Finite Group Databasebeta

← #13165808

Subgroups — #13165808

Order 1728.

410 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 90 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×#88945C211yesmaximal, characteristic
2C2×C3×#3777C211yesmaximal, characteristic
3C2×Dic3⋊#157C211yesmaximal, characteristic
4C2×Dic3⋊#157C211yesmaximal, characteristic
5C2×Dic3×#157C211yesmaximal, characteristic
6C3⋊#63747C211yesmaximal, characteristic
7C3⋊#63747C211yesmaximal, characteristic
8#10593554—31nomaximal
9C22×C3⋊#157C2211yescharacteristic
10C22×C3⋊#157C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C22×C3×#157C2211yescharacteristic
15C2×Dic3×C3⋊D3C2211yescharacteristic
16#88945—22no—
17C3×#3777—22no—
18Dic3⋊#157—22no—
19Dic3⋊#157C414yescomplemented
20Dic3×#157—22no—
21C2×C4⋊#157—31no—
22C2×C4⋊#157—31no—
23C2×C4×#157—31no—
24C2×#3777—31no—
25C2×#3777D311yescharacteristic
26#63747—31no—
27#63747—31no—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3⋊#157—21no—
30C2×C3⋊#157—21no—
31C2×C3⋊#157—21no—
32C2×C3⋊#157—21no—
33C2×C3⋊#157—21no—
34C2×C3⋊#157—21no—
35C2×C3⋊#157—21no—
36C2×C3⋊#157C2×C412yes—
37C2×C3⋊#157Q812yes—
38C2×C3×#157—21no—
39C2×C3×#157—21no—
40C2×C3×#157—21no—
41C2×C3×#157—21no—
42C2×C3×#157—21no—
43C2×C3×#157—21no—
44C2×C3×#157—21no—
45C2×C3×#157C2×C412yes—
46C2×C3×#157D412yes—
47C2×C3⋊(C3⋊Dic3)Q811yescharacteristic
48C2×C32×Dic3Q811yescharacteristic
49Dic3×C3⋊D3C2×C412yes—
50Dic3×C3⋊D3D412yes—
51#4976—91nomaximal
52C22×#157—31no—
53C22×#157—31no—
54C22×#157—31no—
55C22×#157C2×D311yescharacteristic
56C22×#157C2×D311yescharacteristic
57C22×#157C2×D311yescharacteristic
58C2×C4×C3⋊D3—31no—
59C2×D3×Dic3—121no—
60C4⋊#157—34no—
61C4⋊#157—62no—
62C4×#157—62no—
63#3777—22no—
64#3777—62no—
65C22×C33C2×Q811yescharacteristic
66C2×C3×C3⋊D3C22×C411yescharacteristic
67C2×C3×C3⋊D3C2×D411yescharacteristic
68C2×C3×C3⋊D3C2×D411yescharacteristic
69C2×C3×C3⋊D3C2×Q811yescharacteristic
70C2×C3×C3⋊D3#4111yescharacteristic
71C2×C3×C3⋊D3#4111yescharacteristic
72C3⋊#157—22no—
73C3⋊#157—22no—
74C3⋊#157—41no—
75C3×#157—22no—
76C3×#157—22no—
77C3×#157—41no—
78C3⋊(C3⋊Dic3)C4⋊C412yescomplemented
79C32×Dic3C4⋊C412yescomplemented
80C2×C3×C4⋊C4—91no—
81C2×Dic3⋊C4—91no—
82C2×Dic3⋊C4—91no—
83C2×Dic3⋊C4—91no—
84C2×C4×Dic3—91no—
85C3⋊#95—91no—
86C3⋊#95—91no—
87C22×C3⋊D3C22×D311yescharacteristic
88C22×C3×D3—121no—
89C2×#157—21no—
90C2×#157—21no—
91C2×#157—21no—
92C2×#157—21no—
93C2×#157—21no—
94C2×#157—21no—
95C2×#157—21no—
96C2×#157—32no—
97C2×#157—32no—
98C2×#157—61no—
99C2×#157—61no—
100C2×#157—61no—
101C2×#157—61no—
102C2×#157—61no—
103C2×#157—61no—
104C2×#157—61no—
105C2×#157C4×D312yes—
106C2×#157#6712yes—
107C2×C32×C4—31no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—121no—
110C2×C3⋊Dic3—241no—
111C4×C3⋊D3—32no—
112C4×C3⋊D3—32no—
113C2×C3×Dic3—41no—
114D3×Dic3—122no—
115D3×Dic3—122no—
116#389—271noSylow
117C2×C33C2×C4⋊C411yescharacteristic
118C2×C33#12511yescharacteristic
119C2×C33C4⋊Q811yescharacteristic
120C3×C3⋊D3C4×D411yescharacteristic
121C3×C3⋊D3C4×Q811yescharacteristic
122C3×C3⋊D3#12111yescharacteristic
123C3×C3⋊D3C22⋊Q811yescharacteristic
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4—91no—
127C22×Dic3—91no—
128C22×Dic3—91no—
129C22×Dic3—91no—
130C22×Dic3—91no—
131C2×C4×D3—361no—
132C3×C4⋊C4—182no—
133Dic3⋊C4—94no—
134Dic3⋊C4—182no—
135Dic3⋊C4—182no—
136C4×Dic3—182no—
137C22×C32—41no—
138C22×C32—81no—
139C22×C32D3×Q811yescharacteristic
140C2×C3⋊D3C2×C4×D311yescharacteristic
141C2×C3⋊D3#23511yescharacteristic
142C2×C3⋊D3D3×D411yescharacteristic
143C2×C3⋊D3#23711yescharacteristic
144C2×C3⋊D3D3×Q811yescharacteristic
145C2×C3⋊D3C2×#6711yescharacteristic
146C2×C3×D3—121no—
147C2×C3×D3—121no—
148C2×C3×D3—121no—
149C2×C3×D3—121no—
150C2×C3×D3—121no—
151C2×C3×D3—121no—
152#157—22no—
153#157—22no—
154#157—41no—
155#157—62no—
156#157—62no—
157#157—121no—
158C32×C4—32no—
159C3⋊Dic3—32no—
160C3⋊Dic3—122no—
161C3⋊Dic3—242no—
162C3×Dic3—42no—
163C2×C4⋊C4—91no—
164C2×C4⋊C4—271no—
165C2×C4⋊C4—271no—
166C2×C4⋊C4—271no—
167C2×C42—271no—
168#95—271no—
169#95—271no—
170C33#38911yescharacteristic, complemented, Sylow
171C23×C3—91no—
172C22×D3—121no—
173C2×C3×C4—92no—
174C2×C3×C4—92no—
175C2×C3×C4—121no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4—181no—
180C2×C3×C4—181no—
181C2×C3×C4—181no—
182C2×C3×C4—181no—
183C2×Dic3—91no—
184C2×Dic3—92no—
185C2×Dic3—92no—
186C2×Dic3—92no—
187C2×Dic3—92no—
188C2×Dic3—181no—
189C2×Dic3—181no—
190C2×Dic3—181no—
191C2×Dic3—181no—
192C2×Dic3—181no—
193C2×Dic3—181no—
194C2×Dic3—181no—
195C2×Dic3—361no—
196C2×Dic3—721no—
197C2×Dic3PSU(3,𝔽2)11yescharacteristic
198C4×D3—362no—
199C4×D3—362no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—81no—
204C2×C32—81no—
205C2×C32—81no—
206C2×C32#88911yescharacteristic
207C2×C32D3×C4⋊C411yescharacteristic
208C2×C32#94411yescharacteristic
209C3⋊D3#88711yescharacteristic
210C3⋊D3#89611yescharacteristic
211C3⋊D3C4×#6711yescharacteristic
212C3⋊D3#93911yescharacteristic
213C3×D3—121no—
214C3×D3—121no—
215C3×D3—121no—
216C3×D3—121no—
217C22×C4—91no—
218C22×C4—91no—
219C22×C4—91no—
220C22×C4—271no—
221C22×C4—271no—
222C22×C4—271no—
223C22×C4—271no—
224C4⋊C4—182no—
225C4⋊C4—274no—
226C4⋊C4—542no—
227C4⋊C4—542no—
228C42—542no—
229C22×C3—41no—
230C22×C3—81no—
231C22×C3—91no—
232C22×C3—91no—
233C22×C3—91no—
234C22×C3—91no—
235C22×C3—91no—
236C22×C3—91no—
237C22×C3C2×PSU(3,𝔽2)11yescharacteristic
238C2×D3—121no—
239C2×D3—121no—
240C2×D3—121no—
241C2×D3—121no—
242C2×D3—121no—
243C2×D3—121no—
244C3×C4—122no—
245C3×C4—182no—
246C3×C4—182no—
247C3×C4—361no—
248Dic3—92no—
249Dic3—182no—
250Dic3—182no—
251Dic3—362no—
252Dic3—361no—
253Dic3—722no—
254Dic3#377712yescomplemented
255C32—41no—
256C32—81no—
257C32#497611yescharacteristic, complemented
258C23—91no—
259C2×C4—31no—
260C2×C4—92no—
261C2×C4—92no—
262C2×C4—181no—
263C2×C4—181no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4—181no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—272no—
270C2×C4—272no—
271C2×C4—271no—
272C2×C4—272no—
273C2×C4—272no—
274C2×C4—541no—
275C2×C4—541no—
276C2×C4—541no—
277C2×C4—541no—
278C2×C4—541no—
279C2×C4—541no—
280C2×C4—541no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—81no—
285C2×C3—81no—
286C2×C3—81no—
287C2×C3—91no—
288C2×C3—91no—
289C2×C3—91no—
290C2×C3—91no—
291C2×C3#6424311yescharacteristic
292C2×C3C4⋊PSU(3,𝔽2)11yescharacteristic
293C2×C3C2×#377711yescharacteristic
294D3—121no—
295D3—121no—
296D3—121no—
297D3—121no—
298C22—91no—
299C22—91no—
300C22—91no—
301C22—91no—
302C22—91no—
303C22—91no—
304C22D3×PSU(3,𝔽2)11yescharacteristic
305C4—32no—
306C4—182no—
307C4—182no—
308C4—272no—
309C4—361no—
310C4—542no—
311C4—542no—
312C4—1081no—
313C3—41no—
314C3—81no—
315C3#1059355411yescharacteristic, complemented
316C2—91no—
317C2—91no—
318C2—91no—
319C2—91no—
320C2D3×#377711yescharacteristic
321C2#1172415211yescharacteristic
322C2#1172415311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×PSU(3,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×Q8
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
19. Dic3⋊#157C4semidirect
78. C3⋊(C3⋊Dic3)C4⋊C4semidirect
79. C32×Dic3C4⋊C4semidirect
170. C33#389semidirect
254. Dic3#3777semidirect
257. C32#4976semidirect
315. C3#10593554semidirect

Maximal subgroups

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

subgroupindex
C2×#889452
C2×C3×#37772
C2×Dic3⋊#1572
C2×Dic3×#1572
C3⋊#637472
#105935543
#49769