Finite Group Databasebeta

← #13155713

Subgroups — #13155713

Order 1728.

786 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 124 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
1#11723499C212yesmaximal, complemented
2C2×C3×#3713C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic, complemented
4#11722021C211yesmaximal, characteristic, complemented
5C3×#63563C212yesmaximal, complemented
6#10591436—32nomaximal
7#10590741—31nomaximal
8C2×C32×D12C2212yes—
9C2×D3×C3⋊Dic3—22no—
10C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C2×C32×C3⋊C8C2211yescharacteristic
14(C2×D3)⋊(C3⋊Dic3)—22no—
15C3×#3713C414yescomplemented
16D4×C3⋊Dic3—32no—
17C2×C3×#213—32no—
18#63898—32no—
19#63898—32no—
20#63898—62no—
21C2×#3713D311yescharacteristic
22#63642—31no—
23C3×#832—32no—
24C3×#821—32no—
25#63574—31no—
26#63573—32no—
27#63563D312yescomplemented
28#63555—32no—
29#63555—61no—
30C22×C32×D3—22no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)D412yes—
33C2×C3×C3⋊Dic3—22no—
34C32×D12—22no—
35C32×D12C2×C414yescomplemented
36D3×C3⋊Dic3—44no—
37C3×C3⋊(C3⋊C8)C2×C412yescomplemented
38C32×C3⋊C8—21no—
39#5152—91no—
40#4785—92no—
41C2×C32×D4—32no—
42C22×C3⋊Dic3—62no—
43C2×C3×D12—22no—
44C2×C3×D12C2×D312yes—
45C2×C3×D12C2×D312yes—
46C2×D3×Dic3—62no—
47C2×D3×Dic3—62no—
48C2×D3×Dic3—64no—
49C2×C32×C8—31no—
50#3757—62no—
51C4⋊(C3⋊Dic3)C2×D312yes—
52C4×C3⋊Dic3—32no—
53C4×C3⋊Dic3—31no—
54C4×C3⋊Dic3—61no—
55C4×C3⋊Dic3—62no—
56C4×C3⋊Dic3—121no—
57C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
58C3×#213—38no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—32no—
61C3×C4⋊Dic3—62no—
62C2×C3×C3⋊C8—21no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—61no—
65C2×C3×C3⋊C8C2×D312yes—
66(C2×D3)⋊Dic3—62no—
67(C2×D3)⋊Dic3—62no—
68(C2×D3)⋊Dic3—64no—
69#3713Dic314yescomplemented
70C22×C33C2×D411yescharacteristic
71C2×C32×D3—24no—
72C2×C32×D3—42no—
73C2×C32×D3—42no—
74C33×C4C22×C411yescharacteristic
75C33×C4#4111yescharacteristic
76C3⋊(C3⋊Dic3)—22no—
77C3×C3⋊Dic3—42no—
78C2×C3×D8—91no—
79D4×Dic3—92no—
80D4×Dic3—92no—
81D4×Dic3—182no—
82C2×#213—32no—
83#893—92no—
84C3×#102—92no—
85#832—32no—
86#821—32no—
87#817—91no—
88#815—92no—
89#815—91no—
90#815—181no—
91#808—92no—
92C23×C32—62no—
93C22×C3×D3—24no—
94C22×C3×D3—22no—
95C22×C3×D3—22no—
96C32×D4—34no—
97C32×D4—62no—
98C2×C32×C4—21no—
99C2×C32×C4—22no—
100C2×C32×C4—41no—
101C2×C32×C4C22×D312yes—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—32no—
105C2×C3⋊Dic3—32no—
106C2×C3⋊Dic3—32no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—62no—
109C2×C3⋊Dic3—62no—
110C2×C3⋊Dic3—122no—
111C2×C3⋊Dic3—124no—
112C2×C3×Dic3—64no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C3×D12—24no—
116C3×D12—22no—
117C3×D12—22no—
118C3×D12—24no—
119C3×D12C4×D314yescomplemented
120C3×D12C2×Dic314yescomplemented
121D3×Dic3—128no—
122D3×Dic3—124no—
123D3×Dic3—124no—
124C32×C8—61no—
125C3⋊(C3⋊C8)C2×Dic312yescomplemented
126C3×C3⋊C8—22no—
127C3×C3⋊C8—22no—
128C3×C3⋊C8—34no—
129C3×C3⋊C8—62no—
130#442—271noSylow
131C2×C33C4×D411yescharacteristic
132C2×C33AGL(1,ℤ/8)12yes—
133C32×D3—44no—
134C2×C3×D4—32no—
135C2×C3×D4—32no—
136C2×C3×D4—62no—
137C22×Dic3—182no—
138C22×Dic3—182no—
139C22×Dic3—184no—
140C2×D12D3212yes—
141C2×C4×D3—182no—
142C3×D8—94no—
143C2×C3×C8—32no—
144C2×C3×C8—61no—
145C2×C3×C8—91no—
146C3×C4⋊C4—92no—
147#217—182no—
148#217—182no—
149#217—184no—
150#213—38no—
151(C2×D3)⋊C4—182no—
152C4⋊Dic3—32no—
153C4⋊Dic3—32no—
154C4⋊Dic3—62no—
155C4×Dic3—92no—
156C4×Dic3—91no—
157C4×Dic3—181no—
158C4×Dic3—182no—
159C4×Dic3—361no—
160C2×C3⋊C8—32no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8D3211yescharacteristic
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—41no—
166C22×C32—64no—
167C22×C32—122no—
168C22×C32—122no—
169C22×C32D3×D412yes—
170C22×C32D3×D411yescharacteristic
171C2×C3×D3—28no—
172C2×C3×D3—24no—
173C2×C3×D3—24no—
174C2×C3×D3—44no—
175C2×C3×D3—44no—
176C2×C3×D3—42no—
177C2×C3×D3—42no—
178C2×C3×D3—42no—
179C2×C3×D3—42no—
180C32×C4—21no—
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—41no—
185C32×C4—41no—
186C32×C4C2×C4×D312yes—
187C32×C4#23712yes—
188C32×C4#23711yescharacteristic
189C32×C4C22×Dic311yescharacteristic
190C3⋊Dic3—42no—
191C3⋊Dic3—62no—
192C3⋊Dic3—64no—
193C3⋊Dic3—62no—
194C3⋊Dic3—62no—
195C3⋊Dic3—122no—
196C3⋊Dic3—122no—
197C3⋊Dic3—124no—
198C3×Dic3—124no—
199C3×Dic3—122no—
200C3×Dic3—122no—
201C2×D8—91no—
202C4×D4—272no—
203#106—271no—
204#102—92no—
205#97—271no—
206C33#44211yescharacteristic, complemented, Sylow
207C23×C3—64no—
208C23×C3—62no—
209C23×C3—62no—
210C22×D3—22no—
211C3×D4—34no—
212C3×D4—34no—
213C3×D4—64no—
214C3×D4—64no—
215C3×D4—62no—
216C3×D4—62no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—182no—
221C2×C3×C4C2×D3211yescharacteristic
222C2×C3×C4C2×D3212yes—
223C2×Dic3—64no—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C2×Dic3—92no—
227C2×Dic3—92no—
228C2×Dic3—92no—
229C2×Dic3—182no—
230C2×Dic3—182no—
231C2×Dic3—182no—
232C2×Dic3—362no—
233C2×Dic3—368no—
234C2×Dic3—364no—
235C2×Dic3—364no—
236D12—22no—
237D12D3×Dic314yescomplemented
238C4×D3—364no—
239C3×C8—62no—
240C3×C8—62no—
241C3×C8—92no—
242C3⋊C8—21no—
243C3⋊C8—34no—
244C3⋊C8—62no—
245C2×C32—22no—
246C2×C32—21no—
247C2×C32—22no—
248C2×C32—22no—
249C2×C32—22no—
250C2×C32—41no—
251C2×C32—42no—
252C2×C32—124no—
253C2×C32#67⋊C412yes—
254C2×C32#91112yes—
255C2×C32#91112yes—
256C2×C32#91412yes—
257C2×C32D4×Dic311yescharacteristic
258C3×D3—48no—
259C3×D3—44no—
260C3×D3—44no—
261C2×D4—32no—
262C22×C4—542no—
263D8—94no—
264C2×C8—31no—
265C2×C8—91no—
266C4⋊C4—92no—
267C22⋊C4—542no—
268C42—271no—
269C22×C3—22no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—68no—
273C22×C3—64no—
274C22×C3—64no—
275C22×C3—124no—
276C22×C3—124no—
277C22×C3—122no—
278C22×C3—122no—
279C22×C3—122no—
280C22×C3—122no—
281C22×C3#380211yescharacteristic
282C22×C3#381112yes—
283C2×D3—24no—
284C2×D3—42no—
285C2×D3—42no—
286C3×C4—21no—
287C3×C4—22no—
288C3×C4—21no—
289C3×C4—22no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—362no—
293C3×C4C4×D3211yescharacteristic
294C3×C4C2×D3×Dic312yes—
295C3×C4#380511yescharacteristic
296C3×C4#380512yes—
297Dic3—124no—
298Dic3—122no—
299Dic3—122no—
300Dic3—184no—
301Dic3—182no—
302Dic3—182no—
303Dic3—182no—
304Dic3—362no—
305Dic3—362no—
306Dic3—364no—
307C32—21no—
308C32—22no—
309C32—41no—
310C32#478512yescomplemented
311C32#515211yescharacteristic, complemented
312C23—62no—
313D4—34no—
314D4—62no—
315C2×C4—182no—
316C2×C4—272no—
317C2×C4—1084no—
318C2×C4D3311yescharacteristic
319C8—61no—
320C8—92no—
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—22no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—42no—
327C2×C3—41no—
328C2×C3—128no—
329C2×C3—124no—
330C2×C3—124no—
331C2×C3#6379812yes—
332C2×C3#6390211yescharacteristic
333C2×C3#6392712yes—
334C2×C3#6393712yes—
335C2×C3#67⋊Dic312yes—
336D3—44no—
337C22—64no—
338C22—122no—
339C22—122no—
340C22#8895411yescharacteristic
341C4—362no—
342C4—542no—
343C4D32×Dic311yescharacteristic
344C4#8896311yescharacteristic
345C3—22no—
346C3—21no—
347C3—41no—
348C3#1059074111yescharacteristic, complemented
349C3#1059143612yescomplemented
350C2—124no—
351C2#1172287712yes—
352C2#1172292511yescharacteristic
subgroupG/H
centerC22#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×#3713C2
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723499C2semidirect
3. #11722079C2semidirect
4. #11722021C2semidirect
5. C3×#63563C2semidirect
15. C3×#3713C4semidirect
27. #63563D3semidirect
35. C32×D12C2×C4semidirect
37. C3×C3⋊(C3⋊C8)C2×C4semidirect
69. #3713Dic3semidirect
119. C3×D12C4×D3semidirect
120. C3×D12C2×Dic3semidirect
125. C3⋊(C3⋊C8)C2×Dic3semidirect
206. C33#442semidirect
237. D12D3×Dic3semidirect
310. C32#4785semidirect
311. C32#5152semidirect
348. C3#10590741semidirect
349. C3#10591436semidirect

Maximal subgroups

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

subgroupindex
#117234992
C2×C3×#37132
#117220792
#117220212
C3×#635632
#105914363
#105907413