Finite Group Databasebeta

← #13159577

Subgroups — #13159577

Order 1728.

736 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 162 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#11723423C211yesmaximal, characteristic
2#11722185C211yesmaximal, characteristic, complemented
3C3×#63641C211yesmaximal, characteristic, complemented
4C32×(C2×D3)⋊C8C211yesmaximal, characteristic
5#11722051C211yesmaximal, characteristic, complemented
6(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic, complemented
8#10592419—31nomaximal
9#10590699—34nomaximal
10C2×C32×C4×D3C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17Dic3⋊(C3⋊Dic3)C411yescharacteristic
18Dic3⋊(C3⋊Dic3)C411yescharacteristic
19(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
21#64135—31no—
22#63855—34no—
23C32×C22⋊C8—31no—
24#63657—31no—
25#63642—31no—
26#63642—34no—
27#63642—64no—
28#63641—31no—
29#63641D311yescharacteristic, complemented
30C8×C3⋊Dic3—31no—
31C3×(C2×D3)⋊C8D314yes—
32C3×Dic3⋊C8—34no—
33Dic3⋊(C3⋊C8)—34no—
34(C2×D3)⋊(C3⋊C8)—34no—
35#63553—34no—
36C22×C32×D3C2×C411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
39C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
40C2×C3×C3⋊Dic3C2×C411yescharacteristic
41C2×C3×C3⋊Dic3C2×C411yescharacteristic
42C2×C32×Dic3C2×C411yescharacteristic
43C32×C4×D3—21no—
44C32×C4×D3—21no—
45C33×C8—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4840—91no—
50#4720—94no—
51C22×C32×C4—31no—
52C2×C3×C4×D3C2×D314yes—
53C2×C32×C8—24no—
54C2×C32×C8—31no—
55C2×C32×C8C2×D314yes—
56C2×C32×C8C2×D311yescharacteristic
57#3757—31no—
58#3757—31no—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—31no—
61C4×C3⋊Dic3—31no—
62C4×C3⋊Dic3—34no—
63C4×C3⋊Dic3—64no—
64C4×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4×Dic3—34no—
70C2×C3×C3⋊C8—34no—
71C2×C3×C3⋊C8C2×D314yes—
72Dic3⋊Dic3—34no—
73Dic3⋊Dic3—34no—
74(C2×D3)⋊Dic3—34no—
75(C2×D3)⋊Dic3—34no—
76C22×C33C22×C411yescharacteristic
77C2×C32×D3—21no—
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C33×C4#4111yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—21no—
83C3⋊(C3⋊Dic3)—21no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—21no—
86C32×Dic3—21no—
87#926—94no—
88#872—91no—
89C3×C4⋊C8—91no—
90C3×C22⋊C8—34no—
91#830—94no—
92(C2×D3)⋊C8—91no—
93(C2×D3)⋊C8C3⋊D311yescharacteristic
94#815—94no—
95#815—91no—
96#815—184no—
97Dic3⋊C8—34no—
98Dic3⋊C8—94no—
99C8×Dic3—94no—
100C8×Dic3—91no—
101C4⋊(C3⋊C8)—91no—
102C23×C32—31no—
103C22×C3×D3C4×D314yes—
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—31no—
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3C4×D311yescharacteristic
117C2×C3⋊Dic3C4×D311yescharacteristic
118C2×C3×Dic3—34no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3C4×D314yes—
121C3×C4×D3—24no—
122C3×C4×D3—24no—
123C32×C8—28no—
124C32×C8—24no—
125C32×C8—21no—
126C32×C8—61no—
127C3⋊(C3⋊C8)—21no—
128C3⋊(C3⋊C8)—68no—
129C3⋊(C3⋊C8)—64no—
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8—24no—
132C3×C3⋊C8—64no—
133#433—271noSylow
134C2×C33#11711yescharacteristic
135C2×C33#13111yescharacteristic
136C2×C33#13111yescharacteristic
137C32×D3—41no—
138C22×C3×C4—34no—
139C2×C4×D3C2×C3⋊D311yescharacteristic
140C2×C3×C8—24no—
141C2×C3×C8—34no—
142C2×C3×C8—91no—
143C2×C3×C8D3214yes—
144C2×C3×C8C2×C3⋊D311yescharacteristic
145C3×C42—91no—
146#217—94no—
147#217—94no—
148(C2×D3)⋊C4—91no—
149(C2×D3)⋊C4—91no—
150C4⋊Dic3—94no—
151C4⋊Dic3—94no—
152Dic3⋊C4—91no—
153Dic3⋊C4—91no—
154C4×Dic3—34no—
155C4×Dic3—94no—
156C4×Dic3—91no—
157C4×Dic3—184no—
158C2×C3⋊C8—34no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—184no—
162C2×C3⋊C8C2×C3⋊D311yescharacteristic
163C22×C32—24no—
164C22×C32—61no—
165C22×C32—61no—
166C22×C32—61no—
167C22×C32C2×C4×D314yes—
168C22×C32C2×C4×D311yescharacteristic
169C2×C3×D3—24no—
170C2×C3×D3—24no—
171C2×C3×D3—24no—
172C32×C4—24no—
173C32×C4—24no—
174C32×C4—61no—
175C32×C4#23714yes—
176C32×C4#23714yes—
177C32×C4#23711yescharacteristic
178C32×C4#23911yescharacteristic
179C3⋊Dic3—21no—
180C3⋊Dic3—21no—
181C3⋊Dic3—61no—
182C3⋊Dic3—61no—
183C3⋊Dic3—64no—
184C3⋊Dic3—64no—
185C3⋊Dic3—124no—
186C3⋊Dic3—124no—
187C3×Dic3—24no—
188C3×Dic3—64no—
189C3×Dic3—64no—
190#117—271no—
191C4⋊C8—91no—
192C4⋊C8—271no—
193C22⋊C8—31no—
194C22⋊C8—271no—
195#97—271no—
196C4×C8—271no—
197C33#43311yescharacteristic, complemented, Sylow
198C23×C3—34no—
199C22×D3C4×C3⋊D311yescharacteristic
200C2×C3×C4—24no—
201C2×C3×C4—34no—
202C2×C3×C4—64no—
203C2×C3×C4—64no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4C2×D3214yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208C2×Dic3—34no—
209C2×Dic3—34no—
210C2×Dic3—94no—
211C2×Dic3—94no—
212C2×Dic3—91no—
213C2×Dic3—91no—
214C2×Dic3—184no—
215C2×Dic3—184no—
216C2×Dic3C4×C3⋊D311yescharacteristic
217C4×D3—21no—
218C4×D3—21no—
219C3×C8—28no—
220C3×C8—24no—
221C3×C8—21no—
222C3×C8—64no—
223C3×C8—181no—
224C3⋊C8—21no—
225C3⋊C8—64no—
226C3⋊C8—188no—
227C3⋊C8—184no—
228C3⋊C8—181no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32—121no—
233C2×C32#87714yes—
234C2×C32#89211yescharacteristic
235C2×C32#90114yes—
236C2×C32#90111yescharacteristic
237C2×C32#90714yes—
238C2×C32#90711yescharacteristic
239C3×D3—44no—
240C22×C4—31no—
241C2×C8—31no—
242C2×C8—91no—
243C2×C8—271no—
244C2×C8D3×C3⋊D311yescharacteristic
245C4⋊C4—271no—
246C4⋊C4—271no—
247C22⋊C4—271no—
248C22⋊C4—271no—
249C42—91no—
250C42—271no—
251C22×C3—24no—
252C22×C3—64no—
253C22×C3—64no—
254C22×C3—64no—
255C22×C3C4×D3214yes—
256C22×C3C2×C4×C3⋊D311yescharacteristic
257C2×D3—21no—
258C2×D3—21no—
259C2×D3—21no—
260C3×C4—24no—
261C3×C4—24no—
262C3×C4—64no—
263C3×C4—181no—
264C3×C4—181no—
265C3×C4#379614yes—
266C3×C4#380514yes—
267C3×C4#383011yescharacteristic
268C3×C4#383011yescharacteristic
269Dic3—21no—
270Dic3—64no—
271Dic3—64no—
272Dic3—184no—
273Dic3—184no—
274Dic3—181no—
275Dic3—181no—
276Dic3—364no—
277Dic3—364no—
278C32—24no—
279C32#472014yescomplemented
280C32#484011yescharacteristic, complemented
281C23—31no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—271no—
288C2×C4—271no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—21no—
291C8—61no—
292C8—181no—
293C8—541no—
294C2×C3—24no—
295C2×C3—24no—
296C2×C3—24no—
297C2×C3—124no—
298C2×C3#6380414yes—
299C2×C3#6380414yes—
300C2×C3#6387814yes—
301C2×C3#6408611yescharacteristic
302C2×C3#6411011yescharacteristic
303C2×C3#6411611yescharacteristic
304D3—41no—
305C22—61no—
306C22—61no—
307C22—61no—
308C22C4×D3×C3⋊D311yescharacteristic
309C4—61no—
310C4—181no—
311C4—181no—
312C4—541no—
313C4—541no—
314C4#8901511yescharacteristic
315C4#8902911yescharacteristic
316C3—24no—
317C3#1059069914yescomplemented
318C3#1059241911yescharacteristic, complemented
319C2—121no—
320C2#1172333211yescharacteristic
321C2#1172333311yescharacteristic
322C2#1172346011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC22×C32×D3C2×C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11722185C2semidirect
3. C3×#63641C2semidirect
5. #11722051C2semidirect
7. #11722018C2semidirect
29. #63641D3semidirect
197. C33#433semidirect
279. C32#4720semidirect
280. C32#4840semidirect
317. C3#10590699semidirect
318. C3#10592419semidirect

Maximal subgroups

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

subgroupindex
#117234232
#117221852
C3×#636412
C32×(C2×D3)⋊C82
#117220512
(C2×D3)⋊(C3⋊(C3⋊C8))2
#117220182
#105924193
#105906993