Finite Group Databasebeta

← #13146615

Subgroups — #13146615

Order 1728.

455 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 69 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×C4⋊#1088C211yesmaximal, characteristic
2C2×#88590C212yesmaximal
3#11721152C211yesmaximal, characteristic
4C4×#6199C211yesmaximal, characteristic
5#11720707C212yesmaximal
6C22×#1088C2212yes—
7C2×C4×#272C2211yescharacteristic
8C2×#6199C2212yes—
9C2×#6199C2211yescharacteristic
10C2×#6199C2211yescharacteristic
11C4⋊#1088C414yescomplemented
12#88590—24no—
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4×(C3⋊Dic3)⋊C3—21no—
15#88449—21no—
16#88447—21no—
17D3×C4⋊Dic3—61no—
18C4×D3×Dic3—61no—
19C22×#272C2311yescharacteristic
20C2×#1088—22no—
21C2×#1088—22no—
22C2×#1088C2×C414yes—
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×#1084D411yescharacteristic
25C4×#272C2×C412yes—
26C4×#272Q812yes—
27C2×(C3⋊Dic3)⋊C3—21no—
28C2×(C3⋊Dic3)⋊C3—21no—
29C2×(C3⋊Dic3)⋊C3—22no—
30#6199—21no—
31#6199—21no—
32#6199—22no—
33#6199—22no—
34#6199D412yes—
35#6199Q812yes—
36#4984—91nomaximal
37C2×C3×C4×D3—61no—
38C2×C3×C4×D3—61no—
39C2×D3×Dic3—61no—
40C2×D3×Dic3—61no—
41C2×D3×Dic3—62no—
42C4⋊(C3⋊Dic3)—21no—
43C4×C3⋊Dic3—21no—
44C3×C4⋊Dic3—61no—
45C3×C4×Dic3—61no—
46Dic3⋊Dic3—62no—
47Dic32—61no—
48Dic32—61no—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C2×#272C22×C411yescharacteristic
51C2×#272C2×D411yescharacteristic
52C2×#272C2×Q811yescharacteristic
53C2×#272C2×Q811yescharacteristic
54C2×#272#4111yescharacteristic
55C2×#272#4111yescharacteristic
56#1088—24no—
57C4×H(3,𝔽3)C4⋊C412yescomplemented
58#1084C4⋊C412yescomplemented
59(C3⋊Dic3)⋊C3—22no—
60(C3⋊Dic3)⋊C3—22no—
61(C3⋊Dic3)⋊C3—42no—
62C2×C3×C4⋊C4—91no—
63C2×C4⋊Dic3—91no—
64C2×C4⋊Dic3—92no—
65C2×C4×Dic3—91no—
66D3×C4⋊C4—181no—
67C42×D3—181no—
68C3⋊#95—92no—
69C22×C3×D3—61no—
70C22×C3×D3—61no—
71C2×C32×C4—21no—
72C2×C32×C4—21no—
73C2×C3⋊Dic3—21no—
74C2×C3⋊Dic3—21no—
75C2×C3⋊Dic3—22no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83C3×C4×D3—62no—
84C3×C4×D3—62no—
85D3×Dic3—62no—
86D3×Dic3—62no—
87D3×Dic3—62no—
88D3×Dic3—62no—
89D3×Dic3—122no—
90D3×Dic3—122no—
91#389—271noSylow
92C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
93C2×H(3,𝔽3)#12111yescharacteristic
94C2×H(3,𝔽3)C22⋊Q811yescharacteristic
95#272C4×D411yescharacteristic
96#272C4×Q811yescharacteristic
97#272#12511yescharacteristic
98#272C4⋊Q811yescharacteristic
99C22×C3×C4—91no—
100C22×C3×C4—92no—
101C22×Dic3—91no—
102C22×Dic3—91no—
103C22×Dic3—92no—
104C2×C4×D3—61no—
105C2×C4×D3—61no—
106C2×C4×D3—181no—
107C2×C4×D3—181no—
108C2×C4×D3—182no—
109C3×C4⋊C4—94no—
110C3×C4⋊C4—181no—
111C3×C42—181no—
112C4⋊Dic3—61no—
113C4⋊Dic3—181no—
114C4⋊Dic3—181no—
115C4⋊Dic3—184no—
116Dic3⋊C4—182no—
117C4×Dic3—61no—
118C4×Dic3—181no—
119C4×Dic3—181no—
120C4×Dic3—181no—
121C4×Dic3—181no—
122C22×C32—21no—
123C22×C32—21no—
124C2×C3×D3—61no—
125C2×C3×D3—61no—
126C2×C3×D3—61no—
127C2×C3×D3—61no—
128C2×C3×D3—61no—
129C2×C3×D3—61no—
130C2×C3×D3—61no—
131C2×C3×D3—61no—
132C2×C3×D3—61no—
133C2×C3×D3—61no—
134C2×C3×D3—61no—
135C2×C3×D3—61no—
136C32×C4—22no—
137C32×C4—22no—
138C3⋊Dic3—22no—
139C3⋊Dic3—22no—
140C3⋊Dic3—42no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—62no—
144C3×Dic3—62no—
145C3×Dic3—122no—
146C2×C4⋊C4—91no—
147C2×C4⋊C4—271no—
148C2×C4⋊C4—272no—
149C2×C42—271no—
150#95—272no—
151H(3,𝔽3)#38911yescharacteristic, complemented, Sylow
152C23×C3—91no—
153C22×D3—61no—
154C22×D3—61no—
155C2×C3×C4—61no—
156C2×C3×C4—61no—
157C2×C3×C4—91no—
158C2×C3×C4—92no—
159C2×C3×C4—92no—
160C2×C3×C4—94no—
161C2×C3×C4—181no—
162C2×C3×C4—181no—
163C2×C3×C4—182no—
164C2×C3×C4—182no—
165C2×C3×C4—182no—
166C2×C3×C4GO+(4,𝔽2)11yescharacteristic
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170C2×Dic3—61no—
171C2×Dic3—62no—
172C2×Dic3—92no—
173C2×Dic3—92no—
174C2×Dic3—181no—
175C2×Dic3—181no—
176C2×Dic3—182no—
177C2×Dic3—181no—
178C2×Dic3—181no—
179C2×Dic3—182no—
180C2×Dic3—182no—
181C4×D3—62no—
182C4×D3—62no—
183C4×D3—62no—
184C4×D3—62no—
185C4×D3—182no—
186C4×D3—182no—
187C4×D3—182no—
188C4×D3—182no—
189C4×D3—362no—
190C4×D3—362no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C3×D3—61no—
198C3×D3—61no—
199C3×D3—61no—
200C3×D3—61no—
201C3×D3—61no—
202C3×D3—61no—
203C3×D3—61no—
204C3×D3—61no—
205C22×C4—91no—
206C22×C4—92no—
207C22×C4—271no—
208C22×C4—271no—
209C22×C4—272no—
210C4⋊C4—94no—
211C4⋊C4—181no—
212C4⋊C4—541no—
213C4⋊C4—544no—
214C42—181no—
215C42—541no—
216C22×C3—61no—
217C22×C3—61no—
218C22×C3—91no—
219C22×C3—91no—
220C22×C3—91no—
221C22×C3—91no—
222C22×C3—91no—
223C22×C3—91no—
224C22×C3C2×GO+(4,𝔽2)11yescharacteristic
225C2×D3—61no—
226C2×D3—61no—
227C2×D3—61no—
228C2×D3—61no—
229C2×D3—61no—
230C2×D3—61no—
231C2×D3—61no—
232C2×D3—61no—
233C2×D3—61no—
234C2×D3—61no—
235C2×D3—61no—
236C2×D3—61no—
237C3×C4—62no—
238C3×C4—62no—
239C3×C4—92no—
240C3×C4—182no—
241C3×C4—182no—
242C3×C4—184no—
243C3×C4—362no—
244C3×C4#377312yes—
245Dic3—62no—
246Dic3—62no—
247Dic3—62no—
248Dic3—62no—
249Dic3—122no—
250Dic3—182no—
251Dic3—182no—
252Dic3—362no—
253C32—21no—
254C32—21no—
255C23—91no—
256C2×C4—91no—
257C2×C4—92no—
258C2×C4—92no—
259C2×C4—94no—
260C2×C4—182no—
261C2×C4—182no—
262C2×C4—181no—
263C2×C4—181no—
264C2×C4—182no—
265C2×C4—272no—
266C2×C4—272no—
267C2×C4—541no—
268C2×C4—541no—
269C2×C4—542no—
270C2×C4—542no—
271C2×C4#1088⋊C211yescharacteristic
272C2×C3—61no—
273C2×C3—61no—
274C2×C3—61no—
275C2×C3—61no—
276C2×C3—61no—
277C2×C3—61no—
278C2×C3—91no—
279C2×C3—91no—
280C2×C3—91no—
281C2×C3—91no—
282C2×C3D32⋊Q811yescharacteristic
283C2×C3#6423011yescharacteristic
284C2×C3C2×#377311yescharacteristic
285D3—61no—
286D3—61no—
287D3—61no—
288D3—61no—
289D3—61no—
290D3—61no—
291D3—61no—
292D3—61no—
293C22—91no—
294C22—91no—
295C22—91no—
296C22—91no—
297C22—91no—
298C22—91no—
299C22C2×#1088⋊C211yescharacteristic
300C4—92no—
301C4—184no—
302C4—182no—
303C4—182no—
304C4—362no—
305C4#8859012yescomplemented
306C3—61no—
307C3—61no—
308C3#1059341211yescharacteristic
309C2—91no—
310C2—91no—
311C2—91no—
312C2—91no—
313C2(((C3⋊D3)⋊C3)⋊C2)⋊Q811yescharacteristic
314C2#1172230411yescharacteristic
315C2C2×#8859011yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutatorC2×#272C22×C4
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radical, cosocleC22×#272C23
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#389
LCSG ▷ C2×#272 ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
11. C4⋊#1088C4semidirect
57. C4×H(3,𝔽3)C4⋊C4semidirect
58. #1084C4⋊C4semidirect
151. H(3,𝔽3)#389semidirect
305. C4#88590semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
C2×#885902
#117211522
C4×#61992
#117207072
#49849