Finite Group Databasebeta

← #13173417

Subgroups — #13173417

Order 1728.

544 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88993C211yesmaximal, characteristic, complemented
2C22×C3⋊#622C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic
4#11723112C211yesmaximal, characteristic, complemented
5#11723112C211yesmaximal, characteristic, complemented
6#11723104C211yesmaximal, characteristic, complemented
7#11723102C211yesmaximal, characteristic, complemented
8#10593820—31nomaximal
9C2×C33×D4C411yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12#88993—22no—
13C2×C3⋊#622—22no—
14C2×C3⋊#622C2211yescharacteristic
15C2×C3⋊#622C2211yescharacteristic
16C2×C3⋊#622C2211yescharacteristic
17C2×C3⋊#622C2211yescharacteristic
18C3×#3757C411yescharacteristic
19C3×#3757C411yescharacteristic
20C3×C4⋊(C3⋊Dic3)C411yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×#3792—31no—
23C22×#622—31no—
24D4×C3⋊Dic3D311yescharacteristic
25C3×D4×Dic3—62no—
26C22⋊#622—31no—
27C22⋊#622—31no—
28#63779—31no—
29#63777—31no—
30C23×C33C2×C411yescharacteristic
31C23×C33C2×C411yescharacteristic
32C33×D4—22no—
33C2×C33×C4C2×C411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3C2×C411yescharacteristic
39C2×C3×C3⋊Dic3C2×C411yescharacteristic
40C2×C3×C3⋊Dic3C2×C411yescharacteristic
41C2×C3×C3⋊Dic3C2311yescharacteristic
42C3⋊#622—21no—
43C3⋊#622—21no—
44C3⋊#622—21no—
45C3⋊#622D412yescomplemented
46#5011—91nomaximal
47C2×C32×D4—22no—
48C2×C32×D4—42no—
49C2×C32×D4Dic311yescharacteristic
50C22×C3⋊Dic3C2×D311yescharacteristic
51C22×C3⋊Dic3C2×D311yescharacteristic
52C22×C3×Dic3—62no—
53C22×C3×Dic3—62no—
54#3792—62no—
55C2×#622—31no—
56C2×#622—31no—
57C2×#622—31no—
58C2×#622—31no—
59C2×#622—62no—
60#3757Dic311yescharacteristic
61#3757Dic311yescharacteristic
62C4⋊(C3⋊Dic3)Dic311yescharacteristic
63C4×C3⋊Dic3C2×D311yescharacteristic
64C3×#217—62no—
65C3×#217—62no—
66C3×C4⋊Dic3—62no—
67C3×C4×Dic3—62no—
68C22×C33—21no—
69C22×C33—21no—
70C22×C33—21no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33M1612yescomplemented
74C22×C33C22×C411yescharacteristic
75C33×C4—21no—
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3C2×D411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C3×C4×D4—91no—
82D4×Dic3—62no—
83C2×#208—91no—
84C22×C3⋊C8—91no—
85#830—91no—
86#830—91no—
87C4⋊(C3⋊C8)—91no—
88#803—91no—
89C23×C32—22no—
90C23×C32—22no—
91C23×C32—42no—
92C23×C32—42no—
93C23×C32C2×Dic311yescharacteristic
94C23×C32C2×Dic311yescharacteristic
95C32×D4—22no—
96C32×D4—28no—
97C32×D4—48no—
98C2×C32×C4—22no—
99C2×C32×C4—42no—
100C2×C32×C4C2×Dic311yescharacteristic
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3C2×Dic311yescharacteristic
106C2×C3⋊Dic3C2×Dic311yescharacteristic
107C2×C3⋊Dic3C2×Dic311yescharacteristic
108C2×C3⋊Dic3C22×D311yescharacteristic
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—122no—
114C2×C3×Dic3—122no—
115C2×C3×Dic3—122no—
116C2×C3×Dic3—122no—
117#622—32no—
118#622—61no—
119#622—61no—
120#622—61no—
121#435—271noSylow
122C2×C33—22no—
123C2×C33—41no—
124C2×C33C4×D411yescharacteristic
125C2×C33C2×M1611yescharacteristic
126C2×C33#13111yescharacteristic
127C2×C3×D4—22no—
128C2×C3×D4—42no—
129C2×C3×D4#15711yescharacteristic
130C22×C3×C4—91no—
131C22×C3×C4—91no—
132C22×Dic3—62no—
133C22×Dic3—62no—
134C3×C4⋊C4—91no—
135C3×C22⋊C4—91no—
136C3×C22⋊C4—91no—
137C3×C42—91no—
138#217—62no—
139#217—62no—
140C4⋊Dic3—62no—
141C4×Dic3—62no—
142#208—182no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—182no—
148C22×C32—21no—
149C22×C32—24no—
150C22×C32—22no—
151C22×C32—24no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—44no—
157C22×C32—42no—
158C22×C32—44no—
159C22×C32—42no—
160C22×C32—42no—
161C22×C32—42no—
162C22×C32—42no—
163C22×C32—82no—
164C22×C32—82no—
165C22×C32—82no—
166C22×C32—82no—
167C22×C32#20812yescomplemented
168C22×C32C22×Dic311yescharacteristic
169C32×C4—21no—
170C32×C4—24no—
171C32×C4—44no—
172C3⋊Dic3—21no—
173C3⋊Dic3—21no—
174C3⋊Dic3—21no—
175C3⋊Dic3D3×D411yescharacteristic
176C3⋊Dic3#23711yescharacteristic
177C3×Dic3—62no—
178C3×Dic3—62no—
179C3×Dic3—122no—
180C3×Dic3—122no—
181C3×Dic3—122no—
182C2×M16—271no—
183C22×C8—271no—
184C4×D4—91no—
185C4⋊C8—271no—
186C22⋊C8—271no—
187C22⋊C8—271no—
188#97—271no—
189C33#43511yescharacteristic, complemented, Sylow
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—42no—
193C23×C3—42no—
194C23×C3C2×#15711yescharacteristic
195C23×C3C2×#15711yescharacteristic
196C3×D4—28no—
197C3×D4—22no—
198C3×D4—48no—
199C2×C3×C4—22no—
200C2×C3×C4—42no—
201C2×C3×C4—91no—
202C2×C3×C4—91no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4C2×#15711yescharacteristic
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—122no—
215C2×Dic3—122no—
216C2×Dic3—122no—
217C2×Dic3—122no—
218C3⋊C8—92no—
219C3⋊C8—181no—
220C3⋊C8—181no—
221C3⋊C8—181no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—22no—
225C2×C32—22no—
226C2×C32—42no—
227C2×C32—42no—
228C2×C32—42no—
229C2×C32—44no—
230C2×C32—41no—
231C2×C32—44no—
232C2×C32—84no—
233C2×C32—84no—
234C2×C32C2×#20811yescharacteristic
235C2×C32D4×Dic311yescharacteristic
236C2×C32#94811yescharacteristic
237C2×D4C3⋊#15711yescharacteristic
238C22×C4—91no—
239C22×C4—91no—
240M16—542no—
241C2×C8—271no—
242C2×C8—271no—
243C2×C8—271no—
244C2×C8—271no—
245C2×C8—542no—
246C4⋊C4—91no—
247C22⋊C4—91no—
248C22⋊C4—91no—
249C42—91no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—24no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—24no—
258C22×C3—42no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—44no—
263C22×C3—42no—
264C22×C3—44no—
265C22×C3—82no—
266C22×C3—82no—
267C22×C3—82no—
268C22×C3—82no—
269C22×C3#379212yescomplemented
270C22×C3C22×#15711yescharacteristic
271C3×C4—24no—
272C3×C4—21no—
273C3×C4—44no—
274C3×C4—91no—
275C3×C4—91no—
276C3×C4—181no—
277C3×C4—181no—
278C3×C4—181no—
279Dic3—62no—
280Dic3—62no—
281Dic3—122no—
282Dic3—122no—
283Dic3—122no—
284C32—22no—
285C32—42no—
286C32#501111yescharacteristic, complemented
287C23C2×C3⋊#15711yescharacteristic
288C23C2×C3⋊#15711yescharacteristic
289D4—22no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4C2×C3⋊#15711yescharacteristic
299C8—272no—
300C8—541no—
301C8—541no—
302C8—541no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—22no—
306C2×C3—22no—
307C2×C3—44no—
308C2×C3—44no—
309C2×C3—41no—
310C2×C3—42no—
311C2×C3—42no—
312C2×C3—42no—
313C2×C3—84no—
314C2×C3—84no—
315C2×C3#6428711yescharacteristic
316C2×C3D4×#15711yescharacteristic
317C2×C3C2×#379211yescharacteristic
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22#8899312yescomplemented
324C22C22×C3⋊#15711yescharacteristic
325C4—21no—
326C4—91no—
327C4—91no—
328C4—181no—
329C4—181no—
330C4—181no—
331C3—22no—
332C3—42no—
333C3#1059382011yescharacteristic, complemented
334C2—22no—
335C2—41no—
336C2#1172451511yescharacteristic
337C2D4×C3⋊#15711yescharacteristic
338C2C2×#8899311yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
Fitting, prime radicalC2×C33×D4C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×D4C3⋊#157
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88993C2semidirect
2. C22×C3⋊#622C2semidirect
4. #11723112C2semidirect
5. #11723112C2semidirect
6. #11723104C2semidirect
7. #11723102C2semidirect
45. C3⋊#622D4semidirect
73. C22×C33M16semidirect
167. C22×C32#208semidirect
189. C33#435semidirect
269. C22×C3#3792semidirect
286. C32#5011semidirect
323. C22#88993semidirect
333. C3#10593820semidirect

Maximal subgroups

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

subgroupindex
C2×#889932
C22×C3⋊#6222
C3×D4×C3⋊Dic32
#117231122
#117231042
#117231022
#105938203
#50119