Finite Group Databasebeta

← #13159536

Subgroups — #13159536

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
1#11723522C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3#11722188C211yesmaximal, characteristic, complemented
4C3×#63647C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592473—31nomaximal
9#10590680—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×#6290—21no—
13C2×Dic3×C3⋊D3—21no—
14C2×C33×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88807—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#64147—31no—
22C4⋊C4×C3⋊D3—31no—
23#63908—34no—
24D3×C4⋊Dic3—34no—
25C32×#102—31no—
26#63659—31no—
27#63647D311yescharacteristic, complemented
28#63645—31no—
29#63645—34no—
30#63645—64no—
31#63634—31no—
32C3×#821D314yescomplemented
33C3×(C2×D3)⋊C8—34no—
34#63575—34no—
35#63563—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C22×C32×D3—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4×C3⋊D3—21no—
43C3×C4×C3⋊D3—21no—
44C2×C32×Dic3—21no—
45C32×D12—21no—
46C32×D12—21no—
47#6290—42no—
48Dic3×C3⋊D3—42no—
49C33×C8—21no—
50C3×C3⋊(C3⋊C8)—21no—
51#4774—94no—
52#4736—91no—
53C2×C32×D4—31no—
54C2×#638—61no—
55C2×C4×C3⋊D3—61no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×D12C2×D314yes—
58C2×C3×C4×D3—34no—
59C2×#625—64no—
60C2×D3×Dic3—64no—
61C2×C32×C8—24no—
62C2×C32×C8C2×D314yes—
63C2×C32×C8C2×D311yescharacteristic
64C32×C4⋊C4—31no—
65#3757—61no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—34no—
70C4⋊(C3⋊Dic3)—64no—
71C4⋊(C3⋊Dic3)—64no—
72#3750—61no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×C4⋊Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76Dic3⋊Dic3—64no—
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#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3⋊(C3⋊Dic3)—41no—
89C3×C3⋊Dic3—21no—
90C32×Dic3—41no—
91#938—94no—
92#930—91no—
93C2×C4⋊Dic3—91no—
94D3×C4⋊C4—94no—
95C3×#102—34no—
96C3×C22⋊C8—91no—
97#832—94no—
98#821—91no—
99#821C3⋊D311yescharacteristic, complemented
100(C2×D3)⋊C8—34no—
101#818—94no—
102#818—91no—
103#818—91no—
104#818—184no—
105#807—94no—
106C23×C32—61no—
107C22×C3⋊D3D1211yescharacteristic
108C22×C3×D3—24no—
109C22×C3×D3—34no—
110C32×D4—61no—
111C32×D4—61no—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116#638—122no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—124no—
122C2×C3⋊Dic3—124no—
123C2×C3⋊Dic3D1211yescharacteristic
124C4×C3⋊D3—21no—
125C4×C3⋊D3—21no—
126C4×C3⋊D3—122no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—34no—
129C3×D12—24no—
130C3×D12—24no—
131C3×C4×D3—64no—
132C3×C4×D3—64no—
133#625—128no—
134D3×Dic3—128no—
135C32×C8—28no—
136C32×C8—24no—
137C32×C8—21no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—64no—
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—
148C2×#67—181no—
149C2×#67—184no—
150C22×Dic3—181no—
151C2×D12C2×C3⋊D311yescharacteristic
152C2×C4×D3—34no—
153C2×C4×D3—184no—
154C2×C3×C8—24no—
155C2×C3×C8—91no—
156C2×C3×C8D3214yes—
157C2×C3×C8C2×C3⋊D311yescharacteristic
158C3×C4⋊C4—34no—
159#217—184no—
160(C2×D3)⋊C4—181no—
161C4⋊Dic3—94no—
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—91no—
165C4⋊Dic3—184no—
166C4⋊Dic3—184no—
167C4⋊Dic3—181no—
168C4⋊Dic3C2×C3⋊D311yescharacteristic
169Dic3⋊C4—184no—
170C2×C3⋊C8—34no—
171C22×C32—24no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32C2×D1211yescharacteristic
176C22×C32D3×D414yes—
177C2×C3⋊D3—21no—
178C2×C3⋊D3—21no—
179C2×C3⋊D3D2412yes—
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—121no—
189C32×C4#23514yes—
190C32×C4#23714yes—
191C32×C4#23711yescharacteristic
192C32×C4#23711yescharacteristic
193C3⋊Dic3—21no—
194C3⋊Dic3—121no—
195C3⋊Dic3—121no—
196C3⋊Dic3—124no—
197C3⋊Dic3—124no—
198C3⋊Dic3—244no—
199C3⋊Dic3—244no—
200C3×Dic3—44no—
201C3×Dic3—64no—
202#121—271no—
203C2×C4⋊C4—271no—
204#107—271no—
205#107—271no—
206#102—31no—
207#102—271no—
208C22⋊C8—91no—
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—64no—
218C2×C3×C4—91no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4C2×D3214yes—
222C2×C3×C4C22×C3⋊D311yescharacteristic
223#67—368no—
224#67—362no—
225C2×Dic3—21no—
226C2×Dic3—34no—
227C2×Dic3—184no—
228C2×Dic3—184no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231C2×Dic3—364no—
232C2×Dic3—364no—
233C2×Dic3—362no—
234D12—21no—
235D12—21no—
236C4×D3—64no—
237C4×D3—64no—
238C4×D3—368no—
239C3×C8—28no—
240C3×C8—24no—
241C3×C8—21no—
242C3×C8—181no—
243C3⋊C8—64no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—241no—
248C2×C32#88314yes—
249C2×C32#88611yescharacteristic
250C2×C32C2×D2411yescharacteristic
251C2×C32#90911yescharacteristic
252C2×C32D3×D814yes—
253C2×C32#91514yes—
254C3⋊D3—22no—
255C3×D3—68no—
256C3×D3—84no—
257C2×D4—31no—
258C2×D4—541no—
259C22×C4—91no—
260C22×C4—541no—
261C2×C8—91no—
262C2×C8D3×C3⋊D311yescharacteristic
263C4⋊C4—31no—
264C4⋊C4—271no—
265C4⋊C4—271no—
266C4⋊C4—541no—
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×D1214yes—
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—124no—
286C3×C4—181no—
287C3×C4#379514yes—
288C3×C4#380514yes—
289C3×C4#382811yescharacteristic
290C3×C4#383011yescharacteristic
291Dic3—41no—
292Dic3—64no—
293Dic3—364no—
294Dic3—364no—
295Dic3—361no—
296Dic3—361no—
297Dic3—724no—
298Dic3—724no—
299C32—24no—
300C32#473611yescharacteristic, complemented
301C32#477414yescomplemented
302C23—61no—
303C23—91no—
304D4—61no—
305D4—61no—
306D4—1082no—
307C2×C4—61no—
308C2×C4—91no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4—541no—
312C2×C4—541no—
313C2×C4—1082no—
314C2×C4C2×D3×C3⋊D311yescharacteristic
315C8—21no—
316C8—181no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—182no—
321C2×C3—244no—
322C2×C3D3×D2414yes—
323C2×C3#6380014yes—
324C2×C3#6389114yes—
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×D1211yescharacteristic
337C4—121no—
338C4—181no—
339C4—1081no—
340C4—1081no—
341C4#8902111yescharacteristic
342C4#8902911yescharacteristic
343C3—24no—
344C3#1059068014yescomplemented
345C3#1059247311yescharacteristic, complemented
346C2—182no—
347C2—241no—
348C2C3⋊D3×D2411yescharacteristic
349C2#1172334011yescharacteristic
350C2#1172352311yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723522C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#480
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
2. C3⋊D3×C4⋊Dic3C2semidirect
3. #11722188C2semidirect
4. C3×#63647C2semidirect
5. C32×#821C2semidirect
6. #11722057C2semidirect
7. #11722032C2semidirect
27. #63647D3semidirect
32. C3×#821D3semidirect
99. #821C3⋊D3semidirect
209. C33#480semidirect
300. C32#4736semidirect
301. C32#4774semidirect
344. C3#10590680semidirect
345. C3#10592473semidirect

Maximal subgroups

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

subgroupindex
#117235222
C3⋊D3×C4⋊Dic32
#117221882
C3×#636472
C32×#8212
#117220572
#117220322
#105924733
#105906803