Finite Group Databasebeta

← #13144306

Subgroups — #13144306

Order 1728.

1771 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1771 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C22×#6235C212yesmaximal, complemented
2C22×C4⋊#272C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#88542C212yesmaximal, complemented
5#11720574C211yesmaximal, characteristic, complemented
6C3×#4960—34nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C23×#272C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—24no—
12C2×#6235—24no—
13C2×#6235—24no—
14C22×#1084C2212yes—
15C2×C4⋊#272—24no—
16C2×C4⋊#272—22no—
17C22⋊C4×H(3,𝔽3)—22no—
18#88542—24no—
19C22×C3×#67—38no—
20C22×C3×D12—34no—
21C2×C32×C22⋊C4D314yescomplemented
22C2×C3×(C2×D3)⋊C4—38no—
23C3×C3⋊#95—34no—
24C23×H(3,𝔽3)—22no—
25C23×H(3,𝔽3)—22no—
26C23×H(3,𝔽3)—21no—
27C23×H(3,𝔽3)—21no—
28C23×H(3,𝔽3)D412yescomplemented
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×#272—22no—
31C22×#272—22no—
32C22×#272—22no—
33C22×#272—22no—
34C22×#272—22no—
35C22×#272—22no—
36C22×#272D414yescomplemented
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)—22no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)D412yescomplemented
41#6235—216no—
42C2×#1084—22no—
43C2×#1084—22no—
44C2×#1084D414yescomplemented
45C4⋊#272—28no—
46C3×#392—91no—
47#4960—34no—
48C24×C32C2×D314yes—
49C23×C3×D3—38no—
50C22×C32×C4C2×D314yes—
51C22×C32×C4C2×D314yes—
52C2×C3×#67—616no—
53C2×C3×#67—616no—
54C2×C3×#67—616no—
55C22×C3×Dic3—38no—
56C2×C3×D12—616no—
57C2×C3×D12—68no—
58C32×C22⋊C4—28no—
59C3×(C2×D3)⋊C4—616no—
60C22×H(3,𝔽3)—24no—
61C22×H(3,𝔽3)—22no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)—42no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C22×H(3,𝔽3)C2×D412yes—
67C22×H(3,𝔽3)C2×D412yes—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C22×H(3,𝔽3)#4111yescharacteristic
70C2×#272—24no—
71C2×#272—24no—
72C2×#272—24no—
73C2×#272—42no—
74C2×#272—42no—
75C2×#272—42no—
76C2×#272—42no—
77C4×H(3,𝔽3)—22no—
78C4×H(3,𝔽3)—41no—
79#1084—24no—
80C22×C3×D4—92no—
81C22×C3×D4—91no—
82C22×#67—38no—
83C22×D12—34no—
84C2×C3×C22⋊C4—34no—
85C2×C3×C22⋊C4—92no—
86C2×C3×C22⋊C4C3⋊D311yescharacteristic
87C2×(C2×D3)⋊C4—38no—
88C3×#95—91no—
89C3⋊#95—34no—
90C23×C32—24no—
91C23×C32—28no—
92C23×C32—28no—
93C23×C32—24no—
94C23×C32D1218yescomplemented
95C23×C32C22×D314yes—
96C22×C3×D3—316no—
97C22×C3×D3—68no—
98C22×C3×D3—68no—
99C22×C3×D3—68no—
100C22×C3×D3—68no—
101C22×C3×D3—68no—
102C22×C3×D3—68no—
103C2×C32×C4—28no—
104C2×C32×C4—28no—
105C2×C32×C4—24no—
106C2×C32×C4#6718yescomplemented
107C3×#67—664no—
108C2×C3×Dic3—316no—
109C2×C3×Dic3—68no—
110C2×C3×Dic3—68no—
111C3×D12—632no—
112#392—91noSylow
113C2×H(3,𝔽3)—42no—
114C2×H(3,𝔽3)#12012yes—
115C2×H(3,𝔽3)#12011yescharacteristic
116C2×H(3,𝔽3)#12112yes—
117C2×H(3,𝔽3)#12111yescharacteristic
118C2×H(3,𝔽3)#12711yescharacteristic
119#272—44no—
120C24×C3—34no—
121C24×C3—92no—
122C24×C3C2×C3⋊D311yescharacteristic
123C23×D3—38no—
124C2×C3×D4—184no—
125C2×C3×D4—184no—
126C2×C3×D4—184no—
127C2×C3×D4—184no—
128C2×C3×D4—182no—
129C22×C3×C4—34no—
130C22×C3×C4—34no—
131C22×C3×C4—92no—
132C22×C3×C4C2×C3⋊D311yescharacteristic
133C22×C3×C4C2×C3⋊D311yescharacteristic
134C2×#67—616no—
135C2×#67—616no—
136C2×#67—616no—
137C22×Dic3—38no—
138C2×D12—68no—
139C2×D12—616no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—68no—
142C3×C22⋊C4—184no—
143(C2×D3)⋊C4—616no—
144C22×C32—216no—
145C22×C32—28no—
146C22×C32—48no—
147C22×C32—44no—
148C22×C32—44no—
149C22×C32C2×D1214yes—
150C22×C32#23514yes—
151C22×C32D3×D418yes—
152C22×C32D3×D418yes—
153C22×C32C2×#6714yes—
154C2×C3×D3—616no—
155C2×C3×D3—616no—
156C2×C3×D3—616no—
157C2×C3×D3—128no—
158C2×C3×D3—128no—
159C2×C3×D3—128no—
160C2×C3×D3—128no—
161C32×C4—28no—
162C32×C4—44no—
163C3×Dic3—616no—
164C22×D4—92no—
165C22×D4—91no—
166C2×C22⋊C4—92no—
167C2×C22⋊C4#27211yescharacteristic, complemented
168#95—91no—
169H(3,𝔽3)#39211yescharacteristic, complemented, Sylow
170C23×C3—21no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3—21no—
174C23×C3—38no—
175C23×C3—34no—
176C23×C3—64no—
177C23×C3—68no—
178C23×C3—68no—
179C23×C3—64no—
180C23×C3—94no—
181C23×C3—182no—
182C23×C3—182no—
183C23×C3—182no—
184C23×C3—182no—
185C23×C3—182no—
186C23×C3—182no—
187C23×C3C4⋊(C3⋊D3)12yes—
188C23×C3C22×C3⋊D311yescharacteristic
189C22×D3—316no—
190C22×D3—68no—
191C22×D3—68no—
192C22×D3—68no—
193C22×D3—68no—
194C22×D3—68no—
195C22×D3—68no—
196C3×D4—188no—
197C3×D4—1816no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—21no—
201C2×C3×C4—38no—
202C2×C3×C4—68no—
203C2×C3×C4—68no—
204C2×C3×C4—64no—
205C2×C3×C4—94no—
206C2×C3×C4—182no—
207C2×C3×C4—182no—
208C2×C3×C4#63812yes—
209#67—664no—
210C2×Dic3—316no—
211C2×Dic3—68no—
212C2×Dic3—68no—
213D12—632no—
214C2×C32—48no—
215C2×C32#88218yes—
216C2×C32#88418yes—
217C2×C32#93014yes—
218C2×C32#93714yes—
219C2×C32#94014yes—
220C3×D3—1216no—
221C24—92no—
222C24C2×#27211yescharacteristic
223C2×D4—184no—
224C2×D4—184no—
225C2×D4—184no—
226C2×D4—184no—
227C2×D4—182no—
228C22×C4—92no—
229C22×C4C2×#27211yescharacteristic
230C22×C4C2×#27211yescharacteristic
231C22⋊C4—22no—
232C22⋊C4—184no—
233C22×C3—24no—
234C22×C3—22no—
235C22×C3—34no—
236C22×C3—34no—
237C22×C3—38no—
238C22×C3—38no—
239C22×C3—34no—
240C22×C3—42no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—616no—
244C22×C3—68no—
245C22×C3—128no—
246C22×C3—124no—
247C22×C3—124no—
248C22×C3—184no—
249C22×C3—184no—
250C22×C3—184no—
251C22×C3—362no—
252C22×C3—362no—
253C22×C3—362no—
254C22×C3—362no—
255C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
256C22×C3#382811yescharacteristic
257C22×C3D4×C3⋊D312yes—
258C22×C3D4×C3⋊D312yes—
259C22×C3C2×#63811yescharacteristic
260C2×D3—616no—
261C2×D3—616no—
262C2×D3—616no—
263C2×D3—128no—
264C2×D3—128no—
265C2×D3—128no—
266C2×D3—128no—
267C3×C4—22no—
268C3×C4—41no—
269C3×C4—68no—
270C3×C4—124no—
271C3×C4—184no—
272Dic3—616no—
273C32#496014yescomplemented
274C23—21no—
275C23—22no—
276C23—22no—
277C23—21no—
278C23—94no—
279C23—182no—
280C23—182no—
281C23—182no—
282C23—182no—
283C23—182no—
284C23—182no—
285C23C4⋊#27212yescomplemented
286C23C22×#27211yescharacteristic
287D4—188no—
288D4—1816no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—94no—
293C2×C4—182no—
294C2×C4—182no—
295C2×C4#623512yescomplemented
296C2×C3—38no—
297C2×C3—38no—
298C2×C3—34no—
299C2×C3—34no—
300C2×C3—34no—
301C2×C3—42no—
302C2×C3—128no—
303C2×C3—364no—
304C2×C3#6409112yes—
305C2×C3#6409312yes—
306C2×C3#6413911yescharacteristic
307C2×C3#6414611yescharacteristic
308C2×C3#6414911yescharacteristic
309D3—1216no—
310C22—24no—
311C22—22no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—184no—
316C22—184no—
317C22—184no—
318C22—362no—
319C22—362no—
320C22—362no—
321C22—362no—
322C22C2×C4⋊#27211yescharacteristic
323C22#8874011yescharacteristic
324C22D4×#27212yes—
325C22D4×#27212yes—
326C22C2×#623511yescharacteristic
327C4—22no—
328C4—41no—
329C4—184no—
330C3—34no—
331C3#1059265911yescharacteristic
332C2—42no—
333C2—364no—
334C2#1172173812yes—
335C2#1172174012yes—
336C2#1172183011yescharacteristic
337C2#1172184711yescharacteristic
338C2#1172185011yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingC2×C22⋊C4×H(3,𝔽3)C2
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#392
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6235C2semidirect
2. C22×C4⋊#272C2semidirect
3. C2×C22⋊C4×H(3,𝔽3)C2semidirect
4. C2×#88542C2semidirect
5. #11720574C2semidirect
21. C2×C32×C22⋊C4D3semidirect
28. C23×H(3,𝔽3)D4semidirect
36. C22×#272D4semidirect
40. C2×C4×H(3,𝔽3)D4semidirect
44. C2×#1084D4semidirect
94. C23×C32D12semidirect
106. C2×C32×C4#67semidirect
167. C2×C22⋊C4#272semidirect
169. H(3,𝔽3)#392semidirect
273. C32#4960semidirect
285. C23C4⋊#272semidirect
295. C2×C4#6235semidirect

Maximal subgroups

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

subgroupindex
C22×#62352
C22×C4⋊#2722
C2×C22⋊C4×H(3,𝔽3)2
C2×#885422
#117205742
C3×#49603