Finite Group Databasebeta

← #13173426

Subgroups — #13173426

Order 1728.

712 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×C3×#3832C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3#11724511C211yesmaximal, characteristic, complemented
4#11723110C214yesmaximal, complemented
5#10593831—31nomaximal
6C2×C33×Q8C411yescharacteristic, complemented
7C3×#3832—21no—
8C3×#3832—21no—
9C3×#3832C412yescomplemented
10C3×#3832C2212yes—
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15#88994—21no—
16#88993—21no—
17C4⋊(C3⋊#157)—21no—
18C4×C3⋊#157C2212yescomplemented
19#88989C2212yescomplemented
20#88988—21no—
21C2×#3832D311yescharacteristic
22C2×C3×#239—62no—
23#64286—31no—
24#64283—31no—
25#63785—34no—
26C22×C3×C3⋊D3—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3⋊#157—22no—
29C33×Q8—21no—
30C33×Q8C2×C412yes—
31C2×C33×C4—21no—
32C2×C33×C4C2×C411yescharacteristic
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4⋊(C3⋊D3)—22no—
36C3×C4⋊(C3⋊D3)—21no—
37C3×C4⋊(C3⋊D3)C2×C412yes—
38C3×C4×C3⋊D3—22no—
39C3×C4×C3⋊D3—22no—
40C3×C4×C3⋊D3C2×C411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C3×C4×C3⋊D3D411yescharacteristic
43C3×C4×C3⋊D3C2311yescharacteristic
44C3⋊#622—22no—
45#5236—91nomaximal
46C2×C32×Q8—22no—
47C2×C32×Q8—42no—
48C2×C32×Q8Dic311yescharacteristic, complemented
49#3832—21no—
50#3832—21no—
51#3832Dic312yescomplemented
52#3832C2×D312yes—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
55C2×C4×C3⋊D3—21no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C3×#239—64no—
58C3×#239—64no—
59C3×#239—64no—
60C3×#239—64no—
61C2×C3×D12—62no—
62C2×C3×D12—64no—
63C2×C3×C4×D3—62no—
64C2×C3×C4×D3—64no—
65C22⋊#157—61no—
66#3792—61no—
67C4⋊#157—61no—
68C4×#157—32no—
69#3788—32no—
70#3787—61no—
71C22×C33C22⋊C411yescharacteristic
72C2×C3×C3⋊D3—22no—
73C2×C3×C3⋊D3—41no—
74C2×C3×C3⋊D3—42no—
75C2×C3×C3⋊D3C22⋊C411yescharacteristic
76C2×C3×C3⋊D3C2×D411yescharacteristic
77C3⋊#157—42no—
78C33×C4—22no—
79C33×C4C22⋊C411yescharacteristic
80C33×C4C22×C411yescharacteristic
81C3×C3⋊Dic3C22⋊C411yescharacteristic
82C3×C3⋊Dic3C2×D411yescharacteristic
83C2×C3×#41—91no—
84C2×#239—62no—
85#926—91no—
86C2×#208—91no—
87#837—94no—
88C22×C3⋊D3—21no—
89C22×C3⋊D3#6711yescharacteristic
90C22×C3×D3—62no—
91C22×C3×D3—64no—
92C2×#157—62no—
93C32×Q8—21no—
94C32×Q8—24no—
95C32×Q8—24no—
96C32×Q8—44no—
97C32×Q8—44no—
98C32×Q8C2×Dic312yes—
99C2×C32×C4—21no—
100C2×C32×C4—24no—
101C2×C32×C4—22no—
102C2×C32×C4—44no—
103C2×C32×C4—42no—
104C2×C32×C4C2×Dic311yescharacteristic
105C2×C3⋊Dic3#6711yescharacteristic
106C4⋊(C3⋊D3)—22no—
107C4⋊(C3⋊D3)—22no—
108C4⋊(C3⋊D3)—21no—
109C4⋊(C3⋊D3)C2×Dic312yes—
110C4×C3⋊D3—22no—
111C4×C3⋊D3—22no—
112C4×C3⋊D3C2×Dic311yescharacteristic
113C4×C3⋊D3#6711yescharacteristic
114C4×C3⋊D3#6711yescharacteristic
115C4×C3⋊D3C22×D311yescharacteristic
116C2×C3×Dic3—62no—
117C3×D12—64no—
118C3×D12—68no—
119C3×D12—68no—
120C3×D12—64no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×C4×D3—68no—
126C3×C4×D3—68no—
127#622—62no—
128#421—271noSylow
129C2×C33—21no—
130C2×C33C2×C22⋊C411yescharacteristic
131C3×C3⋊D3—21no—
132C3×C3⋊D3—21no—
133C3×C3⋊D3—42no—
134C3×#41—92no—
135C3×#41—92no—
136C3×#41—181no—
137C3×#41—181no—
138C2×C3×Q8—22no—
139C2×C3×Q8—42no—
140C2×C3×Q8#15711yescharacteristic, complemented
141C2×C3×D4—91no—
142C2×C3×D4—181no—
143C22×C3×C4—91no—
144C22×C3×C4—181no—
145#239—64no—
146#239—64no—
147#239—64no—
148#239—64no—
149C2×D12—62no—
150C2×D12—64no—
151C2×C4×D3—62no—
152C2×C4×D3—64no—
153#217—181no—
154C4⋊Dic3—181no—
155C4×Dic3—92no—
156#208—92no—
157#208—181no—
158C2×C3⋊C8—181no—
159C22×C32—22no—
160C22×C32—42no—
161C22×C32#21711yescharacteristic
162C2×C3⋊D3—22no—
163C2×C3⋊D3—41no—
164C2×C3⋊D3—42no—
165C2×C3⋊D3#21711yescharacteristic
166C2×C3⋊D3C2×#6711yescharacteristic
167C2×C3×D3—62no—
168C2×C3×D3—62no—
169C2×C3×D3—68no—
170C2×C3×D3—124no—
171C2×C3×D3—128no—
172#157—122no—
173C32×C4—22no—
174C32×C4—28no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—48no—
178C32×C4—42no—
179C32×C4—42no—
180C32×C4#21711yescharacteristic
181C32×C4C22×Dic311yescharacteristic
182C3⋊Dic3#21711yescharacteristic
183C3⋊Dic3C2×#6711yescharacteristic
184C3×Dic3—62no—
185C3×Dic3—62no—
186C2×#41—91no—
187C2×M16—271no—
188#117—271no—
189#104—274no—
190C33#42111yescharacteristic, complemented, Sylow
191C23×C3—91no—
192C23×C3—181no—
193C22×D3—62no—
194C22×D3—64no—
195C3×Q8—24no—
196C3×Q8—21no—
197C3×Q8—24no—
198C3×Q8—44no—
199C3×Q8—44no—
200C3×Q8C2×#15712yes—
201C3×D4—92no—
202C3×D4—182no—
203C3×D4—182no—
204C3×D4—181no—
205C2×C3×C4—24no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—44no—
209C2×C3×C4—42no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—182no—
216C2×C3×C4—182no—
217C2×C3×C4C2×#15711yescharacteristic
218C2×Dic3—62no—
219C2×Dic3—182no—
220D12—64no—
221D12—68no—
222D12—68no—
223D12—64no—
224C4×D3—62no—
225C4×D3—62no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—68no—
229C4×D3—68no—
230C3⋊C8—182no—
231C2×C32—21no—
232C2×C32—24no—
233C2×C32—22no—
234C2×C32—44no—
235C2×C32—42no—
236C2×C32C2×#21711yescharacteristic
237C3⋊D3—21no—
238C3⋊D3—21no—
239C3⋊D3—42no—
240C3×D3—122no—
241C3×D3—122no—
242C3×D3—128no—
243#41—92no—
244#41—92no—
245#41—181no—
246#41—181no—
247C2×Q8C3⋊#15711yescharacteristic, complemented
248C2×D4—91no—
249C2×D4—181no—
250C22×C4—91no—
251C22×C4—181no—
252M16—272no—
253M16—541no—
254C2×C8—541no—
255C4⋊C4—541no—
256C22⋊C4—541no—
257C42—272no—
258C22×C3—22no—
259C22×C3—42no—
260C22×C3—91no—
261C22×C3—91no—
262C22×C3—182no—
263C22×C3—361no—
264C22×C3—362no—
265C22×C3C22⋊#15711yescharacteristic
266C2×D3—62no—
267C2×D3—62no—
268C2×D3—68no—
269C2×D3—124no—
270C2×D3—128no—
271C3×C4—28no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—48no—
276C3×C4—42no—
277C3×C4—42no—
278C3×C4—91no—
279C3×C4—91no—
280C3×C4C22⋊#15711yescharacteristic
281C3×C4C22×#15711yescharacteristic
282Dic3—62no—
283Dic3—62no—
284Dic3—362no—
285C32—22no—
286C32—42no—
287C32#523611yescharacteristic, complemented
288C23—91no—
289C23—181no—
290Q8—21no—
291Q8C2×C3⋊#15712yes—
292D4—92no—
293D4—182no—
294D4—182no—
295D4—181no—
296C2×C4—21no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—182no—
303C2×C4—182no—
304C2×C4—542no—
305C2×C4C2×C3⋊#15711yescharacteristic
306C8—542no—
307C2×C3—24no—
308C2×C3—21no—
309C2×C3—22no—
310C2×C3—44no—
311C2×C3—42no—
312C2×C3—181no—
313C2×C3—181no—
314C2×C3—362no—
315C2×C3C2×C22⋊#15711yescharacteristic
316D3—122no—
317D3—122no—
318D3—128no—
319C22—91no—
320C22—91no—
321C22—182no—
322C22—361no—
323C22—362no—
324C22#8899411yescharacteristic
325C4—22no—
326C4—91no—
327C4—91no—
328C4—1082no—
329C4#8899411yescharacteristic
330C4C22×C3⋊#15711yescharacteristic
331C3—22no—
332C3—42no—
333C3#1059383111yescharacteristic, complemented
334C2—21no—
335C2—181no—
336C2—181no—
337C2—362no—
338C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC2×C33×Q8C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3832C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×Q8C3⋊#157
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. #11724511C2semidirect
4. #11723110C2semidirect
6. C2×C33×Q8C4semidirect
9. C3×#3832C4semidirect
12. C2×C3×C4⋊(C3⋊D3)C4semidirect
18. C4×C3⋊#157C22semidirect
19. #88989C22semidirect
48. C2×C32×Q8Dic3semidirect
51. #3832Dic3semidirect
54. C2×C4⋊(C3⋊D3)Dic3semidirect
140. C2×C3×Q8#157semidirect
190. C33#421semidirect
247. C2×Q8C3⋊#157semidirect
287. C32#5236semidirect
333. C3#10593831semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
#117245142
#117245112
#117231102
#105938313
#52369