Finite Group Databasebeta

← #13165409

Subgroups — #13165409

Order 1728.

576 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 94 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×C3×#3773C211yesmaximal, characteristic
3C2×Dic3×#157C211yesmaximal, characteristic
4#11722910C212yesmaximal
5#11722769C212yesmaximal
6#10593410—31nomaximal
7C22×C3⋊#157C2211yescharacteristic
8C22×C3×#157C2211yescharacteristic
9C2×#6294C2212yes—
10C2×Dic3×C3⋊D3C2211yescharacteristic
11C2×C3×#624C2212yes—
12#88940C414yescomplemented
13C3×#3773—22no—
14Dic3×#157—22no—
15#88807—22no—
16C3×#3724—22no—
17C2×C4×#157—31no—
18C2×#3773—31no—
19C2×#3773D311yescharacteristic
20#63883—32no—
21D3×Dic3⋊C4—62no—
22#63737—32no—
23C22×C3×C3⋊D3C2311yescharacteristic
24C2×C3⋊#157—21no—
25C2×C3⋊#157—21no—
26C2×C3⋊#157Q812yes—
27C2×C3×#157C2×C412yes—
28C2×C3×#157D412yes—
29C2×C3×#157Q812yes—
30C2×C3⋊(C3⋊Dic3)D411yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C2×C32×Dic3—22no—
33C2×C32×Dic3D411yescharacteristic
34#6294—22no—
35#6294C2×C414yes—
36Dic3×C3⋊D3—21no—
37Dic3×C3⋊D3—21no—
38C3×#624—22no—
39C3×#624—22no—
40#4972—91nomaximal
41C22×#157—31no—
42C22×#157C2×D311yescharacteristic
43C2×C4×C3⋊D3—31no—
44C2×C3×C4×D3—62no—
45C2×#624—32no—
46C2×#624C2×D312yes—
47C2×D3×Dic3—62no—
48C2×D3×Dic3—62no—
49C4×#157—62no—
50#3773—22no—
51#3773—34no—
52#3750—62no—
53C3×Dic3⋊C4—62no—
54C3×Dic3⋊C4—62no—
55#3724—22no—
56Dic3⋊Dic3—62no—
57Dic3⋊Dic3—62no—
58Dic3⋊Dic3—122no—
59C22×C33C2×D411yescharacteristic
60C2×C3×C3⋊D3C22×C411yescharacteristic
61C2×C3×C3⋊D3C2×D411yescharacteristic
62C2×C3×C3⋊D3C2×Q811yescharacteristic
63C2×C3×C3⋊D3C2×Q811yescharacteristic
64C2×C3×C3⋊D3#4111yescharacteristic
65C2×C3×C3⋊D3#4111yescharacteristic
66C3⋊#157—22no—
67C3×#157C4⋊C414yescomplemented
68C3⋊(C3⋊Dic3)—21no—
69C3×C3⋊Dic3—24no—
70C32×Dic3—21no—
71C32×Dic3—42no—
72C2×C3×C4⋊C4—91no—
73C2×C4⋊Dic3—91no—
74C2×Dic3⋊C4—92no—
75C2×C4×Dic3—91no—
76D3×C4⋊C4—182no—
77C3⋊#95—92no—
78C22×C3⋊D3C22×D311yescharacteristic
79C22×C3×D3—62no—
80C2×#157—32no—
81C2×#157—61no—
82C2×#157—61no—
83C2×#157Dic612yes—
84C2×#157C4×D312yes—
85C2×#157#6712yes—
86C2×C32×C4—31no—
87C2×C32×C4—62no—
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3—31no—
90C2×C3⋊Dic3—62no—
91C2×C3⋊Dic3—122no—
92C4×C3⋊D3—61no—
93C4×C3⋊D3—61no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—42no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—122no—
101C3×C4×D3—122no—
102C3×C4×D3—122no—
103#624—22no—
104#624—22no—
105#624—34no—
106#624—62no—
107D3×Dic3—64no—
108D3×Dic3—64no—
109D3×Dic3—122no—
110D3×Dic3—122no—
111#389—271noSylow
112C2×C33C4×D411yescharacteristic
113C2×C33#12111yescharacteristic
114C2×C33C4⋊Q811yescharacteristic
115C3×C3⋊D3C2×C4⋊C411yescharacteristic
116C3×C3⋊D3C4×Q811yescharacteristic
117C3×C3⋊D3C22⋊Q811yescharacteristic
118C3×C3⋊D3#12511yescharacteristic
119C22×C3×C4—92no—
120C22×C3×C4—91no—
121C22×Dic3—92no—
122C22×Dic3—91no—
123C22×Dic3—91no—
124C2×C4×D3—62no—
125C2×C4×D3—182no—
126C2×C4×D3—182no—
127C3×C4⋊C4—182no—
128C3×C4⋊C4—182no—
129C4⋊Dic3—94no—
130C4⋊Dic3—182no—
131Dic3⋊C4—62no—
132Dic3⋊C4—62no—
133Dic3⋊C4—182no—
134Dic3⋊C4—182no—
135Dic3⋊C4—362no—
136C4×Dic3—182no—
137C22×C32—22no—
138C22×C32—42no—
139C22×C32D3×D411yescharacteristic
140C2×C3⋊D3C2×Dic611yescharacteristic
141C2×C3⋊D3C2×C4×D311yescharacteristic
142C2×C3⋊D3#23711yescharacteristic
143C2×C3⋊D3D3×Q811yescharacteristic
144C2×C3⋊D3#23911yescharacteristic
145C2×C3⋊D3C2×#6711yescharacteristic
146C2×C3×D3—62no—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149C2×C3×D3—62no—
150C2×C3×D3—62no—
151C2×C3×D3—62no—
152#157—62no—
153#157Dic3⋊C414yescomplemented
154C32×C4—61no—
155C32×C4—122no—
156C3⋊Dic3—24no—
157C3⋊Dic3—61no—
158C3⋊Dic3—122no—
159C3⋊Dic3—242no—
160C3×Dic3—42no—
161C3×Dic3—42no—
162C3×Dic3—42no—
163C3×Dic3—64no—
164C3×Dic3—64no—
165C3×Dic3—82no—
166C3×Dic3—124no—
167C2×C4⋊C4—91no—
168C2×C4⋊C4—272no—
169C2×C4⋊C4—271no—
170C2×C42—271no—
171#95—272no—
172C33#38911yescharacteristic, complemented, Sylow
173C23×C3—91no—
174C22×D3—62no—
175C2×C3×C4—62no—
176C2×C3×C4—62no—
177C2×C3×C4—62no—
178C2×C3×C4—92no—
179C2×C3×C4—92no—
180C2×C3×C4—92no—
181C2×C3×C4—122no—
182C2×C3×C4—182no—
183C2×C3×C4—182no—
184C2×C3×C4—182no—
185C2×Dic3—22no—
186C2×Dic3—62no—
187C2×Dic3—62no—
188C2×Dic3—94no—
189C2×Dic3—92no—
190C2×Dic3—91no—
191C2×Dic3—122no—
192C2×Dic3—182no—
193C2×Dic3—181no—
194C2×Dic3—181no—
195C2×Dic3—182no—
196C2×Dic3—181no—
197C2×Dic3—181no—
198C2×Dic3—362no—
199C2×Dic3GO+(4,𝔽2)11yescharacteristic
200C4×D3—122no—
201C4×D3—122no—
202C4×D3—184no—
203C4×D3—184no—
204C4×D3—362no—
205C4×D3—362no—
206C2×C32—22no—
207C2×C32—22no—
208C2×C32—22no—
209C2×C32—42no—
210C2×C32—42no—
211C2×C32—42no—
212C2×C32#88811yescharacteristic
213C2×C32#89311yescharacteristic
214C2×C32#93811yescharacteristic
215C3⋊D3#88711yescharacteristic
216C3⋊D3#89011yescharacteristic
217C3⋊D3C2×Dic3⋊C411yescharacteristic
218C3⋊D3#94611yescharacteristic
219C3×D3—62no—
220C3×D3—62no—
221C3×D3—62no—
222C3×D3—62no—
223C22×C4—92no—
224C22×C4—91no—
225C22×C4—271no—
226C22×C4—272no—
227C22×C4—271no—
228C4⋊C4—182no—
229C4⋊C4—182no—
230C4⋊C4—274no—
231C4⋊C4—542no—
232C42—542no—
233C22×C3—22no—
234C22×C3—42no—
235C22×C3—91no—
236C22×C3—91no—
237C22×C3—91no—
238C22×C3—91no—
239C22×C3—91no—
240C22×C3—91no—
241C22×C3C2×GO+(4,𝔽2)11yescharacteristic
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—62no—
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—62no—
248C3×C4—94no—
249C3×C4—122no—
250C3×C4—122no—
251C3×C4—122no—
252C3×C4—184no—
253C3×C4—242no—
254Dic3—21no—
255Dic3—42no—
256Dic3—64no—
257Dic3—64no—
258Dic3—124no—
259Dic3—182no—
260Dic3—181no—
261Dic3—362no—
262Dic3—722no—
263C32—22no—
264C32—42no—
265C32#497211yescharacteristic, complemented
266C23—91no—
267C2×C4—31no—
268C2×C4—62no—
269C2×C4—92no—
270C2×C4—92no—
271C2×C4—92no—
272C2×C4—182no—
273C2×C4—182no—
274C2×C4—182no—
275C2×C4—274no—
276C2×C4—272no—
277C2×C4—271no—
278C2×C4—541no—
279C2×C4—541no—
280C2×C4—542no—
281C2×C4—541no—
282C2×C4—541no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—42no—
287C2×C3—42no—
288C2×C3—42no—
289C2×C3—91no—
290C2×C3—91no—
291C2×C3—91no—
292C2×C3—91no—
293C2×C3#6422211yescharacteristic
294C2×C3C4×GO+(4,𝔽2)11yescharacteristic
295C2×C3#6424211yescharacteristic
296D3—62no—
297D3—62no—
298D3—62no—
299D3—62no—
300C22—91no—
301C22—91no—
302C22—91no—
303C22—91no—
304C22—91no—
305C22—91no—
306C22D3×GO+(4,𝔽2)11yescharacteristic
307C4—61no—
308C4—94no—
309C4—122no—
310C4—184no—
311C4—542no—
312C4—541no—
313C3—22no—
314C3—42no—
315C3#1059341011yescharacteristic, complemented
316C2—91no—
317C2—91no—
318C2—91no—
319C2—91no—
320C2#1172411111yescharacteristic
321C2#1172411411yescharacteristic
322C2#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#389
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
12. #88940C4semidirect
67. C3×#157C4⋊C4semidirect
153. #157Dic3⋊C4semidirect
172. C33#389semidirect
265. C32#4972semidirect
315. C3#10593410semidirect

Maximal subgroups

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

subgroupindex
C2×#889402
C2×C3×#37732
C2×Dic3×#1572
#117229102
#117227692
#105934103
#49729