Finite Group Databasebeta

← #13159386

Subgroups — #13159386

Order 1728.

920 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 174 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×#887C211yesmaximal, characteristic, complemented
2#11723484C211yesmaximal, characteristic
3C2×#88795C211yesmaximal, characteristic
4#11722179C211yesmaximal, characteristic
5C3×#63634C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7#11722019C211yesmaximal, characteristic, complemented
8#10592725—31nomaximal
9#10590457—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×#6289—21no—
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20#88804—21no—
21#88795C414yescomplemented
22C32×C4×Q8—31no—
23C2×#3755—31no—
24C4×C4⋊(C3⋊D3)—31no—
25C3×#887D314yescomplemented
26#63894—34no—
27C2×#3717—34no—
28#63662—31no—
29#63636—31no—
30#63636—34no—
31#63636—64no—
32#63634D311yescharacteristic, complemented
33#63630—31no—
34C3×#807—34no—
35#63565—34no—
36#63554—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))C2×C412yes—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C2×C32×Dic3D411yescharacteristic
46C32×Dic6—21no—
47C32×Dic6C2×C412yescomplemented
48#6289—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C3×C3⋊(C3⋊C8)C2×C412yescomplemented
51#5026—94no—
52#4889—91no—
53C2×C32×Q8—31no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C4⋊(C3⋊D3)—34no—
56C2×C4⋊(C3⋊D3)—64no—
57C2×C4×C3⋊D3—61no—
58C2×C3×Dic6C2×D314yes—
59C2×#624—64no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4—61no—
62C32×C4⋊C4C2×D314yes—
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C42—31no—
65C32×C42—61no—
66#3755—34no—
67#3752—61no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4⋊Dic3—34no—
74C3×Dic3⋊C4—24no—
75C3×C4×Dic3—24no—
76C3×C4×Dic3C2×D314yes—
77C2×C3×C3⋊C8—34no—
78#3722—64no—
79#3717—316no—
80C22×C33C2×D411yescharacteristic
81C2×C3⋊(C3⋊D3)—22no—
82C2×C3⋊(C3⋊D3)—41no—
83C2×C3⋊(C3⋊D3)—41no—
84C33×C4—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—41no—
88C32×Dic3—22no—
89C32×Dic3—21no—
90C32×Dic3—21no—
91C3×C4×Q8—34no—
92C2×#215—94no—
93C2×#204—91no—
94#893—91no—
95#887C3⋊D311yescharacteristic, complemented
96C4×D12—94no—
97C3×#107—91no—
98#835—94no—
99#815—91no—
100#810—91no—
101#809—94no—
102#809—91no—
103#809—184no—
104#807—34no—
105#803—94no—
106C22×C3⋊D3—61no—
107C22×C3⋊D3—64no—
108C22×C3⋊D3—124no—
109C32×Q8—32no—
110C32×Q8—61no—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—31no—
115C2×C32×C4—31no—
116C2×C32×C4—44no—
117C2×C32×C4—61no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C4⋊(C3⋊D3)—32no—
122C4⋊(C3⋊D3)—38no—
123C4⋊(C3⋊D3)—68no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)—64no—
126C4⋊(C3⋊D3)—124no—
127C4×C3⋊D3—122no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3#6714yes—
131C2×C3×Dic3#6714yes—
132C3×Dic6—24no—
133C3×Dic6C4×D318yescomplemented
134#624—128no—
135C3⋊(C3⋊C8)—61no—
136C3⋊(C3⋊C8)—64no—
137C3⋊(C3⋊C8)—124no—
138C3⋊(C3⋊C8)C4×D312yescomplemented
139C3×C3⋊C8—38no—
140#440—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C3⋊(C3⋊D3)—42no—
145C2×C3×Q8—34no—
146C2×D12—94no—
147C2×D12—91no—
148C2×D12—184no—
149C2×C4×D3—184no—
150C2×C4×D3—181no—
151C2×Dic6C2×C3⋊D311yescharacteristic
152C2×C3×C8—91no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—64no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4D3214yes—
157C3×C4⋊C4C2×C3⋊D311yescharacteristic
158C3×C42—34no—
159C3×C42—64no—
160#215—916no—
161(C2×D3)⋊C4—184no—
162(C2×D3)⋊C4—181no—
163C4⋊Dic3—34no—
164Dic3⋊C4—21no—
165C4×Dic3—21no—
166C4×Dic3C2×C3⋊D311yescharacteristic
167C2×C3⋊C8—34no—
168C2×C3⋊C8—94no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—184no—
171#204—94no—
172C22×C32—24no—
173C22×C32D3×D411yescharacteristic
174C22×C32C2×#6714yes—
175C2×C3⋊D3—62no—
176C2×C3⋊D3—68no—
177C2×C3⋊D3—128no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C2×C3⋊D3—124no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—244no—
183C2×C3⋊D3—244no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—41no—
187C32×C4—44no—
188C32×C4—62no—
189C32×C4—61no—
190C32×C4—61no—
191C32×C4—84no—
192C32×C4C2×C4×D314yes—
193C32×C4C2×C4×D311yescharacteristic
194C32×C4#23514yes—
195C32×C4#23911yescharacteristic
196C3⋊Dic3—41no—
197C3×Dic3—28no—
198C3×Dic3—24no—
199C3×Dic3—24no—
200C3×Dic3—124no—
201C2×SD16—271no—
202C4×Q8—31no—
203C4×D4—271no—
204#107—91no—
205#103—271no—
206#102—271no—
207#97—271no—
208C33#44011yescharacteristic, complemented, Sylow
209C22×D3—184no—
210C22×D3—181no—
211C22×D3—364no—
212C3×Q8—38no—
213C3×Q8—64no—
214C2×C3×C4—24no—
215C2×C3×C4—21no—
216C2×C3×C4—24no—
217C2×C3×C4—34no—
218C2×C3×C4—34no—
219C2×C3×C4—44no—
220C2×C3×C4—64no—
221C2×C3×C4—181no—
222C2×C3×C4C2×D3214yes—
223C2×C3×C4C22×C3⋊D311yescharacteristic
224C2×Dic3—21no—
225C2×Dic3—64no—
226C2×Dic3#63811yescharacteristic
227C2×Dic3#63811yescharacteristic
228D12—98no—
229D12—92no—
230D12—188no—
231D12—184no—
232D12—181no—
233D12—364no—
234C4×D3—362no—
235C4×D3—368no—
236Dic6—21no—
237Dic6C4×C3⋊D312yescomplemented
238C3×C8—92no—
239C3⋊C8—38no—
240C3⋊C8—184no—
241C3⋊C8—181no—
242C3⋊C8—364no—
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32#89311yescharacteristic
247C2×C32#91111yescharacteristic
248C2×C32#91811yescharacteristic
249C2×C32C4×#6714yes—
250C2×C32#94214yes—
251C2×C32#94914yes—
252C3⋊D3—122no—
253C3⋊D3—128no—
254C3⋊D3—248no—
255C2×Q8—31no—
256C2×D4—271no—
257C22×C4—541no—
258SD16—274no—
259C2×C8—91no—
260C2×C8—271no—
261C4⋊C4—61no—
262C4⋊C4—91no—
263C4⋊C4D3×C3⋊D311yescharacteristic
264C22⋊C4—541no—
265C42—31no—
266C42—61no—
267C22×C3—24no—
268C22×C3D3×#6714yes—
269C22×C3C2×#63811yescharacteristic
270C2×D3—182no—
271C2×D3—188no—
272C2×D3—368no—
273C2×D3—361no—
274C2×D3—361no—
275C2×D3—364no—
276C2×D3—364no—
277C2×D3—724no—
278C2×D3—724no—
279C3×C4—24no—
280C3×C4—24no—
281C3×C4—41no—
282C3×C4—44no—
283C3×C4—68no—
284C3×C4—64no—
285C3×C4—64no—
286C3×C4—84no—
287C3×C4—361no—
288C3×C4#379914yes—
289C3×C4C4×D3214yes—
290C3×C4C2×C4×C3⋊D311yescharacteristic
291C3×C4#382811yescharacteristic
292Dic3—22no—
293Dic3—21no—
294Dic3—21no—
295Dic3—124no—
296C32—24no—
297C32#488911yescharacteristic, complemented
298C32#502614yescomplemented
299C23—541no—
300Q8—32no—
301Q8—61no—
302D4—272no—
303D4—541no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—61no—
308C2×C4—181no—
309C2×C4—1082no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—92no—
312C8—541no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3#6390414yes—
317C2×C3#6392514yes—
318C2×C3#6394814yes—
319C2×C3C4×#63811yescharacteristic
320C2×C3#6415111yescharacteristic
321C2×C3#6415811yescharacteristic
322D3—362no—
323D3—368no—
324D3—728no—
325C22—542no—
326C22—1081no—
327C22—1081no—
328C22D3×#63811yescharacteristic
329C4—41no—
330C4—61no—
331C4—61no—
332C4—62no—
333C4—361no—
334C4#8902211yescharacteristic
335C4C4×D3×C3⋊D311yescharacteristic
336C3—24no—
337C3#1059045714yescomplemented
338C3#1059272511yescharacteristic, complemented
339C2—1082no—
340C2#1172351011yescharacteristic
341C2#1172354411yescharacteristic
342C2#1172358111yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#440
LCSG ▷ 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×#887C2semidirect
5. C3×#63634C2semidirect
6. #11722037C2semidirect
7. #11722019C2semidirect
21. #88795C4semidirect
25. C3×#887D3semidirect
32. #63634D3semidirect
47. C32×Dic6C2×C4semidirect
50. C3×C3⋊(C3⋊C8)C2×C4semidirect
95. #887C3⋊D3semidirect
133. C3×Dic6C4×D3semidirect
138. C3⋊(C3⋊C8)C4×D3semidirect
208. C33#440semidirect
237. Dic6C4×C3⋊D3semidirect
297. C32#4889semidirect
298. C32#5026semidirect
337. C3#10590457semidirect
338. C3#10592725semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
#117234842
C2×#887952
#117221792
C3×#636342
#117220372
#117220192
#105927253
#105904573