Finite Group Databasebeta

← #13161294

Subgroups — #13161294

Order 1728.

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

normal subgroups only

This group has 840 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×#3756C211yesmaximal, characteristic, complemented
2C32×#947C211yesmaximal, characteristic
3Dic3×#634C211yesmaximal, characteristic, complemented
4C2×#88792C211yesmaximal, characteristic
5#11722205C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic
8#10592753—31nomaximal
9#10591578—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3756—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×(C2×D3)⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20#88807—21no—
21Dic3×C3⋊Dic3—21no—
22#88792—22no—
23C32×#124—31no—
24C2×#3756D311yescharacteristic, complemented
25C2×#3754—31no—
26C4×#634—31no—
27C3×#947D314yes—
28C2×C3×#216—34no—
29Dic3×Dic6—34no—
30C2×#3715—34no—
31#63662—31no—
32#63662—34no—
33#63662—64no—
34#63659—31no—
35C4⋊(C3⋊(C3⋊C8))—31no—
36Dic3⋊(C3⋊C8)—34no—
37#63563—34no—
38C22×C32×D3—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634D412yes—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×D12—21no—
50C32×D12—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53#5185—91no—
54#5054—94no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×D4—31no—
59C2×#634C2×D311yescharacteristic
60C2×C3×D12C2×D314yes—
61C2×C3×Dic6—34no—
62C32×C22⋊C4—61no—
63C32×C22⋊C4—61no—
64C32×C42—31no—
65#3756—22no—
66#3754—62no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—64no—
70#3750—61no—
71C4×C3⋊Dic3—61no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×#216—68no—
77C3×(C2×D3)⋊C4—24no—
78C3×(C2×D3)⋊C4—24no—
79C3×C4×Dic3C2×D314yes—
80C2×C3×C3⋊C8—34no—
81Dic3⋊Dic3—64no—
82Dic32—64no—
83#3715—68no—
84C22×C33C2×D411yescharacteristic
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C33×C4—41no—
89C33×C4C2×D411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—22no—
93C32×Dic3—21no—
94C32×Dic3—21no—
95C2×C3×Q16—91no—
96C3×#124—34no—
97#947C3⋊D311yescharacteristic
98Q8×Dic3—91no—
99C2×#216—34no—
100C2×#215—91no—
101C2×#214—94no—
102C4×Dic6—94no—
103#835—94no—
104#835—91no—
105#835—184no—
106#832—94no—
107#821—91no—
108Dic3⋊C8—91no—
109C4⋊(C3⋊C8)—94no—
110C23×C32—61no—
111C22×C3×D3—24no—
112C32×Q8—28no—
113C32×Q8—24no—
114C32×Q8—21no—
115C32×Q8—21no—
116C32×Q8—24no—
117C32×Q8—44no—
118C32×D4—61no—
119C32×D4—61no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—31no—
124C2×C32×C4—31no—
125C2×C32×C4—44no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—124no—
132#634—21no—
133#634#6712yes—
134C2×C3×Dic3—64no—
135C2×C3×Dic3#6714yes—
136C2×C3×Dic3#6714yes—
137C3×D12—24no—
138C3×D12—24no—
139C3×Dic6—38no—
140C3×Dic6—64no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—61no—
143C3⋊(C3⋊C8)—64no—
144C3⋊(C3⋊C8)—124no—
145C3×C3⋊C8—64no—
146#464—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13511yescharacteristic
149C2×C33#13711yescharacteristic
150C32×D3—81no—
151C2×C3×Q8—24no—
152C2×C3×Q8—91no—
153C2×C3×Q8D3214yes—
154C2×C3×Q8C2×C3⋊D311yescharacteristic
155C2×C3×D4—34no—
156C2×D12C2×C3⋊D311yescharacteristic
157C2×Dic6—34no—
158C3×Q16—182no—
159C2×C3×C8—91no—
160C3×C22⋊C4—64no—
161C3×C22⋊C4—64no—
162C3×C42—34no—
163#216—68no—
164#215—182no—
165#214—188no—
166(C2×D3)⋊C4—21no—
167(C2×D3)⋊C4—21no—
168C4⋊Dic3—94no—
169C4⋊Dic3—91no—
170C4⋊Dic3—184no—
171C4⋊Dic3—181no—
172Dic3⋊C4—184no—
173C4×Dic3—184no—
174C4×Dic3—181no—
175C4×Dic3C2×C3⋊D311yescharacteristic
176C2×C3⋊C8—34no—
177C2×C3⋊C8—94no—
178C2×C3⋊C8—91no—
179C2×C3⋊C8—184no—
180C22×C32—24no—
181C22×C32—121no—
182C22×C32—121no—
183C22×C32—121no—
184C22×C32D3×D411yescharacteristic
185C22×C32C2×#6714yes—
186C2×C3×D3—44no—
187C2×C3×D3—44no—
188C2×C3×D3—44no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—48no—
192C32×C4—44no—
193C32×C4—41no—
194C32×C4—61no—
195C32×C4—61no—
196C32×C4D3×D414yes—
197C32×C4#23714yes—
198C32×C4#23711yescharacteristic
199C32×C4C2×#6711yescharacteristic
200C3⋊Dic3—22no—
201C3⋊Dic3—121no—
202C3⋊Dic3—124no—
203C3⋊Dic3—244no—
204C3×Dic3—24no—
205C3×Dic3—24no—
206C3×Dic3—68no—
207C2×Q16—91no—
208C2×SD16—271no—
209#124—31no—
210C4×Q8—271no—
211C4⋊C8—271no—
212#103—271no—
213#102—271no—
214C33#46411yescharacteristic, complemented, Sylow
215C23×C3—64no—
216C22×D3—21no—
217C3×Q8—28no—
218C3×Q8—24no—
219C3×Q8—24no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—44no—
223C3×Q8—92no—
224C3×Q8—181no—
225C3×D4—64no—
226C3×D4—64no—
227C2×C3×C4—24no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—34no—
231C2×C3×C4—34no—
232C2×C3×C4—44no—
233C2×C3×C4—181no—
234C2×C3×C4C2×D3214yes—
235C2×C3×C4C22×C3⋊D311yescharacteristic
236C2×Dic3—64no—
237C2×Dic3—184no—
238C2×Dic3—181no—
239C2×Dic3—364no—
240C2×Dic3#63811yescharacteristic
241C2×Dic3#63811yescharacteristic
242D12—21no—
243D12—21no—
244Dic6—38no—
245Dic6—64no—
246C3×C8—181no—
247C3⋊C8—64no—
248C3⋊C8—184no—
249C3⋊C8—181no—
250C3⋊C8—364no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—241no—
255C2×C32#91811yescharacteristic
256C2×C32#91911yescharacteristic
257C2×C32#93814yes—
258C2×C32#93911yescharacteristic
259C2×C32#95014yes—
260C2×C32#95114yes—
261C3×D3—84no—
262C2×Q8—91no—
263C2×Q8D3×C3⋊D311yescharacteristic
264C2×D4—31no—
265Q16—182no—
266SD16—542no—
267C2×C8—91no—
268C2×C8—271no—
269C4⋊C4—271no—
270C4⋊C4—541no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C42—31no—
274C42—541no—
275C22×C3—24no—
276C22×C3—124no—
277C22×C3—124no—
278C22×C3—124no—
279C22×C3D3×#6714yes—
280C22×C3C2×#63811yescharacteristic
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—48no—
287C3×C4—41no—
288C3×C4—44no—
289C3×C4—64no—
290C3×C4—64no—
291C3×C4—182no—
292C3×C4#380514yes—
293C3×C4D3×#6714yes—
294C3×C4D4×C3⋊D311yescharacteristic
295C3×C4#383011yescharacteristic
296Dic3—21no—
297Dic3—21no—
298Dic3—68no—
299Dic3—361no—
300Dic3—364no—
301Dic3—724no—
302C32—24no—
303C32#505414yescomplemented
304C32#518511yescharacteristic, complemented
305C23—61no—
306Q8—21no—
307Q8—21no—
308Q8—92no—
309Q8—181no—
310D4—61no—
311D4—61no—
312C2×C4—21no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—181no—
316C2×C4—541no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—181no—
319C8—541no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—244no—
324C2×C3#6394314yes—
325C2×C3#6394914yes—
326C2×C3#6397014yes—
327C2×C3#6414711yescharacteristic
328C2×C3#6415911yescharacteristic
329C2×C3#6416011yescharacteristic
330D3—81no—
331C22—121no—
332C22—121no—
333C22—121no—
334C22D3×#63811yescharacteristic
335C4—41no—
336C4—61no—
337C4—61no—
338C4—182no—
339C4—1081no—
340C4#8902911yescharacteristic
341C4C3⋊D3×#6711yescharacteristic
342C3—24no—
343C3#1059157814yescomplemented
344C3#1059275311yescharacteristic, complemented
345C2—241no—
346C2#1172357411yescharacteristic
347C2#1172358211yescharacteristic
348C2#1172363111yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#464
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3756C2semidirect
3. Dic3×#634C2semidirect
5. #11722205C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
24. C2×#3756D3semidirect
214. C33#464semidirect
303. C32#5054semidirect
304. C32#5185semidirect
343. C3#10591578semidirect
344. C3#10592753semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37562
C32×#9472
Dic3×#6342
C2×#887922
#117222052
Dic3⋊(C3⋊(C3⋊C8))2
#117220322
#105927533
#105915783