
4. C3×#63634C2semidirect
5. #11722038C2semidirect
6. #11722037C2semidirect
31. #63634D3semidirect
196. C33#485semidirect
277. C32#4888semidirect
278. C32#5045semidirect
315. C3#10590448semidirect
316. C3#10592744semidirect

Maximal subgroups

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

subgroupindex
C32×#8972
#117234242
#117221782
C3×#636342
#117220382
#117220372
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927443
#105904483