Finite Group Databasebeta

← #13160353

Subgroups — #13160353

Order 1728.

776 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C3×#64133C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic
3#11722200C211yesmaximal, characteristic
4#11722056C211yesmaximal, characteristic, complemented
5#11722053C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592695—31nomaximal
9#10591102—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×#3757—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C32×#217—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3—21no—
21C32×#117—31no—
22#64133D311yescharacteristic
23C3×#926D314yes—
24C3×#924—34no—
25#63657—31no—
26#63657—34no—
27#63657—64no—
28#63637—31no—
29#63637—31no—
30#63635—31no—
31#63634—31no—
32#63576—34no—
33#63574—34no—
34#63565—34no—
35#63565—34no—
36C23×C33D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C3×#634—21no—
44C3×#634—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49#5045—91no—
50#4996—94no—
51C22×C32×C4—24no—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C2×D311yescharacteristic
54C2×#634C2×D311yescharacteristic
55C2×C3×Dic6—34no—
56C32×C4⋊C4—31no—
57C32×C4⋊C4—31no—
58C32×C22⋊C4—61no—
59C32×C42—61no—
60#3757—21no—
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—21no—
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—31no—
65C2×C3⋊(C3⋊C8)—34no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×#217—24no—
70C3×#217—64no—
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3C2×D314yes—
73C3×C4⋊Dic3C2×D314yes—
74C3×Dic3⋊C4—64no—
75C3×C4×Dic3—24no—
76C22×C33—21no—
77C22×C33—21no—
78C22×C33—21no—
79C22×C33C2×D411yescharacteristic
80C33×C4—21no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3×C3⋊Dic3—41no—
84C3×C3⋊Dic3—41no—
85C32×Dic3—41no—
86C32×Dic3—41no—
87C3×C22⋊Q8—91no—
88C3×#117—34no—
89#926C3⋊D311yescharacteristic
90#924—34no—
91#835—91no—
92#835—91no—
93#830—94no—
94#830—91no—
95#830—184no—
96#810—94no—
97#810—94no—
98#808—94no—
99#808—91no—
100#807—94no—
101#807—91no—
102C23×C32—24no—
103C23×C32#6714yes—
104C23×C32#6711yescharacteristic
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—44no—
112C2×C32×C4—44no—
113C2×C32×C4—61no—
114C2×C32×C4—61no—
115C2×C32×C4#6714yes—
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—21no—
121#634—21no—
122#634—21no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—64no—
127C3×Dic6—64no—
128C3×Dic6—64no—
129C3⋊(C3⋊C8)—61no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—64no—
133C3⋊(C3⋊C8)—124no—
134C3⋊(C3⋊C8)—124no—
135#485—271noSylow
136C2×C33—41no—
137C2×C33#12311yescharacteristic
138C2×C33#13511yescharacteristic
139C2×C33#13711yescharacteristic
140C2×C3×Q8—91no—
141C22×C3×C4—24no—
142C22×C3×C4D3214yes—
143C22×C3×C4C2×C3⋊D311yescharacteristic
144C2×Dic6—34no—
145C3×C4⋊C4—34no—
146C3×C4⋊C4—34no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—181no—
149C3×C22⋊C4—64no—
150C3×C22⋊C4—181no—
151C3×C42—64no—
152#217—21no—
153#217—64no—
154C4⋊Dic3—34no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156C4⋊Dic3C2×C3⋊D311yescharacteristic
157Dic3⋊C4—64no—
158C4×Dic3—21no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—184no—
164C2×C3⋊C8—184no—
165C22×C32—28no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—24no—
169C22×C32—21no—
170C22×C32—24no—
171C22×C32—24no—
172C22×C32—24no—
173C22×C32—44no—
174C22×C32—44no—
175C22×C32C2×#6714yes—
176C22×C32C2×#6711yescharacteristic
177C32×C4—21no—
178C32×C4—24no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—44no—
182C32×C4—121no—
183C32×C4—121no—
184C32×C4#23514yes—
185C32×C4#23514yes—
186C32×C4#23711yescharacteristic
187C32×C4#23711yescharacteristic
188C3⋊Dic3—41no—
189C3⋊Dic3—41no—
190C3×Dic3—44no—
191C3×Dic3—44no—
192C3×Dic3—124no—
193C3×Dic3—124no—
194C22⋊Q8—91no—
195#117—31no—
196#107—271no—
197#106—271no—
198#103—271no—
199#103—271no—
200C22⋊C8—271no—
201C33#48511yescharacteristic, complemented, Sylow
202C23×C3—24no—
203C23×C3#62514yes—
204C23×C3#63811yescharacteristic
205C3×Q8—181no—
206C3×Q8—181no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—44no—
214C2×C3×C4—44no—
215C2×C3×C4—64no—
216C2×C3×C4—64no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4#62514yes—
220C2×C3×C4#63811yescharacteristic
221C2×C3×C4C2×D3214yes—
222C2×C3×C4C22×C3⋊D311yescharacteristic
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—64no—
226C2×Dic3—64no—
227Dic6—64no—
228Dic6—64no—
229C3⋊C8—184no—
230C3⋊C8—184no—
231C3⋊C8—181no—
232C3⋊C8—181no—
233C3⋊C8—364no—
234C3⋊C8—364no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—48no—
239C2×C32—44no—
240C2×C32—41no—
241C2×C32#92914yes—
242C2×C32#93511yescharacteristic
243C2×C32#94211yescharacteristic
244C2×C32#95014yes—
245C2×C32#95011yescharacteristic
246C2×C32#95114yes—
247C2×Q8—91no—
248C22×C4D3×C3⋊D311yescharacteristic
249C2×C8—271no—
250C2×C8—271no—
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—91no—
254C4⋊C4—181no—
255C22⋊C4—61no—
256C22⋊C4—181no—
257C42—61no—
258C22×C3—28no—
259C22×C3—24no—
260C22×C3—24no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—24no—
265C22×C3—24no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3C2×#62514yes—
269C22×C3C2×#63811yescharacteristic
270C3×C4—21no—
271C3×C4—24no—
272C3×C4—24no—
273C3×C4—24no—
274C3×C4—44no—
275C3×C4—124no—
276C3×C4—124no—
277C3×C4—361no—
278C3×C4—361no—
279C3×C4#380414yes—
280C3×C4#380414yes—
281C3×C4#382811yescharacteristic
282C3×C4#382811yescharacteristic
283Dic3—41no—
284Dic3—41no—
285Dic3—124no—
286Dic3—124no—
287C32—24no—
288C32#499614yescomplemented
289C32#504511yescharacteristic, complemented
290C23#629011yescharacteristic
291Q8—181no—
292Q8—181no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—181no—
298C2×C4—181no—
299C2×C4#629011yescharacteristic
300C2×C4C2×D3×C3⋊D311yescharacteristic
301C8—541no—
302C8—541no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3—48no—
307C2×C3—44no—
308C2×C3—41no—
309C2×C3#6382214yes—
310C2×C3#6382714yes—
311C2×C3#6396114yes—
312C2×C3#6413811yescharacteristic
313C2×C3#6415911yescharacteristic
314C2×C3#6416011yescharacteristic
315C22—21no—
316C22—21no—
317C22—21no—
318C22C2×#629011yescharacteristic
319C4—21no—
320C4—121no—
321C4—121no—
322C4—361no—
323C4—361no—
324C4#8903111yescharacteristic
325C4#8903111yescharacteristic
326C3—24no—
327C3#1059110214yescomplemented
328C3#1059269511yescharacteristic, complemented
329C2—41no—
330C2#1172336311yescharacteristic
331C2#1172337311yescharacteristic
332C2#1172360511yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722056C2semidirect
5. #11722053C2semidirect
6. #11722039C2semidirect
7. #11722039C2semidirect
201. C33#485semidirect
288. C32#4996semidirect
289. C32#5045semidirect
327. C3#10591102semidirect
328. C3#10592695semidirect

Maximal subgroups

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

subgroupindex
C3×#641332
C32×#9262
#117222002
#117220562
#117220532
#117220392
#105926953
#105911023