Finite Group Databasebeta

← #13161030

Subgroups — #13161030

Order 1728.

1024 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1024 conjugacy classes of subgroups — too many to draw. Only its 168 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#11723536C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic
4C32×#832C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic, complemented
6#11722052C211yesmaximal, characteristic, complemented
7#11722050C211yesmaximal, characteristic, complemented
8#10592460—31nomaximal
9#10591495—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)—21no—
12C2×#6293—21no—
13#88864—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21D4×C3⋊Dic3—31no—
22D4×C3⋊Dic3—34no—
23D4×C3⋊Dic3—64no—
24#64097—31no—
25#63919—34no—
26C32×#102—31no—
27#63659D311yescharacteristic
28#63644—31no—
29#63641—31no—
30#63635—31no—
31C3×#832—34no—
32C3×#832D314yes—
33#63573—34no—
34#63573—34no—
35#63572—34no—
36C23×C33—21no—
37C33×D4—21no—
38C33×D4Q812yescomplemented
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—
46#6293—42no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4761—94no—
50#4761—91no—
51C2×C32×D4—24no—
52C2×C32×D4C2×D314yes—
53C2×C32×D4C2×D311yescharacteristic
54C22×C3⋊Dic3—61no—
55C22×C3⋊Dic3—64no—
56C22×C3⋊Dic3—68no—
57C2×#634—61no—
58C2×#627—64no—
59C2×C32×C8—31no—
60C32×C4⋊C4—31no—
61#3757—61no—
62#3757—64no—
63#3757—68no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—64no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—61no—
69C4×C3⋊Dic3—31no—
70C4×C3⋊Dic3—34no—
71C4×C3⋊Dic3—64no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76C2×C3×C3⋊C8C2×D314yes—
77#3724—64no—
78C22×C33—22no—
79C22×C33—41no—
80C22×C33—41no—
81C22×C33C2×D411yescharacteristic
82C33×C4C2×Q811yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—21no—
85C3⋊(C3⋊Dic3)—41no—
86C3⋊(C3⋊Dic3)SD1612yes—
87C3×C3⋊Dic3—41no—
88C32×Dic3—41no—
89D4×Dic3—94no—
90D4×Dic3—91no—
91D4×Dic3—184no—
92#888—94no—
93#888—91no—
94C3×#102—34no—
95C3×#102—91no—
96#832—34no—
97#832C3⋊D311yescharacteristic
98#817—94no—
99#817—91no—
100Dic3⋊C8—94no—
101Dic3⋊C8—91no—
102#808—94no—
103#808—91no—
104C23×C32—28no—
105C23×C32—24no—
106C23×C32—21no—
107C32×D4—28no—
108C32×D4—24no—
109C32×D4—21no—
110C32×D4—28no—
111C32×D4Dic618yescomplemented
112C32×D4Dic612yescomplemented
113C2×C32×C4—24no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—34no—
121C2×C3⋊Dic3—34no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—68no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—122no—
128C2×C3⋊Dic3—128no—
129C2×C3⋊Dic3—1216no—
130#634—122no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—64no—
133#627—128no—
134C32×C8—61no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—24no—
137C3×C3⋊C8—64no—
138#476—271noSylow
139C2×C33—42no—
140C2×C33C22⋊Q811yescharacteristic
141C2×C33C2×SD1611yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C3×D4—24no—
144C2×C3×D4D3214yes—
145C2×C3×D4C2×C3⋊D311yescharacteristic
146C22×Dic3—184no—
147C22×Dic3—181no—
148C22×Dic3—188no—
149C2×Dic6—181no—
150C2×Dic6—184no—
151C2×C3×C8—34no—
152C2×C3×C8—91no—
153C3×C4⋊C4—34no—
154C3×C4⋊C4—91no—
155#217—184no—
156#217—181no—
157#217—188no—
158C4⋊Dic3—34no—
159C4⋊Dic3—94no—
160C4⋊Dic3—91no—
161C4⋊Dic3—184no—
162C4⋊Dic3C2×C3⋊D311yescharacteristic
163Dic3⋊C4—181no—
164Dic3⋊C4—184no—
165C4×Dic3—94no—
166C4×Dic3—91no—
167C4×Dic3—184no—
168C2×C3⋊C8—34no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170C22×C32—216no—
171C22×C32—28no—
172C22×C32—22no—
173C22×C32—24no—
174C22×C32—48no—
175C22×C32—48no—
176C22×C32—44no—
177C22×C32—41no—
178C22×C32—44no—
179C22×C32—41no—
180C22×C32D3×D414yes—
181C22×C32D3×D411yescharacteristic
182C32×C4—24no—
183C32×C4—24no—
184C32×C4—121no—
185C32×C4C2×Dic614yes—
186C32×C4C2×Dic611yescharacteristic
187C32×C4#23714yes—
188C32×C4#23711yescharacteristic
189C3⋊Dic3—32no—
190C3⋊Dic3—38no—
191C3⋊Dic3—41no—
192C3⋊Dic3—68no—
193C3⋊Dic3—61no—
194C3⋊Dic3—64no—
195C3⋊Dic3—124no—
196C3⋊Dic3—128no—
197C3⋊Dic3—124no—
198C3⋊Dic3—121no—
199C3×Dic3—44no—
200C3×Dic3—124no—
201C4⋊Q8—271no—
202C4×D4—271no—
203#106—271no—
204#106—271no—
205C4⋊C8—271no—
206#102—31no—
207#102—91no—
208C33#47611yescharacteristic, complemented, Sylow
209C23×C3—28no—
210C23×C3—24no—
211C23×C3—21no—
212C3×D4—28no—
213C3×D4—24no—
214C3×D4—21no—
215C3×D4—28no—
216C3×D4#62718yescomplemented
217C3×D4#63412yescomplemented
218C2×C3×C4—24no—
219C2×C3×C4—64no—
220C2×C3×C4—181no—
221C2×C3×C4C2×D3214yes—
222C2×C3×C4C22×C3⋊D311yescharacteristic
223C2×Dic3—21no—
224C2×Dic3—64no—
225C2×Dic3—94no—
226C2×Dic3—91no—
227C2×Dic3—94no—
228C2×Dic3—91no—
229C2×Dic3—184no—
230C2×Dic3—184no—
231C2×Dic3—188no—
232C2×Dic3—184no—
233C2×Dic3—181no—
234C2×Dic3—362no—
235C2×Dic3—368no—
236C2×Dic3—3616no—
237Dic6—362no—
238Dic6—368no—
239C3×C8—64no—
240C3×C8—181no—
241C3⋊C8—21no—
242C3⋊C8—64no—
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—416no—
247C2×C32—48no—
248C2×C32—42no—
249C2×C32#87814yes—
250C2×C32#87811yescharacteristic
251C2×C32#91114yes—
252C2×C32#91111yescharacteristic
253C2×C32D3×SD1614yes—
254C2×C32D3×SD1611yescharacteristic
255C2×Q8—541no—
256C2×D4D3×C3⋊D311yescharacteristic
257C22×C4—541no—
258C2×C8—31no—
259C2×C8—91no—
260C4⋊C4—31no—
261C4⋊C4—91no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C22⋊C4—541no—
265C42—271no—
266C22×C3—216no—
267C22×C3—22no—
268C22×C3—28no—
269C22×C3—24no—
270C22×C3—48no—
271C22×C3—48no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—44no—
275C22×C3—44no—
276C22×C3#381114yes—
277C22×C3D4×C3⋊D311yescharacteristic
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—124no—
281C3×C4—361no—
282C3×C4#380514yes—
283C3×C4C2×#62714yes—
284C3×C4C2×#63411yescharacteristic
285C3×C4#383011yescharacteristic
286Dic3—41no—
287Dic3—92no—
288Dic3—98no—
289Dic3—124no—
290Dic3—188no—
291Dic3—181no—
292Dic3—184no—
293Dic3—364no—
294Dic3—368no—
295Dic3—361no—
296Dic3—364no—
297C32—24no—
298C32#476114yescomplemented
299C32#476111yescharacteristic, complemented
300C23—21no—
301Q8—1082no—
302D4—21no—
303D4#629312yescomplemented
304C2×C4—61no—
305C2×C4—181no—
306C2×C4—271no—
307C2×C4—271no—
308C2×C4—541no—
309C2×C4—1082no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—61no—
312C8—181no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3—416no—
317C2×C3—48no—
318C2×C3—42no—
319C2×C3#6392714yes—
320C2×C3#6393014yes—
321C2×C3#6398214yes—
322C2×C3#6408711yescharacteristic
323C2×C3#6412011yescharacteristic
324C2×C3SD16×C3⋊D311yescharacteristic
325C22—22no—
326C22—41no—
327C22—41no—
328C22#8903911yescharacteristic
329C4—121no—
330C4—272no—
331C4—361no—
332C4—541no—
333C4—1081no—
334C4#8902911yescharacteristic
335C4C2×#629311yescharacteristic
336C3—24no—
337C3#1059149514yescomplemented
338C3#1059246011yescharacteristic, complemented
339C2—42no—
340C2#1172354811yescharacteristic
341C2#1172355311yescharacteristic
342C2#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#476
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. #11723536C2semidirect
5. #11722054C2semidirect
6. #11722052C2semidirect
7. #11722050C2semidirect
38. C33×D4Q8semidirect
111. C32×D4Dic6semidirect
112. C32×D4Dic6semidirect
208. C33#476semidirect
216. C3×D4#627semidirect
217. C3×D4#634semidirect
298. C32#4761semidirect
299. C32#4761semidirect
303. D4#6293semidirect
337. C3#10591495semidirect
338. C3#10592460semidirect

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
#117235362
C3×#636592
C32×#8322
#117220542
#117220522
#117220502
#105924603
#105914953