Finite Group Databasebeta

← #13167451

Subgroups — #13167451

Order 1728.

386 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 61 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
2C4×#6296C211yesmaximal, characteristic
3C3×C4⋊#622C211yesmaximal, characteristic
4#11722797C212yesmaximal
5#11722794C212yesmaximal
6#10593393—31nomaximal
7C2×#6296—21no—
8C2×#6296C2211yescharacteristic
9C2×C3×#622C2212yes—
10C3×C4⋊(C3⋊Dic3)—21no—
11C3×C4×C3⋊Dic3—21no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13#88815—21no—
14#88815C2212yes—
15#88815C2211yescharacteristic
16#63919—31no—
17#63918—61no—
18C4×#627—31no—
19C4×#627—61no—
20C4⋊#622D311yescharacteristic
21#63743—32no—
22#63740—32no—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29#6296—21no—
30#6296—42no—
31#6296Q812yescomplemented
32C3×#622—22no—
33#4850—91nomaximal
34C2×#627—31no—
35C2×#627—61no—
36C2×#627—62no—
37C2×#627—61no—
38C2×#622C2×D312yes—
39C4⋊(C3⋊Dic3)—21no—
40C4×C3⋊Dic3—21no—
41C4×C3⋊Dic3C2×D311yescharacteristic
42C3×C4⋊Dic3—61no—
43C3×C4⋊Dic3—61no—
44C3×C4⋊Dic3—121no—
45C3×C4×Dic3—61no—
46C3×C4×Dic3—61no—
47C3×C4×Dic3—61no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—121no—
50#3724—31no—
51#3724—32no—
52#3724—61no—
53Dic3⋊Dic3—61no—
54Dic3⋊Dic3—62no—
55Dic3⋊Dic3—61no—
56Dic3⋊Dic3—61no—
57C22×C33C2×D411yescharacteristic
58C33×C4Q1612yes—
59C3×C3⋊Dic3—22no—
60C3×C3⋊Dic3—21no—
61C3×C3⋊Dic3—41no—
62C3×C3⋊Dic3—42no—
63C3×C3⋊Dic3C2×Q811yescharacteristic
64C3×C3⋊Dic3#4111yescharacteristic
65#888—181no—
66#869—91no—
67C4×Dic6—91no—
68C4×Dic6—181no—
69C3×C4⋊C8—91no—
70#818—92no—
71#816—92no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4D1211yescharacteristic
77C2×C3⋊Dic3—21no—
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3D1211yescharacteristic
81C2×C3⋊Dic3C22×D311yescharacteristic
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—121no—
93C2×C3×Dic3—121no—
94C2×C3×Dic3—122no—
95#627—32no—
96#627—62no—
97#627—62no—
98#627—61no—
99#627—124no—
100#627—122no—
101#622—22no—
102#477—271noSylow
103C2×C33C22⋊Q811yescharacteristic
104C2×C33C2×Q1611yescharacteristic
105C2×C33AGL(1,ℤ/8)11yescharacteristic
106C2×Dic6—91no—
107C2×Dic6—181no—
108C2×Dic6—181no—
109C2×Dic6—182no—
110C2×C3×C8—92no—
111C3×C4⋊C4—181no—
112C3×C42—91no—
113C3×C42—181no—
114C4⋊Dic3—61no—
115C4⋊Dic3—61no—
116C4⋊Dic3—92no—
117C4⋊Dic3—91no—
118C4⋊Dic3—121no—
119C4⋊Dic3—181no—
120Dic3⋊C4—181no—
121Dic3⋊C4—182no—
122Dic3⋊C4—181no—
123Dic3⋊C4—181no—
124C4×Dic3—61no—
125C4×Dic3—61no—
126C4×Dic3—61no—
127C4×Dic3—61no—
128C4×Dic3—121no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—41no—
132C22×C32—41no—
133C22×C32C2×D1211yescharacteristic
134C32×C4—22no—
135C32×C4—22no—
136C32×C4—42no—
137C32×C4—42no—
138C32×C4Dic1212yes—
139C3⋊Dic3—22no—
140C3⋊Dic3—21no—
141C3⋊Dic3—41no—
142C3⋊Dic3—42no—
143C3⋊Dic3#23711yescharacteristic
144C3⋊Dic3D3×Q811yescharacteristic
145C3×Dic3—61no—
146C3×Dic3—61no—
147C3×Dic3—61no—
148C3×Dic3—61no—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C3×Dic3—62no—
152C3×Dic3—121no—
153C3×Dic3—122no—
154C3×Dic3—122no—
155C3×Dic3—121no—
156C3×Dic3—122no—
157C3×Dic3—122no—
158C3×Dic3—121no—
159C3×Dic3—242no—
160C4⋊Q8—271no—
161C4×Q8—271no—
162#107—272no—
163C4⋊C8—91no—
164#103—272no—
165C33#47711yescharacteristic, complemented, Sylow
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—91no—
171C2×C3×C4—91no—
172C2×C3×C4—181no—
173C2×C3×C4—181no—
174C2×C3×C4—182no—
175C2×C3×C4GO+(4,𝔽2)11yescharacteristic
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—61no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—62no—
184C2×Dic3—61no—
185C2×Dic3—62no—
186C2×Dic3—121no—
187C2×Dic3—121no—
188C2×Dic3—122no—
189Dic6—92no—
190Dic6—182no—
191Dic6—181no—
192Dic6—182no—
193Dic6—364no—
194Dic6—362no—
195C3×C8—182no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32#89711yescharacteristic
209C2×C32C2×Dic1211yescharacteristic
210C2×C32#90811yescharacteristic
211C2×Q8—271no—
212C2×Q8—541no—
213C2×C8—92no—
214C4⋊C4—181no—
215C4⋊C4—272no—
216C4⋊C4—271no—
217C4⋊C4—541no—
218C42—91no—
219C42—181no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3C2×GO+(4,𝔽2)11yescharacteristic
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—42no—
228C3×C4—42no—
229C3×C4—91no—
230C3×C4—91no—
231C3×C4—182no—
232C3×C4—181no—
233C3×C4—362no—
234C3×C4—361no—
235C3×C4#377612yes—
236Dic3—61no—
237Dic3—61no—
238Dic3—61no—
239Dic3—61no—
240Dic3—62no—
241Dic3—62no—
242Dic3—62no—
243Dic3—121no—
244Dic3—122no—
245Dic3—122no—
246Dic3—121no—
247Dic3—122no—
248Dic3—122no—
249Dic3—121no—
250Dic3—242no—
251C32—21no—
252C32—21no—
253C32—41no—
254C32—41no—
255C32#485011yescharacteristic, complemented
256Q8—272no—
257Q8—541no—
258Q8—1082no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—181no—
262C2×C4—181no—
263C2×C4—182no—
264C2×C4#157⋊D311yescharacteristic
265C8—182no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—41no—
278C2×C3D32⋊Q811yescharacteristic
279C2×C3#6422411yescharacteristic
280C2×C3C2×#377611yescharacteristic
281C22C2×#157⋊D311yescharacteristic
282C4—91no—
283C4—91no—
284C4—182no—
285C4—181no—
286C4—362no—
287C4—361no—
288C4#8894312yes—
289C3—21no—
290C3—21no—
291C3—41no—
292C3—41no—
293C3#1059339311yescharacteristic, complemented
294C2#1172433311yescharacteristic
295C2#1172434011yescharacteristic
296C2C2×#8894311yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#477
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
31. #6296Q8semidirect
165. C33#477semidirect
255. C32#4850semidirect
293. C3#10593393semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C4×#62962
C3×C4⋊#6222
#117227972
#117227942
#105933933
#48509