Finite Group Databasebeta

← #13165380

Subgroups — #13165380

Order 1728.

608 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C2×#88940C211yesmaximal, characteristic
2C2×#88934C212yesmaximal
3C2×C3×#3773C211yesmaximal, characteristic
4C2×Dic3×#157C211yesmaximal, characteristic
5#11722911C212yesmaximal
6#10593413—31nomaximal
7C22×C3⋊#157C2211yescharacteristic
8C22×C3×#157C2211yescharacteristic
9C2×#6294C2212yes—
10C2×Dic3×C3⋊D3C2211yescharacteristic
11C2×C3×#624C2212yes—
12#88940—22no—
13#88934—24no—
14C3×#3773—22no—
15Dic3×#157C414yescomplemented
16#88806—22no—
17C3×Dic3⋊Dic3—22no—
18C2×C4×#157—31no—
19C2×#3773—32no—
20C2×#3773—31no—
21C2×#3773D311yescharacteristic
22#63900—32no—
23D3×Dic3⋊C4—62no—
24C22×C3×C3⋊D3C2311yescharacteristic
25C2×C3⋊#157C2×C412yes—
26C2×C3⋊#157D412yes—
27C2×C3⋊#157Q812yes—
28C2×C3×#157C2×C412yes—
29C2×C3×#157D412yes—
30C2×C3×#157Q812yes—
31C2×C3⋊(C3⋊Dic3)D411yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C32×Dic3—22no—
34C2×C32×Dic3D411yescharacteristic
35#6294—22no—
36#6294—22no—
37Dic3×C3⋊D3C2×C412yes—
38Dic3×C3⋊D3Q812yes—
39C3×#624—22no—
40C3×#624—22no—
41#4975—91nomaximal
42C22×#157—31no—
43C22×#157C2×D311yescharacteristic
44C2×C4×C3⋊D3—31no—
45C2×C3×C4×D3—62no—
46C2×#624—32no—
47C2×#624C2×D312yes—
48C2×D3×Dic3—62no—
49C2×D3×Dic3—62no—
50C4×#157—34no—
51#3773—22no—
52#3773—64no—
53#3773—62no—
54#3750—62no—
55C3×C4⋊Dic3—62no—
56C3×Dic3⋊C4—62no—
57#3724—62no—
58#3724—62no—
59#3724—122no—
60Dic3⋊Dic3—22no—
61C22×C33C2×D411yescharacteristic
62C2×C3×C3⋊D3C22×C411yescharacteristic
63C2×C3×C3⋊D3C2×D411yescharacteristic
64C2×C3×C3⋊D3C2×D411yescharacteristic
65C2×C3×C3⋊D3C2×Q811yescharacteristic
66C2×C3×C3⋊D3C2×Q811yescharacteristic
67C2×C3×C3⋊D3C2×Q811yescharacteristic
68C3⋊#157C4⋊C414yescomplemented
69C3×#157C4⋊C414yescomplemented
70C3⋊(C3⋊Dic3)C4⋊C412yescomplemented
71C3×C3⋊Dic3—42no—
72C32×Dic3—42no—
73C32×Dic3C4⋊C412yescomplemented
74C2×C3×C4⋊C4—91no—
75C2×C4⋊Dic3—91no—
76C2×Dic3⋊C4—92no—
77C2×Dic3⋊C4—92no—
78C2×C4×Dic3—91no—
79D3×C4⋊C4—182no—
80C22×C3⋊D3C22×D311yescharacteristic
81C22×C3×D3—62no—
82C2×#157—32no—
83C2×#157—32no—
84C2×#157—32no—
85C2×#157Dic612yes—
86C2×#157C4×D312yes—
87C2×#157#6712yes—
88C2×C32×C4—31no—
89C2×C32×C4—62no—
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3—31no—
92C2×C3⋊Dic3—62no—
93C2×C3⋊Dic3—122no—
94C4×C3⋊D3—32no—
95C4×C3⋊D3—32no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3—42no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—122no—
103C3×C4×D3—122no—
104C3×C4×D3—122no—
105#624—22no—
106#624—22no—
107#624—62no—
108#624—62no—
109D3×Dic3—64no—
110D3×Dic3—64no—
111D3×Dic3—122no—
112D3×Dic3—122no—
113#384—271noSylow
114C2×C33C2×C4⋊C411yescharacteristic
115C2×C33#12711yescharacteristic
116C2×C33C4⋊Q811yescharacteristic
117C3×C3⋊D3C2×C4⋊C411yescharacteristic
118C3×C3⋊D3C2×C4⋊C411yescharacteristic
119C3×C3⋊D3C4⋊Q811yescharacteristic
120C3×C3⋊D3C4⋊Q811yescharacteristic
121C22×C3×C4—92no—
122C22×C3×C4—91no—
123C22×Dic3—92no—
124C22×Dic3—91no—
125C22×Dic3—91no—
126C2×C4×D3—62no—
127C2×C4×D3—182no—
128C2×C4×D3—182no—
129C3×C4⋊C4—182no—
130C3×C4⋊C4—182no—
131C4⋊Dic3—62no—
132C4⋊Dic3—182no—
133Dic3⋊C4—62no—
134Dic3⋊C4—182no—
135Dic3⋊C4—182no—
136Dic3⋊C4—182no—
137Dic3⋊C4—184no—
138Dic3⋊C4—362no—
139C4×Dic3—94no—
140C22×C32—22no—
141C22×C32—42no—
142C22×C32D3×D411yescharacteristic
143C2×C3⋊D3C2×Dic611yescharacteristic
144C2×C3⋊D3C2×C4×D311yescharacteristic
145C2×C3⋊D3D3×D411yescharacteristic
146C2×C3⋊D3D3×Q811yescharacteristic
147C2×C3⋊D3D3×Q811yescharacteristic
148C2×C3⋊D3C2×#6711yescharacteristic
149C2×C3×D3—62no—
150C2×C3×D3—62no—
151C2×C3×D3—62no—
152C2×C3×D3—62no—
153C2×C3×D3—62no—
154C2×C3×D3—62no—
155#157—34no—
156#157Dic3⋊C414yescomplemented
157C32×C4—32no—
158C32×C4—122no—
159C3⋊Dic3—32no—
160C3⋊Dic3—42no—
161C3⋊Dic3—64no—
162C3⋊Dic3—124no—
163C3×Dic3—24no—
164C3×Dic3—42no—
165C3×Dic3—42no—
166C3×Dic3—82no—
167C3×Dic3—122no—
168C3×Dic3—122no—
169C3×Dic3—242no—
170C2×C4⋊C4—91no—
171C2×C4⋊C4—272no—
172C2×C4⋊C4—272no—
173C2×C4⋊C4—271no—
174C2×C42—271no—
175C33#38411yescharacteristic, complemented, Sylow
176C23×C3—91no—
177C22×D3—62no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4—92no—
182C2×C3×C4—92no—
183C2×C3×C4—92no—
184C2×C3×C4—122no—
185C2×C3×C4—182no—
186C2×C3×C4—182no—
187C2×C3×C4—182no—
188C2×Dic3—22no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—92no—
192C2×Dic3—92no—
193C2×Dic3—92no—
194C2×Dic3—91no—
195C2×Dic3—92no—
196C2×Dic3—92no—
197C2×Dic3—122no—
198C2×Dic3—182no—
199C2×Dic3—182no—
200C2×Dic3—182no—
201C2×Dic3—362no—
202C2×Dic3GO+(4,𝔽2)11yescharacteristic
203C4×D3—122no—
204C4×D3—122no—
205C4×D3—184no—
206C4×D3—184no—
207C4×D3—362no—
208C4×D3—362no—
209C2×C32—22no—
210C2×C32—22no—
211C2×C32—22no—
212C2×C32—42no—
213C2×C32—42no—
214C2×C32—42no—
215C2×C32#88811yescharacteristic
216C2×C32D3×C4⋊C411yescharacteristic
217C2×C32#94011yescharacteristic
218C3⋊D3#88811yescharacteristic
219C3⋊D3D3×C4⋊C411yescharacteristic
220C3⋊D3C2×Dic3⋊C411yescharacteristic
221C3⋊D3#94411yescharacteristic
222C3×D3—62no—
223C3×D3—62no—
224C3×D3—62no—
225C3×D3—62no—
226C22×C4—92no—
227C22×C4—91no—
228C22×C4—271no—
229C22×C4—272no—
230C22×C4—271no—
231C4⋊C4—182no—
232C4⋊C4—182no—
233C4⋊C4—542no—
234C4⋊C4—542no—
235C4⋊C4—544no—
236C42—274no—
237C22×C3—22no—
238C22×C3—42no—
239C22×C3—91no—
240C22×C3—91no—
241C22×C3—91no—
242C22×C3—91no—
243C22×C3—91no—
244C22×C3—91no—
245C22×C3C2×GO+(4,𝔽2)11yescharacteristic
246C2×D3—62no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—62no—
252C3×C4—64no—
253C3×C4—94no—
254C3×C4—122no—
255C3×C4—122no—
256C3×C4—242no—
257C3×C4—362no—
258Dic3—42no—
259Dic3—94no—
260Dic3—92no—
261Dic3—122no—
262Dic3—122no—
263Dic3—184no—
264Dic3—242no—
265Dic3—364no—
266Dic3#377312yescomplemented
267C32—22no—
268C32—42no—
269C32#497511yescharacteristic, complemented
270C23—91no—
271C2×C4—31no—
272C2×C4—62no—
273C2×C4—92no—
274C2×C4—92no—
275C2×C4—92no—
276C2×C4—182no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4—272no—
280C2×C4—272no—
281C2×C4—272no—
282C2×C4—272no—
283C2×C4—272no—
284C2×C4—271no—
285C2×C4—542no—
286C2×C4—542no—
287C2×C3—22no—
288C2×C3—22no—
289C2×C3—22no—
290C2×C3—42no—
291C2×C3—42no—
292C2×C3—42no—
293C2×C3—91no—
294C2×C3—91no—
295C2×C3—91no—
296C2×C3—91no—
297C2×C3#6422211yescharacteristic
298C2×C3#6423111yescharacteristic
299C2×C3C2×#377311yescharacteristic
300D3—62no—
301D3—62no—
302D3—62no—
303D3—62no—
304C22—91no—
305C22—91no—
306C22—91no—
307C22—91no—
308C22—91no—
309C22—91no—
310C22D3×GO+(4,𝔽2)11yescharacteristic
311C4—32no—
312C4—94no—
313C4—122no—
314C4—274no—
315C4—272no—
316C4—362no—
317C3—22no—
318C3—42no—
319C3#1059341311yescharacteristic, complemented
320C2—91no—
321C2—91no—
322C2—91no—
323C2—91no—
324C2D3×#377311yescharacteristic
325C2#1172412411yescharacteristic
326C2#1172412511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×D4
prime radical, cosocleC22×C3×C3⋊D3C23
hypocenterC33#384
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
15. Dic3×#157C4semidirect
68. C3⋊#157C4⋊C4semidirect
69. C3×#157C4⋊C4semidirect
70. C3⋊(C3⋊Dic3)C4⋊C4semidirect
73. C32×Dic3C4⋊C4semidirect
156. #157Dic3⋊C4semidirect
175. C33#384semidirect
266. Dic3#3773semidirect
269. C32#4975semidirect
319. C3#10593413semidirect

Maximal subgroups

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

subgroupindex
C2×#889402
C2×#889342
C2×C3×#37732
C2×Dic3×#1572
#117229112
#105934133
#49759