Finite Group Databasebeta

← #13159964

Subgroups — #13159964

Order 1728.

872 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 156 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×C3×#3746C211yesmaximal, characteristic, complemented
2#11723441C211yesmaximal, characteristic
3Dic3×#634C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic
5#11722191C211yesmaximal, characteristic
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11722040C211yesmaximal, characteristic, complemented
8#10592544—31nomaximal
9#10590801—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×#3746—22no—
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19#88804—21no—
20#88804—21no—
21Dic3×C3⋊Dic3—21no—
22#88797—22no—
23C2×#3746D311yescharacteristic, complemented
24C2×#3744—31no—
25#64084—31no—
26C4×#634—31no—
27C2×C3×Dic12—34no—
28#63864—34no—
29Dic3×Dic6—34no—
30C2×#3717—34no—
31C32×C4⋊C8—31no—
32#63648—31no—
33#63648—34no—
34#63648—64no—
35#63643—31no—
36C3×Dic3⋊C8D314yescomplemented
37#63566—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C3×#634—21no—
48C3×#634D412yes—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51C33×C8—21no—
52C32×C3⋊C8—21no—
53#5185—91no—
54#4845—94no—
55C2×C4⋊(C3⋊D3)—31no—
56C2×C4⋊(C3⋊D3)—34no—
57C2×C4⋊(C3⋊D3)—64no—
58C2×#634C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—34no—
61C2×C3×Dic6—34no—
62C2×C32×C8—24no—
63C2×C32×C8—31no—
64C2×C32×C8C2×D314yes—
65C2×C32×C8C2×D311yescharacteristic
66C32×C42—31no—
67#3752—61no—
68#3752—61no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72#3750—61no—
73C4×C3⋊Dic3—61no—
74#3746—22no—
75#3744—62no—
76C3×C4×Dic3C2×D314yes—
77C2×C3×C3⋊C8C2×D314yes—
78C3×Dic12—68no—
79Dic3⋊Dic3—64no—
80#3722—64no—
81#3722—64no—
82Dic32—64no—
83#3717—68no—
84C22×C33C2×D411yescharacteristic
85C2×C3⋊(C3⋊D3)—41no—
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—41no—
91C3×C3⋊Dic3—22no—
92C3×C3⋊Dic3—41no—
93C32×Dic3—21no—
94C32×Dic3—21no—
95C2×C3×Q16—91no—
96#947—91no—
97Q8×Dic3—91no—
98C2×#215—91no—
99C2×Dic12—34no—
100C2×#204—94no—
101#875—94no—
102C4×Dic6—94no—
103C3×C4⋊C8—34no—
104#835—91no—
105#821—94no—
106#821—91no—
107#821—184no—
108#816—94no—
109Dic3⋊C8C3⋊D311yescharacteristic, complemented
110C22×C3⋊D3—61no—
111C22×C3⋊D3—64no—
112C22×C3⋊D3—124no—
113C2×C32×C4—24no—
114C2×C32×C4—31no—
115C2×C32×C4—31no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—124no—
123C4⋊(C3⋊D3)—68no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)—64no—
126C4⋊(C3⋊D3)—61no—
127C4⋊(C3⋊D3)—64no—
128C4⋊(C3⋊D3)—124no—
129#634—21no—
130#634—21no—
131#634—21no—
132#634#6712yes—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—64no—
135C2×C3×Dic3D1214yes—
136C2×C3×Dic3D1214yes—
137C3×Dic6—38no—
138C3×Dic6—64no—
139C3×Dic6—64no—
140C3×Dic6—64no—
141C32×C8—21no—
142C32×C8—24no—
143C32×C8—44no—
144C32×C8—61no—
145C3×C3⋊C8—24no—
146#464—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13511yescharacteristic
149C2×C33#13711yescharacteristic
150C3⋊(C3⋊D3)—81no—
151C2×C3×Q8—91no—
152C2×C3×Q8—91no—
153C2×D12—94no—
154C2×D12—91no—
155C2×D12—184no—
156C2×Dic6—34no—
157C2×Dic6—34no—
158C3×Q16—182no—
159C2×C3×C8—24no—
160C2×C3×C8—34no—
161C2×C3×C8D3214yes—
162C2×C3×C8C2×C3⋊D311yescharacteristic
163C3×C42—34no—
164#215—182no—
165(C2×D3)⋊C4—184no—
166(C2×D3)⋊C4—184no—
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—94no—
170C4⋊Dic3—91no—
171C4⋊Dic3—184no—
172C4⋊Dic3—181no—
173Dic3⋊C4—184no—
174C4×Dic3—184no—
175C4×Dic3—181no—
176C4×Dic3C2×C3⋊D311yescharacteristic
177C2×C3⋊C8C2×C3⋊D311yescharacteristic
178Dic12—68no—
179#204—188no—
180C22×C32—24no—
181C22×C32C2×D1214yes—
182C22×C32D3×D411yescharacteristic
183C2×C3⋊D3—128no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—121no—
186C2×C3⋊D3—121no—
187C2×C3⋊D3—121no—
188C2×C3⋊D3—124no—
189C2×C3⋊D3—124no—
190C2×C3⋊D3—244no—
191C2×C3⋊D3—244no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—61no—
195C32×C4—61no—
196C32×C4D3×D414yes—
197C32×C4#23711yescharacteristic
198C32×C4#23914yes—
199C32×C4C2×#6711yescharacteristic
200C3⋊Dic3—22no—
201C3⋊Dic3—41no—
202C3⋊Dic3—121no—
203C3⋊Dic3—124no—
204C3⋊Dic3—244no—
205C3×Dic3—24no—
206C3×Dic3—24no—
207C3×Dic3—68no—
208C3×Dic3—124no—
209C2×Q16—91no—
210C2×SD16—271no—
211#124—271no—
212C4×Q8—271no—
213C4⋊C8—31no—
214#103—271no—
215#102—271no—
216C33#46411yescharacteristic, complemented, Sylow
217C22×D3—184no—
218C22×D3—181no—
219C22×D3—364no—
220C3×Q8—92no—
221C3×Q8—181no—
222C3×Q8—181no—
223C3×Q8—181no—
224C2×C3×C4—24no—
225C2×C3×C4—34no—
226C2×C3×C4—34no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4C2×D3214yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231C2×Dic3—64no—
232C2×Dic3—64no—
233C2×Dic3—184no—
234C2×Dic3—181no—
235C2×Dic3—364no—
236C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
237C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
238D12—188no—
239D12—184no—
240D12—184no—
241D12—181no—
242D12—181no—
243D12—364no—
244Dic6—38no—
245Dic6—64no—
246Dic6—64no—
247Dic6—64no—
248C3×C8—24no—
249C3×C8—21no—
250C3×C8—44no—
251C3×C8—64no—
252C3⋊C8—21no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32#89514yes—
257C2×C32#90414yes—
258C2×C32#90914yes—
259C2×C32#91811yescharacteristic
260C2×C32#91911yescharacteristic
261C2×C32#93911yescharacteristic
262C3⋊D3—248no—
263C3⋊D3—241no—
264C3⋊D3—244no—
265C2×Q8—91no—
266C2×Q8—91no—
267C2×D4—271no—
268Q16—182no—
269SD16—542no—
270C2×C8—31no—
271C2×C8D3×C3⋊D311yescharacteristic
272C4⋊C4—271no—
273C4⋊C4—541no—
274C22⋊C4—541no—
275C22⋊C4—541no—
276C42—31no—
277C42—541no—
278C22×C3—24no—
279C22×C3D3×D1214yes—
280C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
281C2×D3—368no—
282C2×D3—364no—
283C2×D3—364no—
284C2×D3—361no—
285C2×D3—361no—
286C2×D3—361no—
287C2×D3—364no—
288C2×D3—724no—
289C2×D3—724no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—64no—
293C3×C4—64no—
294C3×C4—182no—
295C3×C4—361no—
296C3×C4#379614yes—
297C3×C4D3×#6714yes—
298C3×C4D4×C3⋊D311yescharacteristic
299C3×C4#383211yescharacteristic
300Dic3—21no—
301Dic3—21no—
302Dic3—68no—
303Dic3—124no—
304Dic3—364no—
305Dic3—361no—
306Dic3—724no—
307C32—24no—
308C32#484514yescomplemented
309C32#518511yescharacteristic, complemented
310C23—541no—
311Q8—92no—
312Q8—181no—
313Q8—181no—
314Q8—181no—
315D4—541no—
316D4—541no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—541no—
322C2×C4C2×D3×C3⋊D311yescharacteristic
323C8—21no—
324C8—61no—
325C2×C3—24no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3#6380114yes—
329C2×C3#6380814yes—
330C2×C3#6388614yes—
331C2×C3#6410411yescharacteristic
332C2×C3#6411311yescharacteristic
333C2×C3#6411811yescharacteristic
334D3—728no—
335D3—721no—
336D3—724no—
337C22—1081no—
338C22—1081no—
339C22—1081no—
340C22D3×C4⋊(C3⋊D3)11yescharacteristic
341C4—61no—
342C4—61no—
343C4—182no—
344C4—361no—
345C4—1081no—
346C4#8901711yescharacteristic
347C4C3⋊D3×#6711yescharacteristic
348C3—24no—
349C3#1059080114yescomplemented
350C3#1059254411yescharacteristic, complemented
351C2—2161no—
352C2#1172332711yescharacteristic
353C2#1172333711yescharacteristic
354C2#1172348211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#464
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3746C2semidirect
3. Dic3×#634C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. #11722040C2semidirect
23. C2×#3746D3semidirect
36. C3×Dic3⋊C8D3semidirect
109. Dic3⋊C8C3⋊D3semidirect
216. C33#464semidirect
308. C32#4845semidirect
309. C32#5185semidirect
349. C3#10590801semidirect
350. C3#10592544semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37462
#117234412
Dic3×#6342
C2×#887972
#117221912
C32×Dic3⋊C82
#117220402
#105925443
#105908013