Finite Group Databasebeta

← #13167011

Subgroups — #13167011

Order 1728.

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

normal subgroups only

This group has 404 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
1C4⋊#6296C211yesmaximal, characteristic, complemented
2C3×#63915C211yesmaximal, characteristic
3#11723102C211yesmaximal, characteristic, complemented
4#11722775C212yesmaximal, complemented
5#11722773C212yesmaximal
6#10593362—31nomaximal
7C2×#6296—21no—
8C2×C3×#627C2211yescharacteristic
9C2×C3×#625C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×(C2×D3)⋊C4—21no—
14#88815C2212yes—
15C3×(C2×D3)⋊Dic3—21no—
16C3×#885—61no—
17#63919—31no—
18#63918—61no—
19#63915D311yescharacteristic
20#63777—31no—
21#63740—32no—
22#63739—32no—
23C22×C32×D3—21no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—21no—
29#6296—42no—
30C3×#627—22no—
31C3×#625—22no—
32C3⋊#622—22no—
33#5055—91nomaximal
34C2×C3×#67—61no—
35C2×C3×Dic6—61no—
36C2×#627—62no—
37C2×#627—61no—
38C2×#627C2×D311yescharacteristic
39C2×#625C2×D311yescharacteristic
40C2×#622—32no—
41C32×C22⋊C4—61no—
42C4⋊(C3⋊Dic3)—21no—
43C4×C3⋊Dic3C2×D311yescharacteristic
44C3×#217—61no—
45C3×(C2×D3)⋊C4—21no—
46C3×(C2×D3)⋊C4—21no—
47C3×(C2×D3)⋊C4—41no—
48C3×(C2×D3)⋊C4—61no—
49C3×C4⋊Dic3—61no—
50C3×C4⋊Dic3—61no—
51C3×C4⋊Dic3—121no—
52C3×C4×Dic3—61no—
53C3×C4×Dic3—61no—
54#3724—32no—
55Dic3⋊Dic3—62no—
56(C2×D3)⋊Dic3—21no—
57C22×C33C2×D411yescharacteristic
58C2×C32×D3—41no—
59C2×C32×D3—41no—
60C2×C32×D3—41no—
61C33×C4—21no—
62C3×C3⋊Dic3—21no—
63C3×C3⋊Dic3—42no—
64C3×C3⋊Dic3#4112yes—
65C32×Dic3—41no—
66C3×#124—91no—
67#888—181no—
68#885—61no—
69#869—91no—
70#835—92no—
71#832—92no—
72#803—91no—
73C23×C32—61no—
74C22×C3×D3—21no—
75C22×C3×D3—21no—
76C22×C3×D3—41no—
77C2×C32×C4—21no—
78C2×C32×C4—21no—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C32×C4—61no—
82C2×C32×C4#6711yescharacteristic
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3#6711yescharacteristic
85C2×C3⋊Dic3C22×D311yescharacteristic
86C3×#67—122no—
87C2×C3×Dic3—21no—
88C2×C3×Dic3—21no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—122no—
97C3×Dic6—122no—
98#627—22no—
99#627—124no—
100#627—122no—
101#625—22no—
102#622—62no—
103#489—271noSylow
104C2×C33#12411yescharacteristic
105C2×C33AGL(1,ℤ/8)11yescharacteristic
106C2×C33#13711yescharacteristic
107C32×D3—81no—
108C2×C3×Q8—91no—
109C2×C3×D4—91no—
110C2×#67—61no—
111C2×Dic6—61no—
112C2×Dic6—181no—
113C2×Dic6—182no—
114C3×C4⋊C4—181no—
115C3×C22⋊C4—61no—
116C3×C22⋊C4—61no—
117C3×C22⋊C4—121no—
118C3×C22⋊C4—181no—
119C3×C42—91no—
120#217—61no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—61no—
123C4⋊Dic3—61no—
124C4⋊Dic3—61no—
125C4⋊Dic3—92no—
126C4⋊Dic3—121no—
127Dic3⋊C4—182no—
128C4×Dic3—61no—
129C4×Dic3—61no—
130C2×C3⋊C8—92no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32—121no—
136C22×C32—121no—
137C22×C32—121no—
138C22×C32C2×#6711yescharacteristic
139C2×C3×D3—41no—
140C2×C3×D3—41no—
141C2×C3×D3—41no—
142C2×C3×D3—42no—
143C2×C3×D3—41no—
144C2×C3×D3—41no—
145C2×C3×D3—41no—
146C2×C3×D3—81no—
147C2×C3×D3—81no—
148C32×C4—22no—
149C32×C4—21no—
150C32×C4—41no—
151C32×C4—42no—
152C32×C4—81no—
153C32×C4—121no—
154C3⋊Dic3—21no—
155C3⋊Dic3—42no—
156C3⋊Dic3#23712yes—
157C3×Dic3—41no—
158C3×Dic3—41no—
159C3×Dic3—62no—
160C3×Dic3—62no—
161C3×Dic3—62no—
162C3×Dic3—81no—
163C3×Dic3—121no—
164C3×Dic3—122no—
165C3×Dic3—122no—
166C3×Dic3—242no—
167C4⋊Q8—271no—
168#124—91no—
169#103—272no—
170#102—272no—
171#97—271no—
172C33#48911yescharacteristic, complemented, Sylow
173C23×C3—61no—
174C23×C3—61no—
175C23×C3—121no—
176C22×D3—21no—
177C3×Q8—182no—
178C3×D4—182no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—91no—
186C2×C3×C4—91no—
187C2×C3×C4—121no—
188C2×C3×C4—182no—
189C2×C3×C4GO+(4,𝔽2)11yescharacteristic
190#67—122no—
191C2×Dic3—21no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196C2×Dic3—62no—
197C2×Dic3—62no—
198C2×Dic3—122no—
199Dic6—122no—
200Dic6—364no—
201Dic6—362no—
202C3⋊C8—182no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—241no—
216C2×C32#93611yescharacteristic
217C2×C32#94911yescharacteristic
218C2×C32#95111yescharacteristic
219C3×D3—81no—
220C3×D3—82no—
221C3×D3—81no—
222C2×Q8—91no—
223C2×Q8—541no—
224C2×D4—91no—
225C2×C8—272no—
226C4⋊C4—181no—
227C4⋊C4—272no—
228C22⋊C4—61no—
229C22⋊C4—181no—
230C42—91no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—121no—
236C22×C3—121no—
237C22×C3—121no—
238C22×C3—122no—
239C22×C3—121no—
240C22×C3—121no—
241C22×C3—121no—
242C22×C3—241no—
243C22×C3—241no—
244C22×C3C2×GO+(4,𝔽2)11yescharacteristic
245C2×D3—41no—
246C2×D3—41no—
247C2×D3—41no—
248C3×C4—22no—
249C3×C4—21no—
250C3×C4—41no—
251C3×C4—42no—
252C3×C4—81no—
253C3×C4—92no—
254C3×C4—121no—
255C3×C4—121no—
256C3×C4—181no—
257C3×C4—241no—
258C3×C4—362no—
259Dic3—41no—
260Dic3—62no—
261Dic3—62no—
262Dic3—62no—
263Dic3—121no—
264Dic3—122no—
265Dic3—122no—
266Dic3—242no—
267C32—21no—
268C32—21no—
269C32—41no—
270C32—41no—
271C32#505511yescharacteristic, complemented
272C23—61no—
273Q8—182no—
274Q8—1082no—
275D4—182no—
276C2×C4—61no—
277C2×C4—91no—
278C2×C4—91no—
279C2×C4—182no—
280C2×C4D32⋊D311yescharacteristic
281C8—542no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3—241no—
295C2×C3—242no—
296C2×C3—241no—
297C2×C3#6422111yescharacteristic
298C2×C3#6422411yescharacteristic
299C2×C3#6423911yescharacteristic
300D3—81no—
301C22—121no—
302C22—121no—
303C22—121no—
304C22C2×D32⋊D311yescharacteristic
305C4—21no—
306C4—92no—
307C4—121no—
308C4—181no—
309C4—362no—
310C3—21no—
311C3—21no—
312C3—41no—
313C3—41no—
314C3#1059336211yescharacteristic, complemented
315C2—241no—
316C2#1172430111yescharacteristic
317C2#1172430511yescharacteristic
318C2#1172432811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypercenterC2×C4D32⋊D3
hypocenterC33#489
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
1. C4⋊#6296C2semidirect
3. #11723102C2semidirect
4. #11722775C2semidirect
172. C33#489semidirect
271. C32#5055semidirect
314. C3#10593362semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×#639152
#117231022
#117227752
#117227732
#105933623
#50559