Finite Group Databasebeta

← #13167078

Subgroups — #13167078

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
2C3×#63919C211yesmaximal, characteristic
3C4⋊(C3⋊#622)C211yesmaximal, characteristic
4#11722779C212yesmaximal
5#11722774C212yesmaximal
6#10593393—31nomaximal
7C2×#6296C2211yescharacteristic
8C2×C3×#627—21no—
9C2×C3⋊#622C2212yes—
10C3×C4×C3⋊Dic3—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3—21no—
13#88815—21no—
14#88815C2211yescharacteristic
15C3×#3724C2212yes—
16C3×#888—61no—
17#63919D311yescharacteristic
18C4×#627—31no—
19C4×#627—61no—
20C4⋊#622—31no—
21#63743—32no—
22#63740—32no—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—22no—
29#6296—21no—
30#6296Q812yescomplemented
31C3×#627—42no—
32C3⋊#622—22no—
33#5091—91nomaximal
34C2×C3×Dic6—62no—
35C2×#627—21no—
36C2×#627—31no—
37C2×#627—61no—
38C2×#622—32no—
39C32×C4⋊C4—61no—
40C4×C3⋊Dic3—21no—
41C4×C3⋊Dic3C2×D311yescharacteristic
42C3×C4⋊Dic3—21no—
43C3×C4⋊Dic3—21no—
44C3×C4⋊Dic3—41no—
45C3×Dic3⋊C4—62no—
46C3×C4×Dic3—61no—
47C3×C4×Dic3—61no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—61no—
50C3×C4×Dic3—121no—
51#3724—31no—
52#3724—61no—
53#3724C2×D312yes—
54Dic3⋊Dic3—61no—
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⋊Dic3C2×Q811yescharacteristic
63C3×C3⋊Dic3#4111yescharacteristic
64C32×Dic3—42no—
65C3×C4⋊Q8—91no—
66#888—61no—
67C4×Dic6—91no—
68C4×Dic6—181no—
69#810—92no—
70#807—92no—
71C4⋊(C3⋊C8)—91no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4—62no—
77C2×C32×C4#6711yescharacteristic
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3—21no—
80C2×C3⋊Dic3#6711yescharacteristic
81C2×C3⋊Dic3C22×D311yescharacteristic
82C2×C3×Dic3—22no—
83C2×C3×Dic3—22no—
84C2×C3×Dic3—42no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
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—121no—
94C2×C3×Dic3—121no—
95C3×Dic6—124no—
96#627—32no—
97#627—42no—
98#627—62no—
99#627—62no—
100#627—61no—
101#622—62no—
102#477—271noSylow
103C2×C33C22⋊Q811yescharacteristic
104C2×C33C2×Q1611yescharacteristic
105C2×C33AGL(1,ℤ/8)11yescharacteristic
106C2×C3×Q8—181no—
107C2×Dic6—62no—
108C2×Dic6—91no—
109C2×Dic6—181no—
110C3×C4⋊C4—61no—
111C3×C4⋊C4—61no—
112C3×C4⋊C4—92no—
113C3×C4⋊C4—121no—
114C3×C42—91no—
115C3×C42—181no—
116C4⋊Dic3—21no—
117C4⋊Dic3—91no—
118C4⋊Dic3—181no—
119Dic3⋊C4—62no—
120Dic3⋊C4—181no—
121Dic3⋊C4—181no—
122Dic3⋊C4—181no—
123C4×Dic3—61no—
124C4×Dic3—61no—
125C4×Dic3—61no—
126C4×Dic3—61no—
127C4×Dic3—121no—
128C2×C3⋊C8—92no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—41no—
132C22×C32—41no—
133C22×C32C2×#6711yescharacteristic
134C32×C4—22no—
135C32×C4—22no—
136C32×C4—42no—
137C32×C4—42no—
138C32×C4—122no—
139C32×C4#21612yes—
140C3⋊Dic3—22no—
141C3⋊Dic3—21no—
142C3⋊Dic3—41no—
143C3⋊Dic3D3×Q811yescharacteristic
144C3⋊Dic3#23911yescharacteristic
145C3×Dic3—42no—
146C3×Dic3—42no—
147C3×Dic3—61no—
148C3×Dic3—61no—
149C3×Dic3—61no—
150C3×Dic3—61no—
151C3×Dic3—62no—
152C3×Dic3—62no—
153C3×Dic3—62no—
154C3×Dic3—82no—
155C3×Dic3—121no—
156C3×Dic3—122no—
157C3×Dic3—121no—
158C3×Dic3—122no—
159C3×Dic3—121no—
160C4⋊Q8—91no—
161C4×Q8—271no—
162#107—272no—
163C4⋊C8—271no—
164#103—272no—
165C33#47711yescharacteristic, complemented, Sylow
166C3×Q8—362no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—62no—
172C2×C3×C4—62no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—122no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4GO+(4,𝔽2)11yescharacteristic
179C2×Dic3—22no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—121no—
189C2×Dic3—121no—
190Dic6—92no—
191Dic6—124no—
192Dic6—182no—
193Dic6—181no—
194Dic6—182no—
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#93211yescharacteristic
209C2×C32C2×#21611yescharacteristic
210C2×C32#94611yescharacteristic
211C2×Q8—181no—
212C2×Q8—271no—
213C2×C8—272no—
214C4⋊C4—61no—
215C4⋊C4—92no—
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—122no—
232C3×C4—122no—
233C3×C4—182no—
234C3×C4—181no—
235C3×C4—242no—
236C3×C4—361no—
237C3×C4#377612yes—
238Dic3—42no—
239Dic3—61no—
240Dic3—61no—
241Dic3—61no—
242Dic3—61no—
243Dic3—62no—
244Dic3—62no—
245Dic3—62no—
246Dic3—121no—
247Dic3—122no—
248Dic3—121no—
249Dic3—122no—
250Dic3—121no—
251C32—21no—
252C32—21no—
253C32—41no—
254C32—41no—
255C32#509111yescharacteristic, complemented
256Q8—272no—
257Q8—362no—
258Q8—541no—
259C2×C4—62no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—181no—
263C2×C4—181no—
264C2×C4D32⋊D311yescharacteristic
265C8—542no—
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×D32⋊D311yescharacteristic
282C4—91no—
283C4—91no—
284C4—122no—
285C4—182no—
286C4—181no—
287C4—361no—
288C4#8893812yes—
289C3—21no—
290C3—21no—
291C3—41no—
292C3—41no—
293C3#1059339311yescharacteristic, complemented
294C2#1172429811yescharacteristic
295C2#1172430411yescharacteristic
296C2C2×#8893811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4D32⋊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
30. #6296Q8semidirect
165. C33#477semidirect
255. C32#5091semidirect
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
C3×#639192
C4⋊(C3⋊#622)2
#117227792
#117227742
#105933933
#50919