Finite Group Databasebeta

← #13159147

Subgroups — #13159147

Order 1728.

828 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 148 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×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
2#11723522C211yesmaximal, characteristic
3#11722177C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic, complemented
5#11722055C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592473—31nomaximal
9#10590498—34nomaximal
10C2×C3×C4×C3⋊D3—21no—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6290—21no—
14C33×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#64147—31no—
22C4⋊C4×C3⋊D3D311yescharacteristic, complemented
23C3×D3×C4⋊C4—34no—
24#63908—34no—
25C32×#102—31no—
26#63659—31no—
27#63647—31no—
28#63645—31no—
29#63634—31no—
30#63634—34no—
31#63634—64no—
32C3×#809D314yescomplemented
33#63575—34no—
34#63562—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C22×C32×D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3—21no—
44C3×C4×C3⋊D3—21no—
45C3×C4×C3⋊D3—42no—
46C32×D12—21no—
47C32×D12—21no—
48#6290—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C32×C3⋊C8—21no—
51#4967—91no—
52#4774—94no—
53C2×C32×D4—31no—
54C2×#638—61no—
55C2×C4×C3⋊D3—21no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×D12C2×D314yes—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—64no—
60C2×#625—64no—
61C2×C32×C8—31no—
62C32×C4⋊C4—24no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D311yescharacteristic
65#3757—61no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—64no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—21no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C3×C4⋊Dic3—34no—
75C3×Dic3⋊C4—64no—
76C2×C3×C3⋊C8C2×D314yes—
77(C2×D3)⋊Dic3—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3D812yes—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C2×C32×D3—41no—
85C33×C4—41no—
86C33×C4#4111yescharacteristic
87C33×C4#4111yescharacteristic
88C3⋊(C3⋊Dic3)—41no—
89C3×C3⋊Dic3—21no—
90C3×C3⋊Dic3—41no—
91C2×C3×C4⋊C4—91no—
92#938—94no—
93#930—91no—
94D3×C4⋊C4—34no—
95C3×#102—34no—
96#832—94no—
97#830—91no—
98(C2×D3)⋊C8—94no—
99#818—94no—
100#809—91no—
101#809C3⋊D311yescharacteristic, complemented
102#807—94no—
103#807—91no—
104#807—91no—
105#807—184no—
106C23×C32—61no—
107C22×C3⋊D3#6711yescharacteristic
108C22×C3×D3—24no—
109C22×C3×D3—34no—
110C32×D4—61no—
111C32×D4—61no—
112C2×C32×C4—21no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—44no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118#638—122no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—124no—
123C2×C3⋊Dic3#6711yescharacteristic
124C4×C3⋊D3—21no—
125C4×C3⋊D3—21no—
126C4×C3⋊D3—42no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—64no—
129C3×D12—24no—
130C3×D12—24no—
131C3×C4×D3—64no—
132C3×C4×D3—64no—
133C3×C4×D3—128no—
134#625—128no—
135C32×C8—61no—
136C3⋊(C3⋊C8)—68no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—61no—
139C3×C3⋊C8—24no—
140#480—271noSylow
141C2×C33#12311yescharacteristic
142C2×C33C2×D811yescharacteristic
143C2×C33#13711yescharacteristic
144C3×C3⋊D3—22no—
145C32×D3—81no—
146C2×C3×D4—34no—
147C22×C3×C4—91no—
148C22×C3×C4—181no—
149C2×#67—181no—
150C2×#67—184no—
151C2×D12C2×C3⋊D311yescharacteristic
152C2×C4×D3—34no—
153C2×C4×D3—64no—
154C2×C3×C8—34no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4D3214yes—
159C3×C4⋊C4C2×C3⋊D311yescharacteristic
160#217—184no—
161(C2×D3)⋊C4—181no—
162C4⋊Dic3—34no—
163C4⋊Dic3—94no—
164C4⋊Dic3—91no—
165C4⋊Dic3—184no—
166Dic3⋊C4—64no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171C22×C32—24no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32D3×D414yes—
176C22×C32C2×#6711yescharacteristic
177C2×C3⋊D3—21no—
178C2×C3⋊D3—21no—
179C2×C3⋊D3#21312yes—
180C2×C3×D3—38no—
181C2×C3×D3—44no—
182C2×C3×D3—44no—
183C2×C3×D3—44no—
184C2×C3×D3—64no—
185C2×C3×D3—64no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—41no—
189C32×C4—44no—
190C32×C4—84no—
191C32×C4#23514yes—
192C32×C4#23511yescharacteristic
193C32×C4#23511yescharacteristic
194C32×C4#23714yes—
195C3⋊Dic3—21no—
196C3⋊Dic3—41no—
197C3⋊Dic3—121no—
198C3⋊Dic3—124no—
199C3⋊Dic3—244no—
200C3×Dic3—64no—
201C3×Dic3—124no—
202#121—271no—
203C2×C4⋊C4—91no—
204#107—271no—
205#107—271no—
206#102—31no—
207#102—271no—
208C22⋊C8—271no—
209C33#48011yescharacteristic, complemented, Sylow
210C23×C3—64no—
211C23×C3—91no—
212C22×D3—21no—
213C22×D3—34no—
214C3×D4—64no—
215C3×D4—64no—
216C2×C3×C4—24no—
217C2×C3×C4—21no—
218C2×C3×C4—24no—
219C2×C3×C4—44no—
220C2×C3×C4—91no—
221C2×C3×C4—181no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—362no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227#67—368no—
228#67—362no—
229C2×Dic3—34no—
230C2×Dic3—64no—
231C2×Dic3—184no—
232C2×Dic3—181no—
233C2×Dic3—364no—
234D12—21no—
235D12—21no—
236C4×D3—64no—
237C4×D3—64no—
238C4×D3—128no—
239C3×C8—64no—
240C3⋊C8—21no—
241C3⋊C8—188no—
242C3⋊C8—184no—
243C3⋊C8—181no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—241no—
248C2×C32#88314yes—
249C2×C32D3×D814yes—
250C2×C32#91514yes—
251C2×C32#92911yescharacteristic
252C2×C32C2×#21311yescharacteristic
253C2×C32#94211yescharacteristic
254C3⋊D3—22no—
255C3×D3—68no—
256C3×D3—84no—
257C2×D4—31no—
258C2×D4—541no—
259C22×C4—91no—
260C22×C4—181no—
261C2×C8—31no—
262C2×C8—271no—
263C4⋊C4—91no—
264C4⋊C4—181no—
265C4⋊C4—271no—
266C4⋊C4D3×C3⋊D311yescharacteristic
267C22⋊C4—541no—
268C22×C3—24no—
269C22×C3—92no—
270C22×C3—124no—
271C22×C3—124no—
272C22×C3—124no—
273C22×C3—181no—
274C22×C3—181no—
275C22×C3D3×#6714yes—
276C22×C3D4×C3⋊D311yescharacteristic
277C2×D3—38no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—64no—
282C2×D3—64no—
283C3×C4—24no—
284C3×C4—24no—
285C3×C4—41no—
286C3×C4—44no—
287C3×C4—84no—
288C3×C4—181no—
289C3×C4—361no—
290C3×C4#379514yes—
291C3×C4#379814yes—
292C3×C4#382811yescharacteristic
293C3×C4#383011yescharacteristic
294Dic3—64no—
295Dic3—124no—
296Dic3—361no—
297Dic3—364no—
298Dic3—724no—
299C32—24no—
300C32#477414yescomplemented
301C32#496711yescharacteristic, complemented
302C23—61no—
303C23—91no—
304D4—61no—
305D4—61no—
306D4—1082no—
307C2×C4—21no—
308C2×C4—91no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—362no—
313C2×C4—541no—
314C2×C4C2×D3×C3⋊D311yescharacteristic
315C8—61no—
316C8—541no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—182no—
321C2×C3—244no—
322C2×C3#6390514yes—
323C2×C3D3×#21314yes—
324C2×C3#6394614yes—
325C2×C3#6409211yescharacteristic
326C2×C3D8×C3⋊D311yescharacteristic
327C2×C3#6412411yescharacteristic
328D3—68no—
329D3—81no—
330C22—92no—
331C22—121no—
332C22—121no—
333C22—121no—
334C22—181no—
335C22—181no—
336C22C3⋊D3×#6711yescharacteristic
337C4—41no—
338C4—181no—
339C4—361no—
340C4—1081no—
341C4#8901411yescharacteristic
342C4#8902011yescharacteristic
343C3—24no—
344C3#1059049814yescomplemented
345C3#1059247311yescharacteristic, complemented
346C2—182no—
347C2—241no—
348C2#1172351811yescharacteristic
349C2C3⋊D3×#21311yescharacteristic
350C2#1172358411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723522C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#480
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. C3×C4⋊C4×C3⋊D3C2semidirect
3. #11722177C2semidirect
4. C32×#809C2semidirect
5. #11722055C2semidirect
6. #11722031C2semidirect
7. #11722023C2semidirect
22. C4⋊C4×C3⋊D3D3semidirect
32. C3×#809D3semidirect
101. #809C3⋊D3semidirect
209. C33#480semidirect
300. C32#4774semidirect
301. C32#4967semidirect
344. C3#10590498semidirect
345. C3#10592473semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊C4×C3⋊D32
#117235222
#117221772
C32×#8092
#117220552
#117220312
#117220232
#105924733
#105904983