Finite Group Databasebeta

← #13161487

Subgroups — #13161487

Order 1728.

590 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C3×#64084C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic, complemented
3C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722076C212yesmaximal, complemented
5#11722075C212yesmaximal
6#10590293—31nomaximal
7#10590278—31nomaximal
8#10590274—31nomaximal
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C3×#3752—22no—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×#3750—22no—
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#64084D311yescharacteristic
17#64079D311yescharacteristic, complemented
18C3×#875—31no—
19C3×#875—31no—
20C3×#875—61no—
21C3×#870—31no—
22C3×#870—31no—
23C3×#870—61no—
24C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
25C3×C4×C3⋊C8—31no—
26C3×C4×C3⋊C8—31no—
27C3×C4×C3⋊C8—61no—
28#63567—32no—
29#63566—32no—
30#63565—32no—
31#63564—32no—
32#63563—32no—
33#63562—32no—
34C22×C3×C3⋊D3—21no—
35C2×C33×C4D412yes—
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—21no—
39C3×C4⋊(C3⋊D3)—22no—
40C3×#634—22no—
41C3×C3⋊(C3⋊C8)—22no—
42#5068—91no—
43#5056—91no—
44#4697—91no—
45C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
46C2×#634C2×D311yescharacteristic
47C2×C3×D12—31no—
48C2×C3×D12—31no—
49C2×C3×D12—61no—
50C2×C3×Dic6—31no—
51C2×C3×Dic6—31no—
52C2×C3×Dic6—61no—
53C32×C42—21no—
54C32×C42—21no—
55C32×C42—21no—
56C32×C42—41no—
57C32×C42C2×D311yescharacteristic
58C32×C42C2×D311yescharacteristic
59C32×C42C2×D311yescharacteristic
60#3752—22no—
61C4⋊(C3⋊Dic3)C2×D312yes—
62#3750—22no—
63C2×C3⋊(C3⋊C8)C2×D312yes—
64C3×(C2×D3)⋊C4—64no—
65C3×(C2×D3)⋊C4—62no—
66C3×(C2×D3)⋊C4—62no—
67C3×C4⋊Dic3—32no—
68C3×C4⋊Dic3—32no—
69C3×C4⋊Dic3—62no—
70C3×Dic3⋊C4—64no—
71C3×Dic3⋊C4—62no—
72C3×Dic3⋊C4—62no—
73C2×C3×C3⋊C8—32no—
74C2×C3×C3⋊C8—32no—
75C2×C3×C3⋊C8—62no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—42no—
79C33×C4—22no—
80C33×C4#4112yes—
81C3×C3⋊Dic3—41no—
82C3×C3⋊Dic3—42no—
83C3×#125—91no—
84C3×#124—91no—
85#875—31no—
86#875—31no—
87#875—61no—
88#870—31no—
89#870—31no—
90#870—61no—
91C3×C4×C8—91no—
92#835—92no—
93#832—92no—
94#821—92no—
95#816—92no—
96#810—92no—
97#809—92no—
98C4×C3⋊C8—31no—
99C4×C3⋊C8—31no—
100C4×C3⋊C8—61no—
101C22×C3⋊D3—21no—
102C22×C3×D3—62no—
103C22×C3×D3—61no—
104C22×C3×D3—61no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—42no—
112C2×C32×C4—41no—
113C2×C32×C4D1212yes—
114C2×C32×C4#6712yes—
115C2×C32×C4#6712yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—21no—
121C4⋊(C3⋊D3)—22no—
122#634—22no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—61no—
129C3×D12—62no—
130C3×D12—62no—
131C3×D12—122no—
132C3×Dic6—62no—
133C3×Dic6—62no—
134C3×Dic6—122no—
135C3⋊(C3⋊C8)—22no—
136C3×C3⋊C8—64no—
137C3×C3⋊C8—62no—
138C3×C3⋊C8—62no—
139#488—271noSylow
140C2×C33#12411yescharacteristic
141C2×C33#13512yes—
142C3×C3⋊D3—81no—
143C2×C3×Q8—91no—
144C2×C3×D4—91no—
145C2×D12—31no—
146C2×D12—31no—
147C2×D12—61no—
148C2×Dic6—31no—
149C2×Dic6—31no—
150C2×Dic6—61no—
151C2×C3×C8—92no—
152C3×C4⋊C4—92no—
153C3×C4⋊C4—182no—
154C3×C22⋊C4—182no—
155C3×C42—21no—
156C3×C42—21no—
157C3×C42—21no—
158C3×C42—41no—
159C3×C42D3211yescharacteristic
160C3×C42D3211yescharacteristic
161C3×C42D3211yescharacteristic
162(C2×D3)⋊C4—64no—
163(C2×D3)⋊C4—62no—
164(C2×D3)⋊C4—62no—
165C4⋊Dic3—32no—
166C4⋊Dic3—32no—
167C4⋊Dic3—62no—
168Dic3⋊C4—64no—
169Dic3⋊C4—62no—
170Dic3⋊C4—62no—
171C2×C3⋊C8—32no—
172C2×C3⋊C8—32no—
173C2×C3⋊C8—62no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—41no—
178C22×C32C2×D1211yescharacteristic
179C22×C32C2×#6711yescharacteristic
180C22×C32C2×#6711yescharacteristic
181C2×C3⋊D3—41no—
182C2×C3⋊D3—42no—
183C2×C3×D3—122no—
184C2×C3×D3—124no—
185C2×C3×D3—121no—
186C2×C3×D3—122no—
187C2×C3×D3—121no—
188C2×C3×D3—122no—
189C32×C4—22no—
190C32×C4—24no—
191C32×C4—22no—
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—22no—
195C32×C4—22no—
196C32×C4—42no—
197C32×C4—42no—
198C32×C4—42no—
199C32×C4—44no—
200C32×C4#23512yes—
201C32×C4#23712yes—
202C32×C4#23912yes—
203C3⋊Dic3—41no—
204C3⋊Dic3—42no—
205C3×Dic3—122no—
206C3×Dic3—124no—
207C3×Dic3—121no—
208C3×Dic3—122no—
209C3×Dic3—121no—
210C3×Dic3—122no—
211#125—91no—
212#124—91no—
213#103—272no—
214#102—272no—
215C4×C8—91no—
216C33#48811yescharacteristic, complemented, Sylow
217C23×C3—181no—
218C22×D3—62no—
219C22×D3—61no—
220C22×D3—61no—
221C3×Q8—182no—
222C3×D4—182no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—181no—
232C2×C3×C4—182no—
233C2×C3×C4#62512yes—
234C2×C3×C4#62612yes—
235C2×C3×C4#62612yes—
236C2×C3×C4C2×D3211yescharacteristic
237C2×C3×C4C2×D3211yescharacteristic
238C2×C3×C4C2×D3211yescharacteristic
239C2×Dic3—64no—
240C2×Dic3—62no—
241C2×Dic3—62no—
242C2×Dic3—61no—
243C2×Dic3—62no—
244C2×Dic3—61no—
245D12—62no—
246D12—62no—
247D12—122no—
248Dic6—62no—
249Dic6—62no—
250Dic6—122no—
251C3×C8—182no—
252C3⋊C8—64no—
253C3⋊C8—62no—
254C3⋊C8—62no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—42no—
263C2×C32#87511yescharacteristic
264C2×C32#90412yes—
265C2×C32#93611yescharacteristic
266C2×C32#94711yescharacteristic
267C2×C32#95012yes—
268C2×C32#95012yes—
269C3⋊D3—81no—
270C3×D3—242no—
271C3×D3—241no—
272C3×D3—241no—
273C2×Q8—91no—
274C2×D4—91no—
275C2×C8—92no—
276C4⋊C4—92no—
277C4⋊C4—182no—
278C22⋊C4—182no—
279C42#111311yescharacteristic
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—41no—
284C22×C3—362no—
285C22×C3—361no—
286C22×C3C2×#62511yescharacteristic
287C22×C3C2×#62611yescharacteristic
288C22×C3C2×#62611yescharacteristic
289C2×D3—122no—
290C2×D3—124no—
291C2×D3—122no—
292C2×D3—122no—
293C2×D3—121no—
294C2×D3—121no—
295C3×C4—24no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—42no—
303C3×C4—42no—
304C3×C4—44no—
305C3×C4—42no—
306C3×C4—362no—
307C3×C4—361no—
308C3×C4#379512yes—
309C3×C4#379612yes—
310C3×C4#379912yes—
311Dic3—122no—
312Dic3—124no—
313Dic3—122no—
314Dic3—122no—
315Dic3—121no—
316Dic3—121no—
317C32—21no—
318C32—21no—
319C32—21no—
320C32—41no—
321C32#469711yescharacteristic, complemented
322C32#505611yescharacteristic, complemented
323C32#506811yescharacteristic, complemented
324C23—181no—
325Q8—182no—
326D4—182no—
327C2×C4—181no—
328C2×C4—182no—
329C2×C4#629512yes—
330C2×C4C2×#111311yescharacteristic
331C8—182no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—22no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—42no—
339C2×C3—41no—
340C2×C3—721no—
341C2×C3#6382212yes—
342C2×C3#6382912yes—
343C2×C3#6382912yes—
344C2×C3#6386011yescharacteristic
345C2×C3#6386311yescharacteristic
346C2×C3#6386411yescharacteristic
347D3—242no—
348D3—241no—
349D3—241no—
350C22—362no—
351C22—361no—
352C22C2×#629511yescharacteristic
353C4—22no—
354C4—362no—
355C4—361no—
356C4#8904112yes—
357C3—21no—
358C3—21no—
359C3—21no—
360C3—41no—
361C3#1059027411yescharacteristic, complemented
362C3#1059027811yescharacteristic, complemented
363C3#1059029311yescharacteristic, complemented
364C2—721no—
365C2#1172366012yes—
366C2#1172367711yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC42#1113
hypocenterC33#488
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64079C2semidirect
3. C3×C4×C3⋊(C3⋊C8)C2semidirect
4. #11722076C2semidirect
17. #64079D3semidirect
24. C4×C3⋊(C3⋊C8)D3semidirect
216. C33#488semidirect
321. C32#4697semidirect
322. C32#5056semidirect
323. C32#5068semidirect
361. C3#10590274semidirect
362. C3#10590278semidirect
363. C3#10590293semidirect

Maximal subgroups

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

subgroupindex
C3×#640842
C3×#640792
C3×C4×C3⋊(C3⋊C8)2
#117220762
#117220752
#105902933
#105902783
#105902743