Finite Group Databasebeta

← #13159272

Subgroups — #13159272

Order 1728.

908 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C32×#888C211yesmaximal, characteristic, complemented
2#11723517C211yesmaximal, characteristic
3#11722179C211yesmaximal, characteristic, complemented
4C3×#63636C211yesmaximal, characteristic, complemented
5#11722035C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10592787—31nomaximal
9#10590548—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6—21no—
13C2×#6292—21no—
14C2×#6292—21no—
15C33×C4⋊C4C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3C2211yescharacteristic
21C32×C4⋊Q8—31no—
22#64083—31no—
23C3×#888D314yescomplemented
24#63911—34no—
25#63636—31no—
26#63636—31no—
27#63636—31no—
28#63636—34no—
29#63636—64no—
30#63636D311yescharacteristic, complemented
31C4×C3⋊(C3⋊C8)—31no—
32C3×#809—34no—
33#63564—34no—
34#63562—34no—
35Dic3×C3⋊C8—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C22×C3×C3⋊D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C2×C32×Dic3—21no—
45C2×C32×Dic3D411yescharacteristic
46C2×C32×Dic3D411yescharacteristic
47C32×Dic6—42no—
48#6292—42no—
49#6292—42no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3×C3⋊(C3⋊C8)—21no—
52#5088—94no—
53#4760—91no—
54C2×C32×Q8—61no—
55C2×C4⋊(C3⋊D3)—31no—
56C2×C4⋊(C3⋊D3)—34no—
57C2×C4⋊(C3⋊D3)—64no—
58C2×C4⋊(C3⋊D3)—61no—
59C2×C4⋊(C3⋊D3)—61no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C3×D12—34no—
62C2×C3×Dic6—24no—
63C2×#626—64no—
64C2×#626—64no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4—31no—
67C32×C4⋊C4—61no—
68C32×C4⋊C4C2×D314yes—
69C32×C4⋊C4C2×D311yescharacteristic
70C32×C42—31no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×C4⋊Dic3C2×D314yes—
76C3×Dic3⋊C4—24no—
77C3×C4×Dic3C2×D314yes—
78C2×C3×C3⋊C8—34no—
79C22×C33C2×D411yescharacteristic
80C2×C3⋊(C3⋊D3)—41no—
81C2×C3⋊(C3⋊D3)—41no—
82C2×C3⋊(C3⋊D3)—41no—
83C2×C3×C3⋊D3—41no—
84C2×C3×C3⋊D3—41no—
85C2×C3×C3⋊D3—41no—
86C33×C4—41no—
87C33×C4#4111yescharacteristic
88C33×C4#4111yescharacteristic
89C32×Dic3—41no—
90C32×Dic3D812yes—
91C32×Dic3SD1612yes—
92C3×C4⋊Q8—34no—
93#940—91no—
94#888C3⋊D311yescharacteristic, complemented
95#874—94no—
96C3×#102—91no—
97#832—91no—
98#821—91no—
99C8×Dic3—91no—
100#809—34no—
101#809—94no—
102#809—94no—
103#809—94no—
104#809—91no—
105#809—184no—
106C4×C3⋊C8—94no—
107C22×C3⋊D3—21no—
108C22×C3⋊D3—61no—
109C22×C3⋊D3—64no—
110C22×C3⋊D3—124no—
111C22×C3×D3—64no—
112C32×Q8—122no—
113C2×C32×C4—21no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—31no—
117C2×C32×C4—31no—
118C2×C32×C4—44no—
119C2×C32×C4—61no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D311yescharacteristic
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)—21no—
124C4⋊(C3⋊D3)—68no—
125C4⋊(C3⋊D3)—61no—
126C4⋊(C3⋊D3)—61no—
127C4⋊(C3⋊D3)—64no—
128C4⋊(C3⋊D3)—64no—
129C4⋊(C3⋊D3)—124no—
130C4⋊(C3⋊D3)—122no—
131C4⋊(C3⋊D3)—122no—
132C2×C3×Dic3—24no—
133C2×C3×Dic3#6714yes—
134C2×C3×Dic3#6714yes—
135C3×D12—64no—
136C3×D12—64no—
137C3×Dic6—48no—
138#626—128no—
139#626—128no—
140C3⋊(C3⋊C8)—21no—
141C3⋊(C3⋊C8)—61no—
142C3⋊(C3⋊C8)—64no—
143C3⋊(C3⋊C8)—124no—
144C3×C3⋊C8—64no—
145#486—271noSylow
146C2×C33#12411yescharacteristic
147C2×C33C2×D811yescharacteristic
148C2×C33C2×SD1611yescharacteristic
149C3⋊(C3⋊D3)—81no—
150C3×C3⋊D3—81no—
151C2×C3×Q8—64no—
152C2×C3×D4—91no—
153C2×#67—181no—
154C2×#67—181no—
155C2×D12—34no—
156C2×D12—94no—
157C2×D12—91no—
158C2×D12—184no—
159C2×D12—184no—
160C2×D12—184no—
161C2×Dic6—21no—
162C2×C3×C8—91no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—34no—
165C3×C4⋊C4—64no—
166C3×C4⋊C4D3214yes—
167C3×C4⋊C4C2×C3⋊D311yescharacteristic
168C3×C42—34no—
169C4⋊Dic3C2×C3⋊D311yescharacteristic
170Dic3⋊C4—21no—
171C4×Dic3C2×C3⋊D311yescharacteristic
172C2×C3⋊C8—34no—
173C2×C3⋊C8—94no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—184no—
176C22×C32—24no—
177C22×C32D3×D411yescharacteristic
178C22×C32C2×#6714yes—
179C2×C3⋊D3—41no—
180C2×C3⋊D3—41no—
181C2×C3⋊D3—41no—
182C2×C3⋊D3—128no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—121no—
185C2×C3⋊D3—121no—
186C2×C3⋊D3—124no—
187C2×C3⋊D3—124no—
188C2×C3⋊D3—124no—
189C2×C3⋊D3—244no—
190C2×C3⋊D3—244no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194C32×C4—24no—
195C32×C4—24no—
196C32×C4—32no—
197C32×C4—32no—
198C32×C4—41no—
199C32×C4—44no—
200C32×C4—84no—
201C32×C4—121no—
202C32×C4#23511yescharacteristic
203C32×C4#23711yescharacteristic
204C32×C4#23914yes—
205C32×C4#23914yes—
206C3×Dic3—44no—
207C3×Dic3#21318yes—
208C3×Dic3#21518yes—
209C4⋊Q8—31no—
210#127—271no—
211#102—91no—
212#102—271no—
213#102—271no—
214#102—271no—
215C4×C8—271no—
216C33#48611yescharacteristic, complemented, Sylow
217C23×C3—181no—
218C22×D3—64no—
219C22×D3—184no—
220C22×D3—181no—
221C22×D3—364no—
222C3×Q8—128no—
223C3×D4—181no—
224C3×D4—181no—
225C2×C3×C4—24no—
226C2×C3×C4—21no—
227C2×C3×C4—24no—
228C2×C3×C4—34no—
229C2×C3×C4—34no—
230C2×C3×C4—44no—
231C2×C3×C4—64no—
232C2×C3×C4C2×D3214yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234#67—362no—
235#67—362no—
236C2×Dic3—21no—
237C2×Dic3#63811yescharacteristic
238C2×Dic3#63811yescharacteristic
239D12—64no—
240D12—64no—
241D12—188no—
242D12—184no—
243D12—184no—
244D12—181no—
245D12—181no—
246D12—364no—
247D12—368no—
248D12—368no—
249Dic6—42no—
250C3×C8—181no—
251C3⋊C8—64no—
252C3⋊C8—184no—
253C3⋊C8—181no—
254C3⋊C8—364no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32#88511yescharacteristic
259C2×C32D3×D811yescharacteristic
260C2×C32D3×SD1611yescharacteristic
261C2×C32C2×#21314yes—
262C2×C32C2×#21514yes—
263C2×C32#94714yes—
264C3⋊D3—81no—
265C3⋊D3—248no—
266C3⋊D3—241no—
267C3⋊D3—244no—
268C3×D3—244no—
269C2×Q8—61no—
270C2×D4—91no—
271C2×D4—271no—
272C2×D4—541no—
273C2×D4—541no—
274C2×C8—91no—
275C2×C8—271no—
276C4⋊C4—31no—
277C4⋊C4—61no—
278C4⋊C4D3×C3⋊D311yescharacteristic
279C42—31no—
280C22×C3—24no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3D3×#6714yes—
285C22×C3C2×#63811yescharacteristic
286C2×D3—124no—
287C2×D3—124no—
288C2×D3—124no—
289C2×D3—368no—
290C2×D3—361no—
291C2×D3—361no—
292C2×D3—361no—
293C2×D3—364no—
294C2×D3—364no—
295C2×D3—364no—
296C2×D3—724no—
297C2×D3—724no—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—38no—
301C3×C4—38no—
302C3×C4—41no—
303C3×C4—44no—
304C3×C4—84no—
305C3×C4—124no—
306C3×C4#379614yes—
307C3×C4#379914yes—
308C3×C4#383211yescharacteristic
309C3×C4#383211yescharacteristic
310Dic3—41no—
311Dic3#375312yes—
312Dic3#375512yes—
313C32—24no—
314C32#476011yescharacteristic, complemented
315C32#508814yescomplemented
316C23—181no—
317C23—541no—
318Q8—122no—
319D4—181no—
320D4—181no—
321D4—541no—
322D4—541no—
323D4—1082no—
324D4—1082no—
325C2×C4—21no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—61no—
329C2×C4C2×D3×C3⋊D311yescharacteristic
330C8—181no—
331C8—541no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—721no—
336C2×C3#6390714yes—
337C2×C3D3×#21314yes—
338C2×C3D3×#21514yes—
339C2×C3C2×#375311yescharacteristic
340C2×C3C2×#375511yescharacteristic
341C2×C3#6415611yescharacteristic
342D3—244no—
343D3—728no—
344D3—721no—
345D3—724no—
346C22—361no—
347C22—361no—
348C22—361no—
349C22—1081no—
350C22—1081no—
351C22—1081no—
352C22D3×#63811yescharacteristic
353C4—32no—
354C4—32no—
355C4—41no—
356C4—121no—
357C4#8901711yescharacteristic
358C4#8901811yescharacteristic
359C3—24no—
360C3#1059054814yescomplemented
361C3#1059278711yescharacteristic, complemented
362C2—721no—
363C2—2161no—
364C2#1172339211yescharacteristic
365C2D3×#375311yescharacteristic
366C2D3×#375511yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723517C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#486
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#888C2semidirect
3. #11722179C2semidirect
4. C3×#63636C2semidirect
5. #11722035C2semidirect
6. #11722033C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
23. C3×#888D3semidirect
30. #63636D3semidirect
94. #888C3⋊D3semidirect
216. C33#486semidirect
314. C32#4760semidirect
315. C32#5088semidirect
360. C3#10590548semidirect
361. C3#10592787semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
#117235172
#117221792
C3×#636362
#117220352
#117220332
Dic3×C3⋊(C3⋊C8)2
#105927873
#105905483