Finite Group Databasebeta

← #13159168

Subgroups — #13159168

Order 1728.

884 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 164 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×D3×C4⋊C4C211yesmaximal, characteristic, complemented
2#11723516C211yesmaximal, characteristic
3#11722179C211yesmaximal, characteristic, complemented
4C3×#63635C211yesmaximal, characteristic, complemented
5#11722053C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592667—31nomaximal
9#10590423—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×C4×D3—21no—
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6291—21no—
14C33×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20#88804—21no—
21C2×C32×C4⋊C4—31no—
22#64139—31no—
23C3×D3×C4⋊C4D314yescomplemented
24#63909—34no—
25#63657—31no—
26#63636—31no—
27#63636—31no—
28#63636—34no—
29#63636—64no—
30#63635—31no—
31#63635D311yescharacteristic, complemented
32C3×#808—34no—
33#63574—34no—
34#63564—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C22×C32×D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C32×C4×D3—21no—
46C32×C4×D3—21no—
47C32×C4×D3—42no—
48#6291—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C3×C3⋊(C3⋊C8)—21no—
51#4968—94no—
52#4863—91no—
53C22×C32×C4—31no—
54C22×C32×C4—61no—
55C2×#638—61no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C3×C4×D3—24no—
60C2×C3×C4×D3C2×D314yes—
61C2×#626—64no—
62C32×C4⋊C4—24no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—61no—
65C32×C4⋊C4C2×D314yes—
66C32×C4⋊C4C2×D311yescharacteristic
67#3752—61no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3C2×D314yes—
75C3×Dic3⋊C4—24no—
76C2×C3×C3⋊C8—34no—
77#3722—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3⋊(C3⋊D3)—41no—
80C2×C3⋊(C3⋊D3)—41no—
81C2×C3⋊(C3⋊D3)—41no—
82C2×C32×D3—21no—
83C2×C32×D3—21no—
84C2×C32×D3SD1612yes—
85C33×C4—41no—
86C33×C4#4111yescharacteristic
87C33×C4#4111yescharacteristic
88C3×C3⋊Dic3—41no—
89C32×Dic3—21no—
90C32×Dic3—41no—
91C2×C3×C4⋊C4—34no—
92#930—94no—
93#895—91no—
94D3×C4⋊C4C3⋊D311yescharacteristic, complemented
95C3×#106—91no—
96#830—94no—
97#821—91no—
98(C2×D3)⋊C8—91no—
99#809—94no—
100#809—94no—
101#809—91no—
102#809—184no—
103#808—34no—
104#808—94no—
105#808—91no—
106C23×C32—31no—
107C22×C3⋊D3—61no—
108C22×C3⋊D3—64no—
109C22×C3⋊D3—124no—
110C22×C3×D3#6714yes—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—31no—
115C2×C32×C4—44no—
116C2×C32×C4—61no—
117C2×C32×C4—61no—
118C2×C32×C4—61no—
119C2×C32×C4—122no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D311yescharacteristic
122#638—122no—
123C2×C3⋊Dic3—21no—
124C4⋊(C3⋊D3)—68no—
125C4⋊(C3⋊D3)—64no—
126C4⋊(C3⋊D3)—61no—
127C4⋊(C3⋊D3)—61no—
128C4⋊(C3⋊D3)—64no—
129C4⋊(C3⋊D3)—124no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—64no—
132C2×C3×Dic3#6714yes—
133C3×C4×D3—24no—
134C3×C4×D3—24no—
135C3×C4×D3—48no—
136#626—128no—
137C3⋊(C3⋊C8)—21no—
138C3⋊(C3⋊C8)—61no—
139C3⋊(C3⋊C8)—64no—
140C3⋊(C3⋊C8)—124no—
141C3×C3⋊C8—64no—
142#481—271noSylow
143C2×C33#12311yescharacteristic
144C2×C33C2×SD1611yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C3⋊(C3⋊D3)—81no—
147C32×D3—22no—
148C22×C3×C4—34no—
149C22×C3×C4—64no—
150C2×#67—184no—
151C2×D12—94no—
152C2×D12—91no—
153C2×D12—184no—
154C2×D12—181no—
155C2×C4×D3—21no—
156C2×C4×D3C2×C3⋊D311yescharacteristic
157C2×C3×C8—91no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—34no—
160C3×C4⋊C4—64no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4D3214yes—
163C3×C4⋊C4C2×C3⋊D311yescharacteristic
164(C2×D3)⋊C4—184no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—34no—
167C4⋊Dic3C2×C3⋊D311yescharacteristic
168Dic3⋊C4—21no—
169C2×C3⋊C8—34no—
170C2×C3⋊C8—94no—
171C2×C3⋊C8—91no—
172C2×C3⋊C8—184no—
173C22×C32—24no—
174C22×C32—32no—
175C22×C32—61no—
176C22×C32—61no—
177C22×C32D3×D411yescharacteristic
178C22×C32C2×#6714yes—
179C2×C3⋊D3—128no—
180C2×C3⋊D3—121no—
181C2×C3⋊D3—121no—
182C2×C3⋊D3—121no—
183C2×C3⋊D3—124no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3—244no—
187C2×C3⋊D3—244no—
188C2×C3×D3—24no—
189C2×C3×D3—24no—
190C2×C3×D3#21518yes—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—41no—
194C32×C4—44no—
195C32×C4—61no—
196C32×C4—84no—
197C32×C4—121no—
198C32×C4#23514yes—
199C32×C4#23514yes—
200C32×C4#23511yescharacteristic
201C32×C4#23911yescharacteristic
202C3⋊Dic3—41no—
203C3×Dic3—24no—
204C3×Dic3—44no—
205C3×Dic3—124no—
206#121—271no—
207C2×C4⋊C4—31no—
208#106—91no—
209#106—271no—
210#102—271no—
211#102—271no—
212C22⋊C8—271no—
213C33#48111yescharacteristic, complemented, Sylow
214C23×C3—34no—
215C22×D3—184no—
216C22×D3—181no—
217C22×D3—364no—
218C22×D3#63811yescharacteristic
219C2×C3×C4—24no—
220C2×C3×C4—21no—
221C2×C3×C4—24no—
222C2×C3×C4—34no—
223C2×C3×C4—44no—
224C2×C3×C4—64no—
225C2×C3×C4—64no—
226C2×C3×C4—64no—
227C2×C3×C4—128no—
228C2×C3×C4—181no—
229C2×C3×C4C2×D3214yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231#67—368no—
232C2×Dic3—21no—
233C2×Dic3—64no—
234C2×Dic3#63811yescharacteristic
235D12—188no—
236D12—184no—
237D12—184no—
238D12—181no—
239D12—181no—
240D12—364no—
241D12—362no—
242C4×D3—21no—
243C4×D3—21no—
244C4×D3—42no—
245C3×C8—181no—
246C3⋊C8—64no—
247C3⋊C8—184no—
248C3⋊C8—181no—
249C3⋊C8—364no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—62no—
254C2×C32#89411yescharacteristic
255C2×C32D3×SD1611yescharacteristic
256C2×C32#91411yescharacteristic
257C2×C32#92914yes—
258C2×C32#93214yes—
259C2×C32C2×#21514yes—
260C3⋊D3—248no—
261C3⋊D3—244no—
262C3⋊D3—241no—
263C3×D3—28no—
264C2×D4—271no—
265C2×D4—541no—
266C22×C4—31no—
267C22×C4—61no—
268C2×C8—91no—
269C2×C8—271no—
270C4⋊C4—31no—
271C4⋊C4—61no—
272C4⋊C4—91no—
273C4⋊C4D3×C3⋊D311yescharacteristic
274C22⋊C4—541no—
275C22×C3—24no—
276C22×C3—38no—
277C22×C3—64no—
278C22×C3—64no—
279C22×C3D3×#6714yes—
280C22×C3C2×#63811yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—368no—
284C2×D3—364no—
285C2×D3—361no—
286C2×D3—361no—
287C2×D3—361no—
288C2×D3—364no—
289C2×D3—364no—
290C2×D3—724no—
291C2×D3—724no—
292C2×D3#375512yes—
293C3×C4—24no—
294C3×C4—24no—
295C3×C4—41no—
296C3×C4—44no—
297C3×C4—64no—
298C3×C4—84no—
299C3×C4—124no—
300C3×C4—361no—
301C3×C4#379814yes—
302C3×C4#379914yes—
303C3×C4#382811yescharacteristic
304C3×C4#382811yescharacteristic
305Dic3—21no—
306Dic3—41no—
307Dic3—124no—
308C32—24no—
309C32#486311yescharacteristic, complemented
310C32#496814yescomplemented
311C23—31no—
312C23—541no—
313D4—541no—
314D4—541no—
315D4—1082no—
316C2×C4—21no—
317C2×C4—31no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—122no—
322C2×C4—181no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—181no—
325C8—541no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—68no—
330C2×C3#6385014yes—
331C2×C3#6393414yes—
332C2×C3D3×#21514yes—
333C2×C3#6413811yescharacteristic
334C2×C3#6414111yescharacteristic
335C2×C3C2×#375511yescharacteristic
336D3—22no—
337D3—728no—
338D3—721no—
339D3—724no—
340C22—32no—
341C22—61no—
342C22—61no—
343C22—1081no—
344C22—1081no—
345C22—1081no—
346C22D3×#63811yescharacteristic
347C4—41no—
348C4—61no—
349C4—121no—
350C4—361no—
351C4#8902011yescharacteristic
352C4#8902211yescharacteristic
353C3—24no—
354C3#1059042314yescomplemented
355C3#1059266711yescharacteristic, complemented
356C2—62no—
357C2—2161no—
358C2#1172341411yescharacteristic
359C2#1172355711yescharacteristic
360C2D3×#375511yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723516C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#481
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×D3×C4⋊C4C2semidirect
3. #11722179C2semidirect
4. C3×#63635C2semidirect
5. #11722053C2semidirect
6. #11722035C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
23. C3×D3×C4⋊C4D3semidirect
31. #63635D3semidirect
94. D3×C4⋊C4C3⋊D3semidirect
213. C33#481semidirect
309. C32#4863semidirect
310. C32#4968semidirect
354. C3#10590423semidirect
355. C3#10592667semidirect

Maximal subgroups

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

subgroupindex
C32×D3×C4⋊C42
#117235162
#117221792
C3×#636352
#117220532
#117220352
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105926673
#105904233