Finite Group Databasebeta

← #13167404

Subgroups — #13167404

Order 1728.

464 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#11723697C211yesmaximal, characteristic
2#11723679C211yesmaximal, characteristic, complemented
3C3×#63777C211yesmaximal, characteristic, complemented
4#11722793C212yesmaximal, complemented
5#11722793C212yesmaximal, complemented
6#10593356—31nomaximal
7C2×C3×C4⋊(C3⋊D3)—21no—
8C2×#6295—21no—
9C2×#6295C2212yes—
10C2×C3×#622C2212yes—
11C3×#3750—21no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13#88815—21no—
14#88815C2211yescharacteristic
15#88815C2211yescharacteristic
16#63914—31no—
17#63911—61no—
18#63870—31no—
19#63867—61no—
20#63777D311yescharacteristic, complemented
21#63739—32no—
22#63739—32no—
23C22×C3×C3⋊D3—22no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C3×C4⋊(C3⋊D3)—42no—
30#6295—24no—
31#6295—42no—
32C3×#622—22no—
33#4713—91nomaximal
34C2×C4⋊(C3⋊D3)—21no—
35C2×C3×D12—61no—
36C2×C3×D12—61no—
37C2×C3×D12—121no—
38C2×#626—62no—
39C2×#626—62no—
40C2×#625—32no—
41C2×#625—61no—
42C2×#622C2×D312yes—
43#3750—21no—
44C4×C3⋊Dic3C2×D311yescharacteristic
45C3×Dic3⋊C4—61no—
46C3×Dic3⋊C4—61no—
47C3×Dic3⋊C4—121no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—61no—
50#3724—31no—
51#3724—31no—
52#3724—61no—
53Dic3⋊Dic3—61no—
54Dic3⋊Dic3—61no—
55Dic3⋊Dic3—61no—
56Dic3⋊Dic3—61no—
57C22×C33C2×D411yescharacteristic
58C2×C3×C3⋊D3—42no—
59C2×C3×C3⋊D3—42no—
60C2×C3×C3⋊D3—42no—
61C33×C4—21no—
62C3×C3⋊Dic3—21no—
63C3×C3⋊Dic3—41no—
64C3×C3⋊Dic3—41no—
65C3×C3⋊Dic3#4112yes—
66#940—181no—
67#889—181no—
68#874—91no—
69#870—91no—
70C3×#97—91no—
71#821—92no—
72#821—92no—
73C22×C3⋊D3—22no—
74C22×C3×D3—62no—
75C22×C3×D3—62no—
76C22×C3×D3—122no—
77C2×C32×C4—21no—
78C2×C32×C4—21no—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C32×C4D1211yescharacteristic
82C2×C3⋊Dic3—21no—
83C2×C3⋊Dic3—21no—
84C2×C3⋊Dic3D1211yescharacteristic
85C2×C3⋊Dic3C22×D311yescharacteristic
86C4⋊(C3⋊D3)—42no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—121no—
96C2×C3×Dic3—121no—
97C3×D12—122no—
98C3×D12—122no—
99C3×D12—124no—
100#626—124no—
101#626—124no—
102#625—64no—
103#625—122no—
104#622—22no—
105#490—271noSylow
106C2×C33#12411yescharacteristic
107C2×C33AGL(1,ℤ/8)11yescharacteristic
108C2×C33AGL(1,ℤ/8)11yescharacteristic
109C3×C3⋊D3—82no—
110C2×C3×D4—181no—
111C2×#67—182no—
112C2×#67—182no—
113C2×D12—61no—
114C2×D12—61no—
115C2×D12—92no—
116C2×D12—121no—
117C2×D12—181no—
118C2×C3×C8—92no—
119C3×C4⋊C4—181no—
120C3×C42—91no—
121C4⋊Dic3—91no—
122C4⋊Dic3—91no—
123C4⋊Dic3—181no—
124Dic3⋊C4—61no—
125Dic3⋊C4—61no—
126Dic3⋊C4—121no—
127Dic3⋊C4—181no—
128Dic3⋊C4—181no—
129Dic3⋊C4—181no—
130Dic3⋊C4—181no—
131C4×Dic3—61no—
132C4×Dic3—61no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32C2×D1211yescharacteristic
138C2×C3⋊D3—42no—
139C2×C3⋊D3—42no—
140C2×C3⋊D3—42no—
141C2×C3×D3—122no—
142C2×C3×D3—122no—
143C2×C3×D3—122no—
144C2×C3×D3—124no—
145C2×C3×D3—122no—
146C2×C3×D3—122no—
147C2×C3×D3—122no—
148C2×C3×D3—242no—
149C2×C3×D3—242no—
150C32×C4—22no—
151C32×C4—21no—
152C32×C4—41no—
153C32×C4—42no—
154C32×C4—81no—
155C3⋊Dic3—21no—
156C3⋊Dic3—41no—
157C3⋊Dic3—41no—
158C3⋊Dic3#23512yes—
159C3×Dic3—62no—
160C3×Dic3—62no—
161C3×Dic3—62no—
162C3×Dic3—121no—
163C3×Dic3—121no—
164C3×Dic3—121no—
165C3×Dic3—122no—
166C3×Dic3—121no—
167C3×Dic3—121no—
168C3×Dic3—241no—
169#127—271no—
170#125—271no—
171#102—272no—
172#102—272no—
173#97—91no—
174C33#49011yescharacteristic, complemented, Sylow
175C23×C3—182no—
176C22×D3—62no—
177C22×D3—62no—
178C22×D3—122no—
179C3×D4—362no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—91no—
185C2×C3×C4—91no—
186C2×C3×C4—181no—
187C2×C3×C4—181no—
188C2×C3×C4GO+(4,𝔽2)11yescharacteristic
189#67—364no—
190#67—364no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196C2×Dic3—61no—
197C2×Dic3—61no—
198C2×Dic3—61no—
199C2×Dic3—121no—
200C2×Dic3—121no—
201D12—122no—
202D12—122no—
203D12—124no—
204D12—184no—
205D12—362no—
206C3×C8—182no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32#87511yescharacteristic
220C2×C32#90811yescharacteristic
221C2×C32#90811yescharacteristic
222C3⋊D3—82no—
223C3×D3—242no—
224C3×D3—244no—
225C3×D3—242no—
226C2×D4—181no—
227C2×D4—272no—
228C2×D4—541no—
229C2×C8—92no—
230C4⋊C4—181no—
231C4⋊C4—271no—
232C4⋊C4—271no—
233C4⋊C4—541no—
234C42—91no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—362no—
240C22×C3—362no—
241C22×C3—362no—
242C22×C3C2×GO+(4,𝔽2)11yescharacteristic
243C2×D3—122no—
244C2×D3—122no—
245C2×D3—122no—
246C2×D3—124no—
247C2×D3—122no—
248C2×D3—122no—
249C2×D3—122no—
250C2×D3—242no—
251C2×D3—242no—
252C3×C4—22no—
253C3×C4—21no—
254C3×C4—41no—
255C3×C4—42no—
256C3×C4—81no—
257C3×C4—92no—
258C3×C4—181no—
259C3×C4—361no—
260C3×C4—361no—
261Dic3—62no—
262Dic3—62no—
263Dic3—62no—
264Dic3—121no—
265Dic3—121no—
266Dic3—121no—
267Dic3—122no—
268Dic3—121no—
269Dic3—121no—
270Dic3—241no—
271C32—21no—
272C32—21no—
273C32—41no—
274C32—41no—
275C32#471311yescharacteristic, complemented
276C23—182no—
277D4—362no—
278D4—544no—
279D4—1082no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C4#157⋊D311yescharacteristic
285C8—182no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—722no—
299C2×C3#6422111yescharacteristic
300C2×C3#6422411yescharacteristic
301C2×C3#6423611yescharacteristic
302D3—242no—
303D3—244no—
304D3—242no—
305C22—362no—
306C22—362no—
307C22—362no—
308C22C2×#157⋊D311yescharacteristic
309C4—21no—
310C4—92no—
311C4—181no—
312C4—361no—
313C4—361no—
314C3—21no—
315C3—21no—
316C3—41no—
317C3—41no—
318C3#1059335611yescharacteristic, complemented
319C2—722no—
320C2#1172433411yescharacteristic
321C2#1172434011yescharacteristic
322C2#1172435011yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical#11723697C2
hypercenterC2×C4#157⋊D3
hypocenterC33#490
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723679C2semidirect
3. C3×#63777C2semidirect
4. #11722793C2semidirect
5. #11722793C2semidirect
20. #63777D3semidirect
174. C33#490semidirect
275. C32#4713semidirect
318. C3#10593356semidirect

Maximal subgroups

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

subgroupindex
#117236972
#117236792
C3×#637772
#117227932
#105933563
#47139