Finite Group Databasebeta

← #13155726

Subgroups — #13155726

Order 1728.

786 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 112 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×#88810C211yesmaximal, characteristic
2#11723501C212yesmaximal, complemented
3#11722021C211yesmaximal, characteristic, complemented
4C3×#63574C211yesmaximal, characteristic, complemented
5C3×#63564C212yesmaximal, complemented
6#10590741—31nomaximal
7#10590462—32nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×Dic3×C3⋊D3—22no—
10C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14#88810C414yescomplemented
15#88803—22no—
16#64102—32no—
17#63898—32no—
18#63898—32no—
19#63898—62no—
20C2×#3714—32no—
21C2×#3713—31no—
22#63642—31no—
23C3×#821—32no—
24C3×#809—32no—
25C3×#808—32no—
26#63574D311yescharacteristic, complemented
27#63564D312yescomplemented
28#63555—32no—
29#63555—61no—
30C22×C3×C3⋊D3—22no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)D412yes—
33C3×C4⋊(C3⋊D3)—22no—
34C3×C4⋊(C3⋊D3)C2×C414yescomplemented
35C2×C32×Dic3—22no—
36Dic3×C3⋊D3—44no—
37C3×C3⋊(C3⋊C8)C2×C412yescomplemented
38C32×C3⋊C8—21no—
39#4869—91no—
40#4785—92no—
41C2×C4⋊(C3⋊D3)C2×D312yes—
42C2×C4×C3⋊D3—62no—
43C2×C3×D12—32no—
44C2×C3×D12—32no—
45C2×C3×D12—62no—
46C2×D3×Dic3—62no—
47C2×D3×Dic3—62no—
48C2×D3×Dic3—64no—
49C2×C32×C8—31no—
50C32×C4⋊C4—32no—
51#3752—62no—
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×C4⋊Dic3—22no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C2×D312yes—
61C2×C3×C3⋊C8—21no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—61no—
64C2×C3×C3⋊C8C2×D312yes—
65(C2×D3)⋊Dic3—62no—
66(C2×D3)⋊Dic3—62no—
67(C2×D3)⋊Dic3—64no—
68#3714—38no—
69#3713—34no—
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—24no—
72C2×C3×C3⋊D3—42no—
73C2×C3×C3⋊D3—42no—
74C33×C4C22×C411yescharacteristic
75C33×C4#4111yescharacteristic
76C3⋊(C3⋊Dic3)—22no—
77C32×Dic3—42no—
78D4×Dic3—92no—
79C2×#213—92no—
80C2×D24—91no—
81#893—92no—
82#893—92no—
83#893—182no—
84C3×#106—91no—
85C3×#102—92no—
86#821—32no—
87#815—92no—
88#815—91no—
89#815—181no—
90#809—32no—
91#808—32no—
92C22×C3⋊D3—22no—
93C22×C3×D3—64no—
94C22×C3×D3—62no—
95C22×C3×D3—62no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—41no—
99C2×C32×C4—62no—
100C2×C32×C4C22×D312yes—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—32no—
103C2×C3⋊Dic3—32no—
104C2×C3⋊Dic3—32no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—62no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—122no—
109C4⋊(C3⋊D3)—22no—
110C4⋊(C3⋊D3)C4×D314yescomplemented
111C4×C3⋊D3—124no—
112C2×C3×Dic3—24no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—22no—
115C3×D12—34no—
116C3×D12—34no—
117C3×D12—64no—
118C3×D12—64no—
119C3×D12—62no—
120C3×D12—62no—
121D3×Dic3—128no—
122D3×Dic3—124no—
123D3×Dic3—124no—
124C32×C8—61no—
125C3⋊(C3⋊C8)C4×D312yescomplemented
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—
133C3×C3⋊D3—44no—
134C2×C3×D4—92no—
135C22×Dic3—182no—
136C2×D12—32no—
137C2×D12—32no—
138C2×D12—62no—
139C2×C4×D3—182no—
140C2×C4×D3—182no—
141C2×C4×D3—184no—
142C2×C3×C8—32no—
143C2×C3×C8—61no—
144C2×C3×C8—91no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—32no—
147C3×C4⋊C4—62no—
148#217—182no—
149#213—98no—
150(C2×D3)⋊C4—182no—
151(C2×D3)⋊C4—182no—
152(C2×D3)⋊C4—184no—
153C4⋊Dic3D3212yes—
154C4×Dic3—91no—
155C4×Dic3—92no—
156C4×Dic3—181no—
157C4×Dic3—182no—
158C4×Dic3—361no—
159C2×C3⋊C8—32no—
160C2×C3⋊C8—61no—
161C2×C3⋊C8D3211yescharacteristic
162D24—94no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—41no—
166C22×C32D3×D412yes—
167C22×C32D3×D411yescharacteristic
168C2×C3⋊D3—24no—
169C2×C3⋊D3—42no—
170C2×C3⋊D3—42no—
171C2×C3×D3—68no—
172C2×C3×D3—64no—
173C2×C3×D3—64no—
174C2×C3×D3—124no—
175C2×C3×D3—124no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C2×C3×D3—122no—
179C2×C3×D3—122no—
180C32×C4—22no—
181C32×C4—21no—
182C32×C4—22no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4—41no—
186C32×C4—122no—
187C32×C4C2×C4×D312yes—
188C32×C4C2×C4×D311yescharacteristic
189C32×C4#23712yes—
190C32×C4#23911yescharacteristic
191C3⋊Dic3—62no—
192C3⋊Dic3—64no—
193C3⋊Dic3—62no—
194C3⋊Dic3—62no—
195C3⋊Dic3—122no—
196C3⋊Dic3—122no—
197C3⋊Dic3—124no—
198C3×Dic3—44no—
199C3×Dic3—42no—
200C3×Dic3—42no—
201C2×D8—271no—
202C4×D4—272no—
203#106—91no—
204#102—92no—
205#97—271no—
206C33#44211yescharacteristic, complemented, Sylow
207C23×C3—182no—
208C22×D3—64no—
209C22×D3—62no—
210C22×D3—62no—
211C3×D4—94no—
212C3×D4—182no—
213C2×C3×C4—21no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—64no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4C2×D3212yes—
220C2×C3×C4C2×D3211yescharacteristic
221C2×Dic3—22no—
222C2×Dic3—92no—
223C2×Dic3—92no—
224C2×Dic3—92no—
225C2×Dic3—182no—
226C2×Dic3—182no—
227C2×Dic3—182no—
228C2×Dic3—362no—
229C2×Dic3—364no—
230D12—34no—
231D12—34no—
232D12—64no—
233D12—64no—
234D12—62no—
235D12—62no—
236C4×D3—368no—
237C4×D3—364no—
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—22no—
247C2×C32—21no—
248C2×C32—22no—
249C2×C32—22no—
250C2×C32—41no—
251C2×C32—42no—
252C2×C32#67⋊C412yes—
253C2×C32#89311yescharacteristic
254C2×C32#91112yes—
255C2×C32#91412yes—
256C2×C32#91412yes—
257C3⋊D3—44no—
258C3×D3—128no—
259C3×D3—124no—
260C3×D3—124no—
261C2×D4—92no—
262C22×C4—542no—
263D8—274no—
264C2×C8—31no—
265C2×C8—91no—
266C4⋊C4—32no—
267C22⋊C4—542no—
268C42—271no—
269C22×C3—22no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—184no—
273C22×C3—362no—
274C22×C3—362no—
275C22×C3#380211yescharacteristic
276C22×C3#381112yes—
277C2×D3—68no—
278C2×D3—64no—
279C2×D3—64no—
280C2×D3—124no—
281C2×D3—124no—
282C2×D3—122no—
283C2×D3—122no—
284C2×D3—122no—
285C2×D3—122no—
286C3×C4—21no—
287C3×C4—22no—
288C3×C4—21no—
289C3×C4—22no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—124no—
293C3×C4—122no—
294C3×C4—122no—
295C3×C4#379612yes—
296C3×C4C4×D3212yes—
297C3×C4C4×D3211yescharacteristic
298C3×C4#380511yescharacteristic
299Dic3—42no—
300Dic3—184no—
301Dic3—182no—
302Dic3—182no—
303Dic3—182no—
304Dic3—362no—
305Dic3—362no—
306Dic3—364no—
307C32—21no—
308C32—22no—
309C32—41no—
310C32#478512yescomplemented
311C32#486911yescharacteristic, complemented
312C23—182no—
313D4—94no—
314D4—182no—
315C2×C4—62no—
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—364no—
329C2×C3#6379812yes—
330C2×C3#6388112yes—
331C2×C3#6390211yescharacteristic
332C2×C3#6393712yes—
333C2×C3#6394112yes—
334D3—128no—
335D3—124no—
336D3—124no—
337C22—184no—
338C22—362no—
339C22—362no—
340C22#8895411yescharacteristic
341C4—122no—
342C4—542no—
343C4#8894911yescharacteristic
344C4#8896411yescharacteristic
345C3—22no—
346C3—21no—
347C3—41no—
348C3#1059046212yescomplemented
349C3#1059074111yescharacteristic, complemented
350C2—364no—
351C2#1172287812yes—
352C2#1172293811yescharacteristic
subgroupG/H
centerC22#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#88810C2
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
2. #11723501C2semidirect
3. #11722021C2semidirect
4. C3×#63574C2semidirect
5. C3×#63564C2semidirect
14. #88810C4semidirect
26. #63574D3semidirect
27. #63564D3semidirect
34. C3×C4⋊(C3⋊D3)C2×C4semidirect
37. C3×C3⋊(C3⋊C8)C2×C4semidirect
110. C4⋊(C3⋊D3)C4×D3semidirect
125. C3⋊(C3⋊C8)C4×D3semidirect
206. C33#442semidirect
310. C32#4785semidirect
311. C32#4869semidirect
348. C3#10590462semidirect
349. C3#10590741semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
#117235012
#117220212
C3×#635742
C3×#635642
#105907413
#105904623