Finite Group Databasebeta

← #13161009

Subgroups — #13161009

Order 1728.

988 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 166 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
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic
4C32×#832C211yesmaximal, characteristic
5#11722055C211yesmaximal, characteristic, complemented
6#11722054C211yesmaximal, characteristic, complemented
7#11722050C211yesmaximal, characteristic, complemented
8#10592465—31nomaximal
9#10591479—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)—21no—
12#88864—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21D4×C3⋊Dic3—31no—
22D4×C3⋊Dic3—34no—
23D4×C3⋊Dic3—64no—
24#64099—31no—
25#63872—34no—
26C32×#102—31no—
27#63659D311yescharacteristic
28#63645—31no—
29#63641—31no—
30#63635—31no—
31C3×#832—34no—
32C3×#832D314yes—
33#63575—34no—
34#63573—34no—
35#63572—34no—
36C23×C33—21no—
37C33×D4—21no—
38C33×D4Q812yes—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)—42no—
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3×C3⋊Dic3—21no—
45C2×C32×Dic3—21no—
46C3×C3⋊(C3⋊C8)—21no—
47C32×C3⋊C8—21no—
48#4766—94no—
49#4763—91no—
50C2×C32×D4—24no—
51C2×C32×D4C2×D314yes—
52C2×C32×D4C2×D311yescharacteristic
53C22×C3⋊Dic3—61no—
54C22×C3⋊Dic3—64no—
55C22×C3⋊Dic3—68no—
56C2×C32×C8—31no—
57C32×C4⋊C4—31no—
58#3757—61no—
59#3757—64no—
60#3757—68no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—61no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—61no—
67C4×C3⋊Dic3—31no—
68C4×C3⋊Dic3—34no—
69C4×C3⋊Dic3—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3C2×D314yes—
73C2×C3×C3⋊C8—34no—
74C2×C3×C3⋊C8C2×D314yes—
75Dic3⋊Dic3—64no—
76Dic3⋊Dic3—64no—
77C22×C33—22no—
78C22×C33—41no—
79C22×C33—41no—
80C22×C33C2×D411yescharacteristic
81C33×C4C2×Q811yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—21no—
84C3⋊(C3⋊Dic3)—21no—
85C3⋊(C3⋊Dic3)—41no—
86C3×C3⋊Dic3—41no—
87C32×Dic3—41no—
88D4×Dic3—94no—
89D4×Dic3—91no—
90D4×Dic3—184no—
91#890—94no—
92#890—91no—
93C3×#102—34no—
94C3×#102—91no—
95#832—34no—
96#832C3⋊D311yescharacteristic
97#818—94no—
98#817—91no—
99Dic3⋊C8—94no—
100Dic3⋊C8—91no—
101#808—94no—
102#807—91no—
103C23×C32—28no—
104C23×C32—24no—
105C23×C32—21no—
106C32×D4—28no—
107C32×D4—24no—
108C32×D4—21no—
109C32×D4—28no—
110C32×D4Dic618yes—
111C32×D4Dic612yes—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—34no—
120C2×C3⋊Dic3—34no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—68no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—1216no—
127C2×C3⋊Dic3—122no—
128C2×C3⋊Dic3—128no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—64no—
131C32×C8—61no—
132C3⋊(C3⋊C8)—21no—
133C3×C3⋊C8—24no—
134C3×C3⋊C8—64no—
135#478—271noSylow
136C2×C33—42no—
137C2×C33C22⋊Q811yescharacteristic
138C2×C33#13511yescharacteristic
139C2×C33AGL(1,ℤ/8)11yescharacteristic
140C2×C3×D4—24no—
141C2×C3×D4D3214yes—
142C2×C3×D4C2×C3⋊D311yescharacteristic
143C22×Dic3—184no—
144C22×Dic3—181no—
145C22×Dic3—188no—
146C2×C3×C8—34no—
147C2×C3×C8—91no—
148C3×C4⋊C4—34no—
149C3×C4⋊C4—91no—
150#217—184no—
151#217—181no—
152#217—188no—
153C4⋊Dic3—34no—
154C4⋊Dic3—94no—
155C4⋊Dic3—91no—
156C4⋊Dic3—184no—
157C4⋊Dic3—181no—
158C4⋊Dic3—184no—
159C4⋊Dic3C2×C3⋊D311yescharacteristic
160Dic3⋊C4—181no—
161Dic3⋊C4—184no—
162C4×Dic3—94no—
163C4×Dic3—91no—
164C4×Dic3—184no—
165C2×C3⋊C8—34no—
166C2×C3⋊C8C2×C3⋊D311yescharacteristic
167C22×C32—216no—
168C22×C32—28no—
169C22×C32—22no—
170C22×C32—24no—
171C22×C32—48no—
172C22×C32—48no—
173C22×C32—41no—
174C22×C32—44no—
175C22×C32—44no—
176C22×C32—41no—
177C22×C32D3×D414yes—
178C22×C32D3×D411yescharacteristic
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—121no—
182C32×C4C2×Dic614yes—
183C32×C4C2×Dic611yescharacteristic
184C32×C4#23714yes—
185C32×C4#23711yescharacteristic
186C3⋊Dic3—41no—
187C3⋊Dic3—68no—
188C3⋊Dic3—61no—
189C3⋊Dic3—64no—
190C3⋊Dic3—61no—
191C3⋊Dic3—64no—
192C3⋊Dic3—124no—
193C3⋊Dic3—128no—
194C3⋊Dic3—121no—
195C3⋊Dic3—124no—
196C3×Dic3—44no—
197C3×Dic3—124no—
198#125—271no—
199C4×D4—271no—
200#107—271no—
201#106—271no—
202C4⋊C8—271no—
203#102—31no—
204#102—91no—
205C33#47811yescharacteristic, complemented, Sylow
206C23×C3—28no—
207C23×C3—24no—
208C23×C3—21no—
209C3×D4—28no—
210C3×D4—24no—
211C3×D4—21no—
212C3×D4—28no—
213C3×D4#62718yes—
214C3×D4#63412yes—
215C2×C3×C4—24no—
216C2×C3×C4—64no—
217C2×C3×C4—181no—
218C2×C3×C4C2×D3214yes—
219C2×C3×C4C22×C3⋊D311yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—64no—
222C2×Dic3—94no—
223C2×Dic3—94no—
224C2×Dic3—91no—
225C2×Dic3—91no—
226C2×Dic3—184no—
227C2×Dic3—184no—
228C2×Dic3—188no—
229C2×Dic3—184no—
230C2×Dic3—181no—
231C2×Dic3—3616no—
232C2×Dic3—362no—
233C2×Dic3—368no—
234C3×C8—64no—
235C3×C8—181no—
236C3⋊C8—21no—
237C3⋊C8—64no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—416no—
242C2×C32—42no—
243C2×C32—48no—
244C2×C32#87814yes—
245C2×C32#87811yescharacteristic
246C2×C32#91111yescharacteristic
247C2×C32#91214yes—
248C2×C32#91414yes—
249C2×C32#91611yescharacteristic
250C2×D4D3×C3⋊D311yescharacteristic
251C22×C4—541no—
252C2×C8—31no—
253C2×C8—91no—
254C4⋊C4—31no—
255C4⋊C4—91no—
256C4⋊C4—271no—
257C4⋊C4—541no—
258C4⋊C4—541no—
259C22⋊C4—541no—
260C42—271no—
261C22×C3—216no—
262C22×C3—22no—
263C22×C3—28no—
264C22×C3—24no—
265C22×C3—48no—
266C22×C3—48no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—44no—
270C22×C3—44no—
271C22×C3#381114yes—
272C22×C3D4×C3⋊D311yescharacteristic
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—124no—
276C3×C4—361no—
277C3×C4#380514yes—
278C3×C4C2×#62714yes—
279C3×C4C2×#63411yescharacteristic
280C3×C4#383011yescharacteristic
281Dic3—41no—
282Dic3—124no—
283Dic3—188no—
284Dic3—181no—
285Dic3—181no—
286Dic3—184no—
287Dic3—184no—
288Dic3—364no—
289Dic3—368no—
290Dic3—361no—
291Dic3—364no—
292C32—24no—
293C32#476311yescharacteristic, complemented
294C32#476614yescomplemented
295C23—21no—
296D4—21no—
297D4#629312yes—
298C2×C4—61no—
299C2×C4—181no—
300C2×C4—271no—
301C2×C4—271no—
302C2×C4—541no—
303C2×C4—1082no—
304C2×C4C2×D3×C3⋊D311yescharacteristic
305C8—61no—
306C8—181no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3—416no—
311C2×C3—42no—
312C2×C3—48no—
313C2×C3#6393614yes—
314C2×C3#6393714yes—
315C2×C3#6398214yes—
316C2×C3#6408711yescharacteristic
317C2×C3#6412111yescharacteristic
318C2×C3#6412311yescharacteristic
319C22—22no—
320C22—41no—
321C22—41no—
322C22#8903911yescharacteristic
323C4—121no—
324C4—361no—
325C4—541no—
326C4—541no—
327C4—1081no—
328C4#8902911yescharacteristic
329C4C2×#629311yescharacteristic
330C3—24no—
331C3#1059147914yescomplemented
332C3#1059246511yescharacteristic, complemented
333C2—42no—
334C2#1172356311yescharacteristic
335C2#1172356611yescharacteristic
336C2#1172364211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#478
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723450C2semidirect
5. #11722055C2semidirect
6. #11722054C2semidirect
7. #11722050C2semidirect
205. C33#478semidirect
293. C32#4763semidirect
294. C32#4766semidirect
331. C3#10591479semidirect
332. C3#10592465semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
#117234502
C3×#636592
C32×#8322
#117220552
#117220542
#117220502
#105924653
#105914793