Finite Group Databasebeta

← #13173347

Subgroups — #13173347

Order 1728.

488 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 79 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×#88989C211yesmaximal, characteristic, complemented
2C2×#88988C211yesmaximal, characteristic, complemented
3C3×#64102C211yesmaximal, characteristic
4#11723104C211yesmaximal, characteristic, complemented
5#11723103C211yesmaximal, characteristic, complemented
6#11723103C211yesmaximal, characteristic, complemented
7#11723102C211yesmaximal, characteristic, complemented
8#10593751—31nomaximal
9C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C2×C3⋊#622C2211yescharacteristic
15C2×C3⋊#622C2211yescharacteristic
16#88989—22no—
17#88988—22no—
18C33×C4⋊C4C411yescharacteristic
19C3×#3752C411yescharacteristic
20C3×#3752C411yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×#3788—31no—
23C2×#3787—31no—
24#64102D311yescharacteristic
25C3×#893—62no—
26#63779—31no—
27#63778—31no—
28#63778—31no—
29#63777—31no—
30C22×C3×C3⋊D3C2×C411yescharacteristic
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4×C3⋊D3—21no—
39C3×C4×C3⋊D3—21no—
40C3×C4×C3⋊D3—21no—
41C3×C4×C3⋊D3—21no—
42C3⋊#622—21no—
43C3⋊#622—21no—
44C3⋊#622—21no—
45C3⋊#622D412yescomplemented
46#5011—91nomaximal
47C2×C4⋊(C3⋊D3)Dic311yescharacteristic
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×C3×D12—62no—
51C2×C3×C4×D3—62no—
52C2×C3×C4×D3—62no—
53C2×#622—31no—
54C2×#622—31no—
55C2×#622—31no—
56C2×#622—31no—
57#3788—62no—
58#3787—62no—
59C32×C4⋊C4—22no—
60C32×C4⋊C4—42no—
61C32×C4⋊C4Dic311yescharacteristic
62#3752Dic311yescharacteristic
63#3752Dic311yescharacteristic
64C4×C3⋊Dic3C2×D311yescharacteristic
65C3×(C2×D3)⋊C4—62no—
66C3×(C2×D3)⋊C4—62no—
67C3×C4×Dic3—62no—
68C22×C33C22×C411yescharacteristic
69C2×C3×C3⋊D3—21no—
70C2×C3×C3⋊D3—21no—
71C2×C3×C3⋊D3—21no—
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3M1612yescomplemented
75C33×C4—21no—
76C33×C4—21no—
77C33×C4—21no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3C2×D411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C3×C4×D4—91no—
82C2×#208—91no—
83C22×C3⋊C8—91no—
84#893—62no—
85#830—91no—
86#830—91no—
87C4⋊(C3⋊C8)—91no—
88#803—91no—
89C22×C3⋊D3C2×Dic311yescharacteristic
90C22×C3⋊D3C2×Dic311yescharacteristic
91C22×C3×D3—62no—
92C22×C3×D3—62no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4—42no—
98C2×C32×C4—42no—
99C2×C32×C4C2×Dic311yescharacteristic
100C2×C32×C4C2×Dic311yescharacteristic
101C2×C32×C4C2×Dic311yescharacteristic
102C2×C3⋊Dic3C2×Dic311yescharacteristic
103C2×C3⋊Dic3C22×D311yescharacteristic
104C4⋊(C3⋊D3)—22no—
105C4×C3⋊D3—21no—
106C4×C3⋊D3—21no—
107C4×C3⋊D3—21no—
108C4×C3⋊D3—21no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C3×D12—68no—
112C3×C4×D3—122no—
113C3×C4×D3—122no—
114C3×C4×D3—122no—
115C3×C4×D3—122no—
116#622—32no—
117#622—61no—
118#622—61no—
119#622—61no—
120#435—271noSylow
121C2×C33C4×D411yescharacteristic
122C2×C33C2×M1611yescharacteristic
123C2×C33#13111yescharacteristic
124C3×C3⋊D3—22no—
125C3×C3⋊D3—41no—
126C2×C3×D4—91no—
127C22×C3×C4—91no—
128C22×C3×C4—91no—
129C2×D12—62no—
130C2×C4×D3—62no—
131C2×C4×D3—62no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—42no—
134C3×C4⋊C4#15711yescharacteristic
135C3×C22⋊C4—91no—
136C3×C22⋊C4—91no—
137C3×C42—91no—
138(C2×D3)⋊C4—62no—
139(C2×D3)⋊C4—62no—
140C4×Dic3—62no—
141#208—182no—
142C2×C3⋊C8—91no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—182no—
147C22×C32—22no—
148C22×C32—42no—
149C22×C32C22×Dic311yescharacteristic
150C2×C3⋊D3—21no—
151C2×C3⋊D3—21no—
152C2×C3⋊D3—21no—
153C2×C3⋊D3—21no—
154C2×C3⋊D3—21no—
155C2×C3⋊D3#20812yescomplemented
156C2×C3×D3—64no—
157C2×C3×D3—64no—
158C2×C3×D3—122no—
159C2×C3×D3—122no—
160C2×C3×D3—122no—
161C2×C3×D3—122no—
162C32×C4—21no—
163C32×C4—24no—
164C32×C4—21no—
165C32×C4—21no—
166C32×C4—44no—
167C32×C4—42no—
168C32×C4—42no—
169C32×C4—82no—
170C32×C4—82no—
171C3⋊Dic3—21no—
172C3⋊Dic3D3×D411yescharacteristic
173C3⋊Dic3#23711yescharacteristic
174C3×Dic3—62no—
175C3×Dic3—62no—
176C3×Dic3—122no—
177C2×M16—271no—
178C22×C8—271no—
179C4×D4—91no—
180C4⋊C8—271no—
181C22⋊C8—271no—
182C22⋊C8—271no—
183#97—271no—
184C33#43511yescharacteristic, complemented, Sylow
185C23×C3—91no—
186C23×C3—91no—
187C22×D3—62no—
188C22×D3—62no—
189C3×D4—182no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4—181no—
199C2×C3×C4—181no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4C2×#15711yescharacteristic
203C2×C3×C4C2×#15711yescharacteristic
204C2×C3×C4C2×#15711yescharacteristic
205C2×Dic3—62no—
206C2×Dic3—62no—
207D12—68no—
208C4×D3—122no—
209C4×D3—122no—
210C4×D3—122no—
211C4×D3—122no—
212C3⋊C8—92no—
213C3⋊C8—181no—
214C3⋊C8—181no—
215C3⋊C8—181no—
216C2×C32—22no—
217C2×C32—22no—
218C2×C32—22no—
219C2×C32—42no—
220C2×C32—42no—
221C2×C32—42no—
222C2×C32C2×#20811yescharacteristic
223C2×C32D4×Dic311yescharacteristic
224C2×C32#94811yescharacteristic
225C3⋊D3—22no—
226C3⋊D3—41no—
227C3×D3—124no—
228C3×D3—124no—
229C2×D4—91no—
230C22×C4—91no—
231C22×C4—91no—
232M16—542no—
233C2×C8—271no—
234C2×C8—271no—
235C2×C8—271no—
236C2×C8—271no—
237C2×C8—542no—
238C4⋊C4C3⋊#15711yescharacteristic
239C22⋊C4—91no—
240C22⋊C4—91no—
241C42—91no—
242C22×C3—22no—
243C22×C3—42no—
244C22×C3—92no—
245C22×C3—181no—
246C22×C3—181no—
247C22×C3—181no—
248C22×C3—181no—
249C22×C3—181no—
250C22×C3C22×#15711yescharacteristic
251C2×D3—64no—
252C2×D3—64no—
253C2×D3—122no—
254C2×D3—122no—
255C2×D3—122no—
256C2×D3—122no—
257C3×C4—24no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—42no—
262C3×C4—42no—
263C3×C4—44no—
264C3×C4—82no—
265C3×C4—82no—
266C3×C4—91no—
267C3×C4—91no—
268C3×C4—181no—
269Dic3—62no—
270Dic3—62no—
271Dic3—122no—
272C32—22no—
273C32—42no—
274C32#501111yescharacteristic, complemented
275C23—91no—
276C23—91no—
277D4—182no—
278C2×C4—91no—
279C2×C4—91no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C4C2×C3⋊#15711yescharacteristic
285C2×C4C2×C3⋊#15711yescharacteristic
286C2×C4C2×C3⋊#15711yescharacteristic
287C8—272no—
288C8—541no—
289C8—541no—
290C8—541no—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—22no—
294C2×C3—42no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3—182no—
298C2×C3—361no—
299C2×C3#6428311yescharacteristic
300C2×C3D4×#15711yescharacteristic
301C2×C3#6428911yescharacteristic
302D3—124no—
303D3—124no—
304C22—92no—
305C22—181no—
306C22—181no—
307C22—181no—
308C22—181no—
309C22—181no—
310C22C22×C3⋊#15711yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—91no—
315C4—91no—
316C4—181no—
317C3—22no—
318C3—42no—
319C3#1059375111yescharacteristic, complemented
320C2—182no—
321C2—361no—
322C2#1172451111yescharacteristic
323C2D4×C3⋊#15711yescharacteristic
324C2#1172451711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC33×C4⋊C4C4
prime radicalC2×C3×C4⋊(C3⋊D3)C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC4⋊C4C3⋊#157
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88989C2semidirect
2. C2×#88988C2semidirect
4. #11723104C2semidirect
5. #11723103C2semidirect
6. #11723103C2semidirect
7. #11723102C2semidirect
45. C3⋊#622D4semidirect
74. C2×C3×C3⋊D3M16semidirect
155. C2×C3⋊D3#208semidirect
184. C33#435semidirect
274. C32#5011semidirect
319. C3#10593751semidirect

Maximal subgroups

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

subgroupindex
C2×#889892
C2×#889882
C3×#641022
#117231042
#117231032
#117231022
#105937513
#50119