Finite Group Databasebeta

← #13166639

Subgroups — #13166639

Order 1728.

504 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C22×C3⋊#622C211yesmaximal, characteristic, complemented
2C2×#88925C211yesmaximal, characteristic, complemented
3#67×C3⋊Dic3C211yesmaximal, characteristic
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5#11722732C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
7#11722724C211yesmaximal, characteristic, complemented
8#10593820—31nomaximal
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C32×#67C411yescharacteristic
11C2×D3×C3⋊Dic3C2211yescharacteristic
12C2×C3⋊#622—22no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3⋊#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17#88925—22no—
18#88864C411yescharacteristic
19Dic3⋊(C3⋊Dic3)C411yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
21Dic3×C3⋊Dic3C2211yescharacteristic
22C2×#3792—31no—
23C22×#622—31no—
24D4×C3⋊Dic3—31no—
25Dic3×#67—62no—
26C22⋊#622—31no—
27C22⋊#622D311yescharacteristic, complemented
28#63779—31no—
29#63777—31no—
30C23×C33C2×C411yescharacteristic
31C22×C32×D3C2×C411yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C32×#67—22no—
39C2×C32×Dic3C2×C411yescharacteristic
40D3×C3⋊Dic3—21no—
41D3×C3⋊Dic3—21no—
42C3⋊#622—21no—
43C3⋊#622D412yescomplemented
44C3×#622—21no—
45C3×#622—21no—
46#4730—91nomaximal
47C2×C32×D4—31no—
48C22×C3⋊Dic3—31no—
49C22×C3⋊Dic3C2×D311yescharacteristic
50C2×C3×#67—22no—
51C22×C3×Dic3—62no—
52C2×D3×Dic3—62no—
53#3792—62no—
54C2×#622—31no—
55C2×#622—31no—
56C2×#622—62no—
57C2×#622C2×D311yescharacteristic
58C2×#622C2×D311yescharacteristic
59#3757—31no—
60#3757—31no—
61#3757—62no—
62#3757—122no—
63C4⋊(C3⋊Dic3)—31no—
64C4×C3⋊Dic3—31no—
65Dic3⋊Dic3—62no—
66(C2×D3)⋊Dic3—62no—
67Dic32—62no—
68C22×C33—21no—
69C22×C33—21no—
70C22×C33M1612yescomplemented
71C22×C33C22×C411yescharacteristic
72C2×C32×D3—21no—
73C2×C32×D3—21no—
74C2×C32×D3—21no—
75C3⋊(C3⋊Dic3)—21no—
76C3⋊(C3⋊Dic3)—21no—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3C2×D411yescharacteristic
79C3×C3⋊Dic3#4111yescharacteristic
80C32×Dic3—21no—
81D4×Dic3—182no—
82C4×#67—91no—
83C22×C3⋊C8—91no—
84C2×ΓL(1,𝔽25)—91no—
85C3×C22⋊C8—91no—
86(C2×D3)⋊C8—91no—
87#815—91no—
88Dic3⋊C8—91no—
89C23×C32—22no—
90C23×C32—31no—
91C23×C32—42no—
92C23×C32C4×D311yescharacteristic
93C22×C3×D3—22no—
94C32×D4—62no—
95C2×C32×C4—31no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—31no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—122no—
105C2×C3⋊Dic3—122no—
106C2×C3⋊Dic3C4×D311yescharacteristic
107C2×C3⋊Dic3C22×D311yescharacteristic
108C3×#67—28no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—122no—
113C2×C3×Dic3—122no—
114D3×Dic3—122no—
115D3×Dic3—122no—
116#622—21no—
117#622—21no—
118#622—32no—
119#622—61no—
120#435—271noSylow
121C2×C33—22no—
122C2×C33C4×D411yescharacteristic
123C2×C33C2×M1611yescharacteristic
124C2×C33#13111yescharacteristic
125C32×D3—41no—
126C2×C3×D4—62no—
127C22×C3×C4—91no—
128C2×#67#15711yescharacteristic
129C22×Dic3—62no—
130C22×Dic3—182no—
131C2×C4×D3—91no—
132C2×C3×C8—91no—
133C2×C3×C8—91no—
134#217—91no—
135#217—182no—
136#217—182no—
137#217—362no—
138(C2×D3)⋊C4—91no—
139C4⋊Dic3—182no—
140Dic3⋊C4—91no—
141C4×Dic3—91no—
142C4×Dic3—182no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—182no—
146ΓL(1,𝔽25)—182no—
147C22×C32—22no—
148C22×C32—24no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—42no—
152C22×C32—44no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32—61no—
156C22×C32—61no—
157C22×C32—61no—
158C22×C32—82no—
159C22×C32—82no—
160C22×C32ΓL(1,𝔽25)12yescomplemented
161C22×C32C2×C4×D311yescharacteristic
162C2×C3×D3—24no—
163C2×C3×D3—42no—
164C2×C3×D3—42no—
165C32×C4—61no—
166C3⋊Dic3—21no—
167C3⋊Dic3—61no—
168C3⋊Dic3—61no—
169C3⋊Dic3—122no—
170C3⋊Dic3—122no—
171C3⋊Dic3—242no—
172C3⋊Dic3—242no—
173C3⋊Dic3D3×D411yescharacteristic
174C3⋊Dic3#23711yescharacteristic
175C3×Dic3—24no—
176C3×Dic3—62no—
177C3×Dic3—62no—
178C3×Dic3—122no—
179C2×M16—271no—
180C22×C8—271no—
181C4×D4—271no—
182C4⋊C8—271no—
183C22⋊C8—91no—
184C22⋊C8—271no—
185#97—271no—
186C33#43511yescharacteristic, complemented, Sylow
187C23×C3—22no—
188C23×C3—42no—
189C23×C3—62no—
190C23×C3C2×#15711yescharacteristic
191C22×D3C2×#15711yescharacteristic
192C3×D4—68no—
193C2×C3×C4—62no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198#67—22no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—91no—
202C2×Dic3—91no—
203C2×Dic3—122no—
204C2×Dic3—122no—
205C2×Dic3—182no—
206C2×Dic3—182no—
207C2×Dic3—362no—
208C2×Dic3—362no—
209C2×Dic3—362no—
210C2×Dic3—362no—
211C2×Dic3C2×#15711yescharacteristic
212C4×D3—181no—
213C4×D3—181no—
214C3×C8—181no—
215C3×C8—181no—
216C3⋊C8—92no—
217C3⋊C8—181no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—22no—
222C2×C32—42no—
223C2×C32—42no—
224C2×C32—42no—
225C2×C32—44no—
226C2×C32—84no—
227C2×C32—121no—
228C2×C32#67⋊C411yescharacteristic
229C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
230C2×C32#90711yescharacteristic
231C3×D3—44no—
232C2×D4—31no—
233C22×C4—91no—
234C22×C4—271no—
235M16—542no—
236C2×C8—91no—
237C2×C8—91no—
238C2×C8—271no—
239C2×C8—271no—
240C2×C8—542no—
241C4⋊C4—271no—
242C22⋊C4—271no—
243C22⋊C4—271no—
244C42—271no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3—44no—
253C22×C3—64no—
254C22×C3—82no—
255C22×C3—82no—
256C22×C3—122no—
257C22×C3—122no—
258C22×C3#379212yescomplemented
259C22×C3C22×#15711yescharacteristic
260C2×D3—21no—
261C2×D3—21no—
262C2×D3—21no—
263C3×C4—64no—
264C3×C4—91no—
265C3×C4—91no—
266C3×C4—181no—
267Dic3—21no—
268Dic3—62no—
269Dic3—62no—
270Dic3—122no—
271Dic3—181no—
272Dic3—181no—
273Dic3—362no—
274Dic3—362no—
275Dic3—722no—
276Dic3—722no—
277C32—22no—
278C32—42no—
279C32#473011yescharacteristic, complemented
280C23—31no—
281C23D3×#15711yescharacteristic
282D4—62no—
283C2×C4—31no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—271no—
289C2×C4—271no—
290C2×C4—541no—
291C2×C4—541no—
292C8—181no—
293C8—181no—
294C8—272no—
295C8—541no—
296C2×C3—22no—
297C2×C3—22no—
298C2×C3—22no—
299C2×C3—22no—
300C2×C3—44no—
301C2×C3—42no—
302C2×C3—42no—
303C2×C3—42no—
304C2×C3—84no—
305C2×C3—124no—
306C2×C3#6428711yescharacteristic
307C2×C3D4×#15711yescharacteristic
308C2×C3C2×#379211yescharacteristic
309D3—41no—
310C22—21no—
311C22—21no—
312C22—61no—
313C22—61no—
314C22—61no—
315C22#8892512yescomplemented
316C22C2×D3×#15711yescharacteristic
317C4—61no—
318C4—91no—
319C4—91no—
320C4—181no—
321C4—541no—
322C4—541no—
323C3—22no—
324C3—42no—
325C3#1059382011yescharacteristic, complemented
326C2—22no—
327C2—121no—
328C2#1172426711yescharacteristic
329C2C2×#8892511yescharacteristic
330C2#67⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC23×C33C2×C4
prime radicalC2×C32×#67C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D3×#157
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊#622C2semidirect
2. C2×#88925C2semidirect
4. C3×C22⋊#622C2semidirect
5. #11722732C2semidirect
6. (C2×D3)⋊#622C2semidirect
7. #11722724C2semidirect
27. C22⋊#622D3semidirect
43. C3⋊#622D4semidirect
70. C22×C33M16semidirect
160. C22×C32ΓL(1,𝔽25)semidirect
186. C33#435semidirect
258. C22×C3#3792semidirect
279. C32#4730semidirect
315. C22#88925semidirect
325. C3#10593820semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C2×#889252
#67×C3⋊Dic32
C3×C22⋊#6222
#117227322
(C2×D3)⋊#6222
#117227242
#105938203
#47309