Finite Group Databasebeta

← #13155004

Subgroups — #13155004

Order 1728.

404 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 67 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
2C2×#88809C211yesmaximal, characteristic
3C3×#63900C211yesmaximal, characteristic
4#10590052—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C2×C3×#624C412yes—
7C2×C3⋊#622—21no—
8#88989—21no—
9#88989C412yescomplemented
10C2×C3×C3⋊(C3⋊C8)—21no—
11C32×C4⋊Dic3—21no—
12#88809—21no—
13#88809C412yescomplemented
14C3×#3724—21no—
15C2×#3788—31no—
16C3×D3×C4⋊C4—61no—
17#63900D311yescharacteristic
18C2×#3712—31no—
19C2×#3711—61no—
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3Q811yescharacteristic
27C2×C32×Dic3—22no—
28C3×#624—42no—
29C3⋊#622—22no—
30C3×C3⋊(C3⋊C8)—22no—
31#4530—91nomaximal
32C2×C4×C3⋊D3C2×D311yescharacteristic
33C2×C3×C4×D3—61no—
34C2×C3×C4×D3—61no—
35C2×C3×C4×D3—62no—
36C2×#624Dic312yes—
37C2×#622—61no—
38#3788—32no—
39#3788—61no—
40C32×C4⋊C4—61no—
41C2×C3⋊(C3⋊C8)—21no—
42C3×C4⋊Dic3—21no—
43C3×C4⋊Dic3—21no—
44C3×C4⋊Dic3—41no—
45C3×Dic3⋊C4—62no—
46C2×C3×C3⋊C8—61no—
47C2×C3×C3⋊C8—61no—
48C2×C3×C3⋊C8—121no—
49#3724—21no—
50#3712—32no—
51#3712—61no—
52#3711—62no—
53#3711—62no—
54C22×C33C22⋊C411yescharacteristic
55C2×C3×C3⋊D3—21no—
56C2×C3×C3⋊D3—21no—
57C2×C3×C3⋊D3C4211yescharacteristic
58C2×C3×C3⋊D3C4⋊C411yescharacteristic
59C33×C4C22⋊C411yescharacteristic
60C33×C4C4⋊C411yescharacteristic
61C3×C3⋊Dic3C22⋊C411yescharacteristic
62C3×C3⋊Dic3C4⋊C411yescharacteristic
63C32×Dic3—42no—
64C2×C3×C4⋊C4—91no—
65C2×#208—91no—
66C2×#208—91no—
67C2×ΓL(1,𝔽25)—181no—
68D3×C4⋊C4—61no—
69C22×C3⋊D3C2×Dic311yescharacteristic
70C22×C3×D3—61no—
71C22×C3×D3—61no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4—62no—
77C2×C32×C4#6711yescharacteristic
78C2×C3⋊Dic3#6711yescharacteristic
79C4×C3⋊D3Dic611yescharacteristic
80C4×C3⋊D3C4×D311yescharacteristic
81C4×C3⋊D3C4×D311yescharacteristic
82C4×C3⋊D3D1211yescharacteristic
83C2×C3×Dic3—22no—
84C2×C3×Dic3—22no—
85C2×C3×Dic3—42no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92C3×C4×D3—61no—
93C3×C4×D3—61no—
94C3×C4×D3—61no—
95C3×C4×D3—61no—
96C3×C4×D3—124no—
97#624—42no—
98#622—62no—
99C3⋊(C3⋊C8)—22no—
100C3×C3⋊C8—62no—
101C3×C3⋊C8—62no—
102C3×C3⋊C8—122no—
103#343—271noSylow
104C2×C33#9511yescharacteristic
105C2×C33#10011yescharacteristic
106C2×C33#10111yescharacteristic
107C3×C3⋊D3—21no—
108C3×C3⋊D3—21no—
109C22×C3×C4—92no—
110C22×C3×C4—91no—
111C2×C4×D3—61no—
112C2×C4×D3—61no—
113C2×C4×D3—62no—
114C2×C3×C8—181no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4—61no—
117C3×C4⋊C4—121no—
118C3×C4⋊C4—181no—
119C4⋊Dic3—21no—
120Dic3⋊C4—62no—
121#208—92no—
122#208—92no—
123#208—181no—
124#208—181no—
125C2×C3⋊C8—61no—
126C2×C3⋊C8—61no—
127C2×C3⋊C8—121no—
128C2×C3⋊C8—181no—
129ΓL(1,𝔽25)—182no—
130ΓL(1,𝔽25)—182no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32#21711yescharacteristic
136C2×C3⋊D3—21no—
137C2×C3⋊D3—21no—
138C2×C3⋊D3C4×Dic311yescharacteristic
139C2×C3⋊D3C4⋊Dic311yescharacteristic
140C2×C3×D3—61no—
141C2×C3×D3—61no—
142C2×C3×D3—61no—
143C2×C3×D3—61no—
144C2×C3×D3—62no—
145C2×C3×D3—62no—
146C2×C3×D3—121no—
147C2×C3×D3—121no—
148C32×C4—21no—
149C32×C4—21no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4—41no—
153C32×C4—41no—
154C32×C4—41no—
155C32×C4—41no—
156C32×C4—122no—
157C32×C4Dic3⋊C411yescharacteristic
158C32×C4(C2×D3)⋊C411yescharacteristic
159C3⋊Dic3Dic3⋊C411yescharacteristic
160C3⋊Dic3(C2×D3)⋊C411yescharacteristic
161C3×Dic3—42no—
162C3×Dic3—42no—
163C3×Dic3—61no—
164C3×Dic3—61no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—82no—
168C2×M16—271no—
169C2×M16—271no—
170C2×C4⋊C4—91no—
171C33#34311yescharacteristic, complemented, Sylow
172C23×C3—91no—
173C22×D3—61no—
174C22×D3—61no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4—91no—
182C2×C3×C4—91no—
183C2×C3×C4—91no—
184C2×C3×C4—91no—
185C2×C3×C4—91no—
186C2×C3×C4—122no—
187C2×C3×C4—362no—
188C2×C3×C4GO+(4,𝔽2)11yescharacteristic
189C2×Dic3—22no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192C4×D3—61no—
193C4×D3—61no—
194C4×D3—61no—
195C4×D3—61no—
196C4×D3—61no—
197C4×D3—61no—
198C4×D3—61no—
199C4×D3—61no—
200C4×D3—124no—
201C3×C8—182no—
202C3⋊C8—62no—
203C3⋊C8—62no—
204C3⋊C8—122no—
205C3⋊C8—182no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32C3⋊#9511yescharacteristic
219C2×C32#83311yescharacteristic
220C2×C32#83611yescharacteristic
221C3⋊D3—21no—
222C3⋊D3—21no—
223C3×D3—62no—
224C3×D3—62no—
225C3×D3—121no—
226C3×D3—121no—
227C22×C4—92no—
228C22×C4—91no—
229M16—272no—
230M16—272no—
231M16—541no—
232M16—541no—
233C2×C8—181no—
234C2×C8—541no—
235C4⋊C4—61no—
236C4⋊C4—181no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—91no—
242C22×C3—91no—
243C22×C3—181no—
244C22×C3—181no—
245C22×C3D32⋊C411yescharacteristic
246C2×D3—61no—
247C2×D3—61no—
248C2×D3—61no—
249C2×D3—61no—
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—121no—
253C2×D3—121no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—91no—
263C3×C4—91no—
264C3×C4—122no—
265C3×C4—122no—
266C3×C4—242no—
267C3×C4D32⋊C411yescharacteristic
268C3×C4#377311yescharacteristic
269Dic3—42no—
270Dic3—61no—
271Dic3—61no—
272Dic3—61no—
273Dic3—61no—
274C32—21no—
275C32—21no—
276C32—41no—
277C32—41no—
278C32#453011yescharacteristic, complemented
279C23—91no—
280C2×C4—62no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—362no—
287C2×C4D32⋊D311yescharacteristic
288C8—182no—
289C8—542no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—181no—
303C2×C3—181no—
304C2×C3#6372711yescharacteristic
305C2×C3#6372811yescharacteristic
306C2×C3#6373711yescharacteristic
307D3—62no—
308D3—62no—
309D3—121no—
310D3—121no—
311C22—91no—
312C22—91no—
313C22—181no—
314C22—181no—
315C22D32⋊Dic311yescharacteristic
316C4—91no—
317C4—91no—
318C4—122no—
319C4#8893211yescharacteristic
320C4#8893411yescharacteristic
321C3—21no—
322C3—21no—
323C3—41no—
324C3—41no—
325C3#1059005211yescharacteristic, complemented
326C2—181no—
327C2—181no—
328C2#1172275411yescharacteristic
329C2#1172275611yescharacteristic
330C2#1172276911yescharacteristic
subgroupG/H
centerC22D32⋊Dic3
commutatorC2×C3×C3⋊D3C42
Frattini, hypercenterC2×C4D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#343
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
9. #88989C4semidirect
13. #88809C4semidirect
171. C33#343semidirect
278. C32#4530semidirect
325. C3#10590052semidirect

Maximal subgroups

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

subgroupindex
C2×#889892
C2×#888092
C3×#639002
#105900523
#45309