Finite Group Databasebeta

← #13159374

Subgroups — #13159374

Order 1728.

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

normal subgroups only

This group has 864 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
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C2×#88793C211yesmaximal, characteristic
4#11722177C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic
6#11722038C211yesmaximal, characteristic, complemented
7#11722019C211yesmaximal, characteristic, complemented
8#10592725—31nomaximal
9#10590454—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
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×Dic3⋊C4—21no—
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20#88803—21no—
21#88793C414yescomplemented
22C32×C4×Q8—31no—
23C2×#3755—31no—
24C4×C4⋊(C3⋊D3)—31no—
25C3×#887D314yescomplemented
26Dic3×D12—34no—
27C2×#3715—34no—
28#63662—31no—
29#63636D311yescharacteristic
30#63634—31no—
31#63634—34no—
32#63634—64no—
33#63630—31no—
34C3×#809—34no—
35#63566—34no—
36#63554—34no—
37C22×C3×C3⋊D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)C2×C412yes—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C2×C32×Dic3D411yescharacteristic
46C32×Dic6—21no—
47C32×Dic6C2×C412yescomplemented
48Dic3×C3⋊D3—42no—
49C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
50C3×C3⋊(C3⋊C8)—21no—
51#5026—94no—
52#4757—91no—
53C2×C32×Q8—31no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C4×C3⋊D3—61no—
56C2×C3×D12—34no—
57C2×C3×Dic6C2×D314yes—
58C2×D3×Dic3—64no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4—61no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C42—31no—
64C32×C42—61no—
65#3755—34no—
66#3752—61no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—64no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—64no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×Dic3⋊C4—24no—
75C3×C4×Dic3—24no—
76C3×C4×Dic3C2×D314yes—
77C2×C3×C3⋊C8—34no—
78(C2×D3)⋊Dic3—64no—
79#3715—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—
93D4×Dic3—91no—
94C2×#214—91no—
95#887C3⋊D311yescharacteristic, complemented
96C4×D12—94no—
97C3×#102—91no—
98#835—94no—
99#816—91no—
100#815—91no—
101#809—34no—
102#807—94no—
103#807—91no—
104#807—184no—
105#803—94no—
106C22×C3⋊D3—21no—
107C22×C3×D3—64no—
108C32×Q8—32no—
109C32×Q8—61no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4—44no—
116C2×C32×C4—61no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—124no—
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)C4×D312yes—
124C4×C3⋊D3—122no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3#6714yes—
127C2×C3×Dic3#6714yes—
128C3×D12—38no—
129C3×D12—64no—
130C3×Dic6—24no—
131C3×Dic6C4×D318yescomplemented
132D3×Dic3—128no—
133C3⋊(C3⋊C8)—21no—
134C3⋊(C3⋊C8)—32no—
135C3⋊(C3⋊C8)—38no—
136C3⋊(C3⋊C8)—68no—
137C3×C3⋊C8—64no—
138#440—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C2×C33#13711yescharacteristic
142C3×C3⋊D3—42no—
143C2×C3×Q8—34no—
144C2×C3×D4—91no—
145C22×Dic3—181no—
146C2×D12—34no—
147C2×C4×D3—184no—
148C2×Dic6C2×C3⋊D311yescharacteristic
149C2×C3×C8—91no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—64no—
152C3×C4⋊C4D3214yes—
153C3×C4⋊C4C2×C3⋊D311yescharacteristic
154C3×C42—34no—
155C3×C42—64no—
156#217—181no—
157#215—916no—
158#214—94no—
159(C2×D3)⋊C4—184no—
160C4⋊Dic3—94no—
161C4⋊Dic3—91no—
162C4⋊Dic3—184no—
163Dic3⋊C4—21no—
164C4×Dic3—21no—
165C4×Dic3C2×C3⋊D311yescharacteristic
166C2×C3⋊C8—34no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C22×C32—24no—
171C22×C32D3×D411yescharacteristic
172C22×C32C2×#6714yes—
173C2×C3⋊D3—22no—
174C2×C3⋊D3—41no—
175C2×C3⋊D3—41no—
176C2×C3×D3—68no—
177C2×C3×D3—124no—
178C2×C3×D3—124no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—41no—
182C32×C4—44no—
183C32×C4—62no—
184C32×C4—61no—
185C32×C4—61no—
186C32×C4—84no—
187C32×C4C2×C4×D314yes—
188C32×C4C2×C4×D311yescharacteristic
189C32×C4#23514yes—
190C32×C4#23711yescharacteristic
191C3⋊Dic3—121no—
192C3⋊Dic3—124no—
193C3⋊Dic3—244no—
194C3×Dic3—28no—
195C3×Dic3—24no—
196C3×Dic3—24no—
197C2×SD16—271no—
198C4×Q8—31no—
199C4×D4—271no—
200#107—271no—
201#103—271no—
202#102—91no—
203#97—271no—
204C33#44011yescharacteristic, complemented, Sylow
205C23×C3—181no—
206C22×D3—64no—
207C3×Q8—38no—
208C3×Q8—64no—
209C3×D4—92no—
210C3×D4—181no—
211C2×C3×C4—24no—
212C2×C3×C4—21no—
213C2×C3×C4—24no—
214C2×C3×C4—34no—
215C2×C3×C4—34no—
216C2×C3×C4—44no—
217C2×C3×C4—64no—
218C2×C3×C4C2×D3214yes—
219C2×C3×C4C22×C3⋊D311yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—184no—
222C2×Dic3—181no—
223C2×Dic3—364no—
224C2×Dic3—362no—
225C2×Dic3#63811yescharacteristic
226C2×Dic3#63811yescharacteristic
227D12—38no—
228D12—64no—
229C4×D3—368no—
230Dic6—21no—
231Dic6C4×C3⋊D312yescomplemented
232C3×C8—181no—
233C3⋊C8—64no—
234C3⋊C8—98no—
235C3⋊C8—92no—
236C3⋊C8—188no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32#67⋊C411yescharacteristic
241C2×C32#91111yescharacteristic
242C2×C32#91511yescharacteristic
243C2×C32C4×#6714yes—
244C2×C32#94214yes—
245C2×C32#94914yes—
246C3⋊D3—42no—
247C3×D3—128no—
248C2×Q8—31no—
249C2×D4—91no—
250C22×C4—541no—
251SD16—274no—
252C2×C8—91no—
253C2×C8—271no—
254C4⋊C4—61no—
255C4⋊C4—271no—
256C4⋊C4D3×C3⋊D311yescharacteristic
257C22⋊C4—541no—
258C42—31no—
259C42—61no—
260C22×C3—24no—
261C22×C3—182no—
262C22×C3—361no—
263C22×C3—361no—
264C22×C3D3×#6714yes—
265C22×C3C2×#63811yescharacteristic
266C2×D3—68no—
267C2×D3—124no—
268C2×D3—124no—
269C3×C4—24no—
270C3×C4—24no—
271C3×C4—41no—
272C3×C4—44no—
273C3×C4—68no—
274C3×C4—64no—
275C3×C4—64no—
276C3×C4—84no—
277C3×C4#379514yes—
278C3×C4C4×D3214yes—
279C3×C4C2×C4×C3⋊D311yescharacteristic
280C3×C4#382811yescharacteristic
281Dic3—22no—
282Dic3—21no—
283Dic3—21no—
284Dic3—361no—
285Dic3—364no—
286Dic3—724no—
287C32—24no—
288C32#475711yescharacteristic, complemented
289C32#502614yescomplemented
290C23—181no—
291Q8—32no—
292Q8—61no—
293D4—92no—
294D4—181no—
295C2×C4—21no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—61no—
299C2×C4—541no—
300C2×C4—1082no—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—181no—
303C8—272no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—362no—
308C2×C3#6387914yes—
309C2×C3#6392514yes—
310C2×C3#6394614yes—
311C2×C3C4×#63811yescharacteristic
312C2×C3#6415111yescharacteristic
313C2×C3#6415811yescharacteristic
314D3—128no—
315C22—182no—
316C22—361no—
317C22—361no—
318C22D3×#63811yescharacteristic
319C4—41no—
320C4—61no—
321C4—61no—
322C4—62no—
323C4—1081no—
324C4#8902111yescharacteristic
325C4C4×D3×C3⋊D311yescharacteristic
326C3—24no—
327C3#1059045414yescomplemented
328C3#1059272511yescharacteristic, complemented
329C2—362no—
330C2#1172350611yescharacteristic
331C2#1172354411yescharacteristic
332C2#1172357911yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(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
4. #11722177C2semidirect
6. #11722038C2semidirect
7. #11722019C2semidirect
21. #88793C4semidirect
25. C3×#887D3semidirect
47. C32×Dic6C2×C4semidirect
49. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
95. #887C3⋊D3semidirect
131. C3×Dic6C4×D3semidirect
204. C33#440semidirect
231. Dic6C4×C3⋊D3semidirect
288. C32#4757semidirect
289. C32#5026semidirect
327. C3#10590454semidirect
328. C3#10592725semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
Dic3×C4⋊(C3⋊D3)2
C2×#887932
#117221772
C3×#636362
#117220382
#117220192
#105927253
#105904543