Finite Group Databasebeta

← #13159596

Subgroups — #13159596

Order 1728.

744 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 152 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#11723426C211yesmaximal, characteristic
2#11723424C211yesmaximal, characteristic
3#11722186C211yesmaximal, characteristic, complemented
4C3×#63645C211yesmaximal, characteristic, complemented
5C32×(C2×D3)⋊C8C211yesmaximal, characteristic
6#11722052C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592422—31nomaximal
9#10590708—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17#88807—21no—
18(C2×D3)⋊(C3⋊Dic3)—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21#64135—31no—
22#64133—31no—
23#63857—34no—
24#63841—34no—
25C32×C22⋊C8—31no—
26#63645D311yescharacteristic, complemented
27#63644—31no—
28#63643—31no—
29#63643—31no—
30#63643—34no—
31#63643—64no—
32C3×(C2×D3)⋊C8D314yes—
33C3×#818—34no—
34#63573—34no—
35#63567—34no—
36C22×C32×D3D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40#6308—21no—
41#6308—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C32×C4×D3—21no—
46C32×C4×D3—21no—
47C33×C8—21no—
48C32×C3⋊C8—21no—
49#4888—91no—
50#4723—94no—
51C22×C32×C4—31no—
52C2×#634—31no—
53C2×#634—34no—
54C2×#634—64no—
55C2×C3×C4×D3C2×D314yes—
56C2×C32×C8—24no—
57C2×C32×C8—31no—
58C2×C32×C8C2×D314yes—
59C2×C32×C8C2×D311yescharacteristic
60#3757—61no—
61#3757—61no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67#3750—61no—
68C4×C3⋊Dic3—61no—
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3—34no—
71C2×C3×C3⋊C8C2×D314yes—
72Dic3⋊Dic3—64no—
73(C2×D3)⋊Dic3—64no—
74(C2×D3)⋊Dic3—64no—
75Dic32—64no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—21no—
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C33×C4#4111yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—41no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—21no—
87#926—94no—
88#924—94no—
89#897—91no—
90#892—91no—
91C3×#107—91no—
92C3×C22⋊C8—34no—
93(C2×D3)⋊C8C3⋊D311yescharacteristic
94#818—34no—
95#817—94no—
96#816—94no—
97#816—94no—
98#816—91no—
99#816—184no—
100#810—91no—
101#808—91no—
102C23×C32—31no—
103C22×C3×D3D1214yes—
104C2×C32×C4—24no—
105C2×C32×C4—31no—
106C2×C32×C4—61no—
107C2×C32×C4—61no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—124no—
117C2×C3⋊Dic3—124no—
118#634—68no—
119#634—61no—
120#634—64no—
121#634—61no—
122#634—64no—
123#634—124no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3D1214yes—
127C3×C4×D3—24no—
128C3×C4×D3—24no—
129C32×C8—21no—
130C32×C8—24no—
131C32×C8—44no—
132C32×C8—61no—
133C3×C3⋊C8—24no—
134#485—271noSylow
135C2×C33#12311yescharacteristic
136C2×C33#13511yescharacteristic
137C2×C33#13711yescharacteristic
138C32×D3—41no—
139C22×C3×C4—34no—
140C2×C4×D3C2×C3⋊D311yescharacteristic
141C2×Dic6—94no—
142C2×Dic6—91no—
143C2×Dic6—184no—
144C2×C3×C8—24no—
145C2×C3×C8—34no—
146C2×C3×C8D3214yes—
147C2×C3×C8C2×C3⋊D311yescharacteristic
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150#217—184no—
151#217—184no—
152(C2×D3)⋊C4—181no—
153(C2×D3)⋊C4—181no—
154C4⋊Dic3—34no—
155C4⋊Dic3—34no—
156C4⋊Dic3—94no—
157C4⋊Dic3—91no—
158C4⋊Dic3—184no—
159C4⋊Dic3—181no—
160Dic3⋊C4—184no—
161C4×Dic3—184no—
162C4×Dic3—181no—
163C2×C3⋊C8C2×C3⋊D311yescharacteristic
164C22×C32—24no—
165C22×C32—61no—
166C22×C32—61no—
167C22×C32—61no—
168C22×C32C2×D1214yes—
169C22×C32D3×D411yescharacteristic
170C2×C3×D3—24no—
171C2×C3×D3—24no—
172C2×C3×D3—24no—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—61no—
176C32×C4#23511yescharacteristic
177C32×C4#23714yes—
178C32×C4#23714yes—
179C32×C4#23911yescharacteristic
180C3⋊Dic3—41no—
181C3⋊Dic3—41no—
182C3⋊Dic3—128no—
183C3⋊Dic3—124no—
184C3⋊Dic3—121no—
185C3⋊Dic3—121no—
186C3⋊Dic3—124no—
187C3⋊Dic3—244no—
188C3×Dic3—24no—
189C3×Dic3—124no—
190C3×Dic3—124no—
191C22⋊Q8—271no—
192#117—271no—
193#107—91no—
194#106—271no—
195#103—271no—
196#103—271no—
197C22⋊C8—31no—
198C33#48511yescharacteristic, complemented, Sylow
199C23×C3—34no—
200C22×D3C4⋊(C3⋊D3)11yescharacteristic
201C2×C3×C4—24no—
202C2×C3×C4—34no—
203C2×C3×C4—64no—
204C2×C3×C4—64no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—64no—
210C2×Dic3—64no—
211C2×Dic3—184no—
212C2×Dic3—184no—
213C2×Dic3—181no—
214C2×Dic3—181no—
215C2×Dic3—364no—
216C2×Dic3—364no—
217C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
218C4×D3—21no—
219C4×D3—21no—
220Dic6—188no—
221Dic6—184no—
222Dic6—184no—
223Dic6—181no—
224Dic6—181no—
225Dic6—364no—
226C3×C8—24no—
227C3×C8—21no—
228C3×C8—44no—
229C3×C8—64no—
230C3⋊C8—21no—
231C2×C32—24no—
232C2×C32—24no—
233C2×C32—24no—
234C2×C32—121no—
235C2×C32#88614yes—
236C2×C32#89411yescharacteristic
237C2×C32#90414yes—
238C2×C32#90914yes—
239C2×C32#91211yescharacteristic
240C2×C32#91811yescharacteristic
241C3×D3—44no—
242C2×Q8—271no—
243C22×C4—31no—
244C2×C8—31no—
245C2×C8D3×C3⋊D311yescharacteristic
246C4⋊C4—91no—
247C4⋊C4—91no—
248C4⋊C4—271no—
249C4⋊C4—541no—
250C22⋊C4—541no—
251C22⋊C4—541no—
252C42—541no—
253C22×C3—24no—
254C22×C3—64no—
255C22×C3—64no—
256C22×C3—64no—
257C22×C3D3×D1214yes—
258C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—21no—
262C3×C4—24no—
263C3×C4—24no—
264C3×C4—64no—
265C3×C4—361no—
266C3×C4—361no—
267C3×C4#379614yes—
268C3×C4#380414yes—
269C3×C4#383011yescharacteristic
270C3×C4#383011yescharacteristic
271Dic3—21no—
272Dic3—124no—
273Dic3—124no—
274Dic3—368no—
275Dic3—364no—
276Dic3—361no—
277Dic3—361no—
278Dic3—364no—
279Dic3—724no—
280C32—24no—
281C32#472314yescomplemented
282C32#488811yescharacteristic, complemented
283C23—31no—
284Q8—541no—
285Q8—541no—
286C2×C4—31no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—181no—
290C2×C4—181no—
291C2×C4—541no—
292C2×C4—541no—
293C2×C4C2×D3×C3⋊D311yescharacteristic
294C8—21no—
295C8—61no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—124no—
300C2×C3#6380114yes—
301C2×C3#6380714yes—
302C2×C3#6389014yes—
303C2×C3#6409511yescharacteristic
304C2×C3#6411311yescharacteristic
305C2×C3#6411811yescharacteristic
306D3—41no—
307C22—61no—
308C22—61no—
309C22—61no—
310C22D3×C4⋊(C3⋊D3)11yescharacteristic
311C4—61no—
312C4—361no—
313C4—361no—
314C4—1081no—
315C4—1081no—
316C4#8901511yescharacteristic
317C4#8902811yescharacteristic
318C3—24no—
319C3#1059070814yescomplemented
320C3#1059242211yescharacteristic, complemented
321C2—121no—
322C2#1172332711yescharacteristic
323C2#1172333611yescharacteristic
324C2#1172347911yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#485
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
3. #11722186C2semidirect
4. C3×#63645C2semidirect
6. #11722052C2semidirect
7. #11722042C2semidirect
26. #63645D3semidirect
198. C33#485semidirect
281. C32#4723semidirect
282. C32#4888semidirect
319. C3#10590708semidirect
320. C3#10592422semidirect

Maximal subgroups

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

subgroupindex
#117234262
#117234242
#117221862
C3×#636452
C32×(C2×D3)⋊C82
#117220522
#117220422
#105924223
#105907083