Finite Group Databasebeta

← #13159021

Subgroups — #13159021

Order 1728.

808 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 186 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×C4×#634C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic
3C2×#88798C211yesmaximal, characteristic
4#11722173C211yesmaximal, characteristic
5#11722053C211yesmaximal, characteristic
6#11722039C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592729—31nomaximal
9#10590381—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3—21no—
21#88798C414yescomplemented
22C32×C4×Q8—31no—
23C2×#3756—31no—
24C4×#634D311yescharacteristic
25C3×C4×Dic6—34no—
26C3×C4×Dic6D314yes—
27C2×#3718—34no—
28#63662—31no—
29#63637—31no—
30#63635—31no—
31#63630—31no—
32#63630—34no—
33#63630—64no—
34#63574—34no—
35#63565—34no—
36#63565—34no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C3×#634—21no—
43C3×#634C2×C412yes—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46C32×Dic6—21no—
47C32×Dic6C2×C412yes—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3⋊(C3⋊(C3⋊C8))C2×C412yes—
50#5030—94no—
51#5030—91no—
52C2×C32×Q8—31no—
53C2×#634C2×D311yescharacteristic
54C2×C3×Dic6—34no—
55C2×C3×Dic6C2×D314yes—
56C32×C4⋊C4—31no—
57C32×C4⋊C4—61no—
58C32×C42—24no—
59C32×C42—61no—
60C32×C42C2×D314yes—
61C32×C42C2×D311yescharacteristic
62#3756—34no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—21no—
65C4×C3⋊Dic3—21no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C2×C3⋊(C3⋊C8)—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C3×Dic3⋊C4—24no—
75C3×Dic3⋊C4—64no—
76C3×C4×Dic3—24no—
77C3×C4×Dic3—64no—
78#3718—316no—
79C22×C33C2×D411yescharacteristic
80C33×C4—21no—
81C33×C4—21no—
82C33×C4C22×C411yescharacteristic
83C33×C4#4111yescharacteristic
84C3×C3⋊Dic3—22no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—22no—
87C32×Dic3—41no—
88C3×C4×Q8—34no—
89C3×C4×Q8—91no—
90C2×#216—94no—
91C2×#216—91no—
92C4×Dic6—34no—
93C4×Dic6C3⋊D311yescharacteristic
94#835—94no—
95#835—91no—
96#810—94no—
97#810—91no—
98#808—94no—
99#808—91no—
100#803—94no—
101#803—91no—
102#803—184no—
103C32×Q8—32no—
104C32×Q8—61no—
105C2×C32×C4—24no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—61no—
109C2×C32×C4—61no—
110C2×C32×C4#6714yes—
111C2×C32×C4#6714yes—
112C2×C32×C4#6711yescharacteristic
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118#634—21no—
119#634C4×D312yes—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C3×Dic6—24no—
125C3×Dic6—38no—
126C3×Dic6—64no—
127C3×Dic6C4×D318yes—
128C3⋊(C3⋊C8)—32no—
129C3⋊(C3⋊C8)—38no—
130C3⋊(C3⋊C8)—68no—
131C3⋊(C3⋊C8)—68no—
132C3⋊(C3⋊C8)—64no—
133C3⋊(C3⋊C8)—61no—
134#441—271noSylow
135C2×C33C4×D411yescharacteristic
136C2×C33#13711yescharacteristic
137C2×C33#13711yescharacteristic
138C2×C3×Q8—34no—
139C2×C3×Q8—91no—
140C2×Dic6—34no—
141C2×Dic6C2×C3⋊D311yescharacteristic
142C3×C4⋊C4—34no—
143C3×C4⋊C4—64no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—181no—
146C3×C42—24no—
147C3×C42—64no—
148C3×C42—181no—
149C3×C42D3214yes—
150C3×C42C2×C3⋊D311yescharacteristic
151#216—916no—
152#216—94no—
153C4⋊Dic3—34no—
154C4⋊Dic3C2×C3⋊D311yescharacteristic
155Dic3⋊C4—21no—
156Dic3⋊C4—64no—
157C4×Dic3—21no—
158C4×Dic3—64no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—94no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—184no—
164C2×C3⋊C8—184no—
165C22×C32—24no—
166C22×C32C2×#6714yes—
167C22×C32C2×#6711yescharacteristic
168C32×C4—21no—
169C32×C4—21no—
170C32×C4—24no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—24no—
174C32×C4—44no—
175C32×C4—44no—
176C32×C4—62no—
177C32×C4—121no—
178C32×C4C2×C4×D314yes—
179C32×C4C2×C4×D311yescharacteristic
180C32×C4#23514yes—
181C32×C4#23511yescharacteristic
182C3⋊Dic3—22no—
183C3⋊Dic3—41no—
184C3×Dic3—28no—
185C3×Dic3—44no—
186C3×Dic3—68no—
187C3×Dic3—124no—
188C2×Q16—271no—
189C4×Q8—31no—
190C4×Q8—91no—
191#106—271no—
192#103—271no—
193#103—271no—
194#97—271no—
195C33#44111yescharacteristic, complemented, Sylow
196C3×Q8—38no—
197C3×Q8—64no—
198C3×Q8—92no—
199C3×Q8—181no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—64no—
204C2×C3×C4—64no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4#62514yes—
208C2×C3×C4#62514yes—
209C2×C3×C4#63811yescharacteristic
210C2×C3×C4#63811yescharacteristic
211C2×C3×C4C2×D3214yes—
212C2×C3×C4C22×C3⋊D311yescharacteristic
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—64no—
216C2×Dic3—64no—
217Dic6—21no—
218Dic6—38no—
219Dic6—64no—
220Dic6C4×C3⋊D312yes—
221C3⋊C8—98no—
222C3⋊C8—92no—
223C3⋊C8—188no—
224C3⋊C8—188no—
225C3⋊C8—184no—
226C3⋊C8—181no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32C4×#6714yes—
231C2×C32C4×#6711yescharacteristic
232C2×C32#94214yes—
233C2×C32#94211yescharacteristic
234C2×C32#95114yes—
235C2×C32#95111yescharacteristic
236C2×Q8—31no—
237C2×Q8—91no—
238Q16—274no—
239C2×C8—271no—
240C2×C8—271no—
241C4⋊C4—31no—
242C4⋊C4—61no—
243C4⋊C4—91no—
244C4⋊C4—181no—
245C42—61no—
246C42—181no—
247C42D3×C3⋊D311yescharacteristic
248C22×C3—24no—
249C22×C3C2×#62514yes—
250C22×C3C2×#63811yescharacteristic
251C3×C4—24no—
252C3×C4—24no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—24no—
256C3×C4—24no—
257C3×C4—44no—
258C3×C4—44no—
259C3×C4—68no—
260C3×C4—124no—
261C3×C4—182no—
262C3×C4—361no—
263C3×C4#379814yes—
264C3×C4C4×D3214yes—
265C3×C4C2×C4×C3⋊D311yescharacteristic
266C3×C4#382811yescharacteristic
267Dic3—22no—
268Dic3—41no—
269Dic3—68no—
270Dic3—124no—
271C32—24no—
272C32#503014yescomplemented
273C32#503011yescharacteristic, complemented
274Q8—32no—
275Q8—61no—
276Q8—92no—
277Q8—181no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4#629011yescharacteristic
283C2×C4#629011yescharacteristic
284C2×C4C2×D3×C3⋊D311yescharacteristic
285C8—272no—
286C8—541no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—24no—
290C2×C3#6382714yes—
291C2×C3#6382714yes—
292C2×C3C4×#62514yes—
293C2×C3C4×#63811yescharacteristic
294C2×C3#6415111yescharacteristic
295C2×C3#6416011yescharacteristic
296C22C2×#629011yescharacteristic
297C4—21no—
298C4—21no—
299C4—62no—
300C4—121no—
301C4—182no—
302C4—361no—
303C4#8902011yescharacteristic
304C4C4×D3×C3⋊D311yescharacteristic
305C3—24no—
306C3#1059038114yescomplemented
307C3#1059272911yescharacteristic, complemented
308C2#1172337311yescharacteristic
309C2#1172337511yescharacteristic
310C2C4×#629011yescharacteristic
subgroupG/H
centerC22C2×#6290
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
21. #88798C4semidirect
195. C33#441semidirect
272. C32#5030semidirect
273. C32#5030semidirect
306. C3#10590381semidirect
307. C3#10592729semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6342
C32×C4×Dic62
C2×#887982
#117221732
#117220532
#117220392
#117220372
#105927293
#105903813