Finite Group Databasebeta

← #13159107

Subgroups — #13159107

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
1C32×#888C211yesmaximal, characteristic, complemented
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3#11722177C211yesmaximal, characteristic, complemented
4C3×#63636C211yesmaximal, characteristic
5#11722057C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic
8#10592786—31nomaximal
9#10590463—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×Dic6—21no—
12C2×Dic3×C3⋊D3—21no—
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20#88803—21no—
21C32×C4⋊Q8—31no—
22C4×C4⋊(C3⋊D3)—31no—
23C3×#888D314yescomplemented
24Dic3×D12—34no—
25#63636—31no—
26#63636D311yescharacteristic
27#63634—31no—
28#63634—31no—
29#63634—34no—
30#63634—64no—
31C4⋊(C3⋊(C3⋊C8))—31no—
32C3×#809—34no—
33#63575—34no—
34Dic3⋊(C3⋊C8)—34no—
35#63562—34no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C3×C4⋊(C3⋊D3)—21no—
41C3×C4⋊(C3⋊D3)Q812yescomplemented
42C2×C32×Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45C32×Dic6—42no—
46Dic3×C3⋊D3—42no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#5087—94no—
50#4759—91no—
51C2×C32×Q8—61no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C4×C3⋊D3—61no—
54C2×C3×D12—34no—
55C2×C3×Dic6—24no—
56C2×D3×Dic3—64no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4—31no—
59C32×C4⋊C4—61no—
60C32×C4⋊C4C2×D314yes—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C42—31no—
63#3752—61no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—64no—
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C4⋊Dic3C2×D314yes—
72C3×Dic3⋊C4—24no—
73C3×C4×Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75(C2×D3)⋊Dic3—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)—41no—
84C32×Dic3—21no—
85C32×Dic3—41no—
86C32×Dic3D812yes—
87C3×C4⋊Q8—34no—
88D4×Dic3—91no—
89#888C3⋊D311yescharacteristic, complemented
90C4×D12—94no—
91C3×#102—91no—
92#832—91no—
93#818—91no—
94Dic3⋊C8—91no—
95#809—34no—
96#809—94no—
97#807—94no—
98#807—94no—
99#807—91no—
100#807—184no—
101C4⋊(C3⋊C8)—94no—
102C22×C3⋊D3—21no—
103C22×C3×D3—64no—
104C32×Q8—122no—
105C2×C32×C4—21no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4—44no—
111C2×C32×C4—61no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—124no—
117C4⋊(C3⋊D3)—21no—
118C4⋊(C3⋊D3)Dic612yescomplemented
119C4×C3⋊D3—122no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3#6714yes—
122C2×C3×Dic3#6714yes—
123C3×D12—38no—
124C3×D12—64no—
125C3×Dic6—48no—
126D3×Dic3—128no—
127C3⋊(C3⋊C8)—21no—
128C3⋊(C3⋊C8)—68no—
129C3⋊(C3⋊C8)—61no—
130C3⋊(C3⋊C8)—64no—
131C3×C3⋊C8—64no—
132#474—271noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33C2×D811yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—42no—
137C2×C3×Q8—64no—
138C2×C3×D4—91no—
139C22×Dic3—181no—
140C2×D12—34no—
141C2×C4×D3—184no—
142C2×Dic6—21no—
143C2×C3×C8—91no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—34no—
146C3×C4⋊C4—64no—
147C3×C4⋊C4D3214yes—
148C3×C4⋊C4C2×C3⋊D311yescharacteristic
149C3×C42—34no—
150#217—181no—
151(C2×D3)⋊C4—184no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—184no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156Dic3⋊C4—21no—
157C4×Dic3C2×C3⋊D311yescharacteristic
158C2×C3⋊C8—34no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—184no—
162C22×C32—24no—
163C22×C32D3×D411yescharacteristic
164C22×C32C2×#6714yes—
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—41no—
175C32×C4—44no—
176C32×C4—61no—
177C32×C4—84no—
178C32×C4—121no—
179C32×C4C2×Dic611yescharacteristic
180C32×C4#23711yescharacteristic
181C32×C4D3×Q814yes—
182C32×C4#23914yes—
183C3⋊Dic3—121no—
184C3⋊Dic3—124no—
185C3⋊Dic3—244no—
186C3×Dic3—24no—
187C3×Dic3—44no—
188C3×Dic3#21318yes—
189C4⋊Q8—31no—
190C4×D4—271no—
191#107—271no—
192#107—271no—
193C4⋊C8—271no—
194#102—91no—
195#102—271no—
196C33#47411yescharacteristic, complemented, Sylow
197C23×C3—181no—
198C22×D3—64no—
199C3×Q8—128no—
200C3×D4—92no—
201C3×D4—181no—
202C2×C3×C4—24no—
203C2×C3×C4—21no—
204C2×C3×C4—24no—
205C2×C3×C4—34no—
206C2×C3×C4—34no—
207C2×C3×C4—44no—
208C2×C3×C4—64no—
209C2×C3×C4C2×D3214yes—
210C2×C3×C4C22×C3⋊D311yescharacteristic
211C2×Dic3—21no—
212C2×Dic3—184no—
213C2×Dic3—181no—
214C2×Dic3—364no—
215C2×Dic3—362no—
216C2×Dic3#63811yescharacteristic
217C2×Dic3#63811yescharacteristic
218D12—38no—
219D12—64no—
220C4×D3—368no—
221Dic6—42no—
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#87811yescharacteristic
231C2×C32D3×D811yescharacteristic
232C2×C32#91511yescharacteristic
233C2×C32C2×#21314yes—
234C2×C32#94214yes—
235C2×C32#94614yes—
236C3⋊D3—42no—
237C3×D3—128no—
238C2×Q8—61no—
239C2×D4—91no—
240C22×C4—541no—
241C2×C8—91no—
242C2×C8—271no—
243C4⋊C4—31no—
244C4⋊C4—61no—
245C4⋊C4—271no—
246C4⋊C4D3×C3⋊D311yescharacteristic
247C22⋊C4—541no—
248C42—31no—
249C22×C3—24no—
250C22×C3—182no—
251C22×C3—361no—
252C22×C3—361no—
253C22×C3D3×#6714yes—
254C22×C3C2×#63811yescharacteristic
255C2×D3—68no—
256C2×D3—124no—
257C2×D3—124no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—38no—
261C3×C4—41no—
262C3×C4—44no—
263C3×C4—64no—
264C3×C4—84no—
265C3×C4—124no—
266C3×C4D3×Dic614yes—
267C3×C4#379614yes—
268C3×C4Q8×C3⋊D311yescharacteristic
269C3×C4#383211yescharacteristic
270Dic3—21no—
271Dic3—41no—
272Dic3—361no—
273Dic3—364no—
274Dic3—724no—
275Dic3#375312yes—
276C32—24no—
277C32#475911yescharacteristic, complemented
278C32#508714yescomplemented
279C23—181no—
280Q8—122no—
281D4—92no—
282D4—181no—
283C2×C4—21no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—61no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—181no—
291C8—541no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—362no—
296C2×C3#6388714yes—
297C2×C3D3×#21314yes—
298C2×C3#6394614yes—
299C2×C3C2×#375311yescharacteristic
300C2×C3#6415111yescharacteristic
301C2×C3#6415511yescharacteristic
302D3—128no—
303C22—182no—
304C22—361no—
305C22—361no—
306C22D3×#63811yescharacteristic
307C4—32no—
308C4—41no—
309C4—61no—
310C4—121no—
311C4—1081no—
312C4C3⋊D3×Dic611yescharacteristic
313C4#8901711yescharacteristic
314C3—24no—
315C3#1059046314yescomplemented
316C3#1059278611yescharacteristic, complemented
317C2—362no—
318C2#1172347611yescharacteristic
319C2D3×#375311yescharacteristic
320C2#1172357911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(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. C32×#888C2semidirect
3. #11722177C2semidirect
5. #11722057C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
23. C3×#888D3semidirect
41. C3×C4⋊(C3⋊D3)Q8semidirect
89. #888C3⋊D3semidirect
118. C4⋊(C3⋊D3)Dic6semidirect
196. C33#474semidirect
277. C32#4759semidirect
278. C32#5087semidirect
315. C3#10590463semidirect
316. C3#10592786semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
Dic3×C4⋊(C3⋊D3)2
#117221772
C3×#636362
#117220572
Dic3⋊(C3⋊(C3⋊C8))2
#117220332
#105927863
#105904633