Finite Group Databasebeta

← #13166515

Subgroups — #13166515

Order 1728.

500 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 68 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×#88926C211yesmaximal, characteristic, complemented
2C2×#88925C211yesmaximal, characteristic, complemented
3#11723461C211yesmaximal, characteristic
4C4×C3⋊#622C211yesmaximal, characteristic, complemented
5C3×#63779C211yesmaximal, characteristic, complemented
6#11722726C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
8#10593742—31nomaximal
9C2×#6291C411yescharacteristic
10C2×#6289C2211yescharacteristic
11C2×D3×C3⋊Dic3C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16#88926—22no—
17#88925—22no—
18#88859C411yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C32×(C2×D3)⋊C4C411yescharacteristic
21#88807C411yescharacteristic
22C2×#3792—31no—
23C2×#3788—31no—
24#64090—31no—
25#63880—62no—
26C22⋊#622—31no—
27#63779D311yescharacteristic, complemented
28#63778—31no—
29C4×#622—31no—
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C22×C32×D3C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3C2×C411yescharacteristic
37#6291—22no—
38#6289—21no—
39#6289—21no—
40D3×C3⋊Dic3—21no—
41D3×C3⋊Dic3—21no—
42C3⋊#622—21no—
43C3⋊#622D412yescomplemented
44C3×#622—21no—
45C3×#622—21no—
46#4838—91nomaximal
47C2×#638—31no—
48C22×C3⋊Dic3—31no—
49C2×C4×C3⋊D3—31no—
50C2×#626—62no—
51C2×#624—62no—
52C2×D3×Dic3—62no—
53#3792—62no—
54C2×#622—31no—
55C2×#622—31no—
56C2×#622C2×D311yescharacteristic
57C2×#622C2×D311yescharacteristic
58#3788—62no—
59C32×C22⋊C4—31no—
60#3752—31no—
61#3752—62no—
62#3752—122no—
63#3750—31no—
64C4×C3⋊Dic3C2×D311yescharacteristic
65C3×(C2×D3)⋊C4—22no—
66C3×C4×Dic3—62no—
67Dic3⋊Dic3—62no—
68C22×C33C22×C411yescharacteristic
69C2×C3⋊(C3⋊D3)—21no—
70C2×C3⋊(C3⋊D3)—21no—
71C2×C3⋊(C3⋊D3)—21no—
72C2×C32×D3—21no—
73C2×C32×D3—21no—
74C2×C32×D3—21no—
75C33×C4—21no—
76C3⋊(C3⋊Dic3)—21no—
77C3×C3⋊Dic3M1612yescomplemented
78C3×C3⋊Dic3C2×D411yescharacteristic
79C3×C3⋊Dic3#4111yescharacteristic
80C32×Dic3—21no—
81C2×ΓL(1,𝔽25)—91no—
82C2×ΓL(1,𝔽25)—91no—
83#67⋊C4—182no—
84C4×D12—91no—
85C3×C4⋊C8—91no—
86(C2×D3)⋊C8—91no—
87(C2×D3)⋊C8—91no—
88C4×C3⋊C8—91no—
89C23×C32—31no—
90C22×C3⋊D3—31no—
91C22×C3⋊D3—62no—
92C22×C3⋊D3—122no—
93C22×C3×D3—22no—
94C2×C32×C4—22no—
95C2×C32×C4—31no—
96C2×C32×C4—42no—
97C2×C32×C4C4×D311yescharacteristic
98#638—62no—
99C2×C3⋊Dic3—31no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—61no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—122no—
104C2×C3⋊Dic3C4×D311yescharacteristic
105C2×C3⋊Dic3C22×D311yescharacteristic
106C4×C3⋊D3—61no—
107C4×C3⋊D3—61no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111#626—68no—
112#624—122no—
113#624—122no—
114D3×Dic3—122no—
115D3×Dic3—122no—
116#622—21no—
117#622—21no—
118#622—32no—
119#622—61no—
120#436—271noSylow
121C2×C33C4×D411yescharacteristic
122C2×C33C2×M1611yescharacteristic
123C2×C33#13111yescharacteristic
124C3⋊(C3⋊D3)—41no—
125C32×D3—41no—
126C2×#67—182no—
127C22×Dic3—182no—
128C2×D12—91no—
129C2×C4×D3—91no—
130C2×C4×D3—91no—
131C2×C4×D3—182no—
132C2×C3×C8—91no—
133C2×C3×C8—91no—
134C3×C22⋊C4—62no—
135C3×C42—91no—
136(C2×D3)⋊C4—91no—
137(C2×D3)⋊C4—182no—
138(C2×D3)⋊C4—362no—
139(C2×D3)⋊C4#15711yescharacteristic
140C4⋊Dic3—91no—
141Dic3⋊C4—182no—
142C4×Dic3—62no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—91no—
145ΓL(1,𝔽25)—182no—
146ΓL(1,𝔽25)—182no—
147C22×C32—22no—
148C22×C32—42no—
149C22×C32—61no—
150C22×C32—61no—
151C22×C32—61no—
152C22×C32C2×C4×D311yescharacteristic
153C2×C3⋊D3—61no—
154C2×C3⋊D3—64no—
155C2×C3⋊D3—61no—
156C2×C3⋊D3—61no—
157C2×C3⋊D3—124no—
158C2×C3⋊D3—122no—
159C2×C3⋊D3—122no—
160C2×C3⋊D3—242no—
161C2×C3⋊D3—242no—
162C2×C3×D3—24no—
163C2×C3×D3—42no—
164C2×C3×D3—42no—
165C32×C4—21no—
166C32×C4—42no—
167C32×C4—61no—
168C32×C4—82no—
169C3⋊Dic3—61no—
170C3⋊Dic3—122no—
171C3⋊Dic3—242no—
172C3⋊Dic3ΓL(1,𝔽25)12yescomplemented
173C3⋊Dic3D3×D411yescharacteristic
174C3⋊Dic3#23911yescharacteristic
175C3×Dic3—42no—
176C3×Dic3—64no—
177C3×Dic3—62no—
178C3×Dic3—62no—
179C2×M16—271no—
180C2×M16—271no—
181C4×D4—271no—
182C4⋊C8—91no—
183C22⋊C8—271no—
184C22⋊C8—271no—
185C4×C8—271no—
186C33#43611yescharacteristic, complemented, Sylow
187C23×C3—62no—
188C22×D3—91no—
189C22×D3—182no—
190C22×D3—362no—
191C22×D3C2×#15711yescharacteristic
192C2×C3×C4—22no—
193C2×C3×C4—42no—
194C2×C3×C4—62no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4C2×#15711yescharacteristic
198#67—188no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—91no—
202C2×Dic3—182no—
203C2×Dic3—362no—
204C2×Dic3—362no—
205C2×Dic3—362no—
206C2×Dic3C2×#15711yescharacteristic
207D12—182no—
208C4×D3—181no—
209C4×D3—181no—
210C4×D3—181no—
211C4×D3—181no—
212C4×D3—362no—
213C4×D3—362no—
214C3×C8—181no—
215C3×C8—181no—
216C3⋊C8—92no—
217C3⋊C8—181no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—42no—
222C2×C32—42no—
223C2×C32—42no—
224C2×C32—121no—
225C2×C32#89311yescharacteristic
226C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
227C2×C32#90711yescharacteristic
228C3⋊D3—121no—
229C3⋊D3—124no—
230C3⋊D3—244no—
231C3×D3—44no—
232C2×D4—271no—
233C22×C4—271no—
234C22×C4—271no—
235M16—542no—
236M16—542no—
237C2×C8—91no—
238C2×C8—91no—
239C2×C8—271no—
240C2×C8—271no—
241C4⋊C4—271no—
242C22⋊C4—31no—
243C22⋊C4—271no—
244C42—91no—
245C22×C3—22no—
246C22×C3—42no—
247C22×C3—64no—
248C22×C3—122no—
249C22×C3—122no—
250C22×C3C22×#15711yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—21no—
254C2×D3—184no—
255C2×D3—181no—
256C2×D3—181no—
257C2×D3—181no—
258C2×D3—362no—
259C2×D3—362no—
260C2×D3—364no—
261C2×D3—722no—
262C2×D3—722no—
263C3×C4—21no—
264C3×C4—42no—
265C3×C4—82no—
266C3×C4—92no—
267C3×C4—91no—
268C3×C4—91no—
269C3×C4—122no—
270Dic3—21no—
271Dic3—62no—
272Dic3—62no—
273Dic3—64no—
274Dic3—181no—
275Dic3—362no—
276Dic3—722no—
277C32—22no—
278C32—42no—
279C32#483811yescharacteristic, complemented
280C23—31no—
281C23—271no—
282D4—542no—
283C2×C4—31no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—271no—
287C2×C4—541no—
288C2×C4—541no—
289C2×C4—541no—
290C2×C4—541no—
291C2×C4D3×#15711yescharacteristic
292C8—181no—
293C8—181no—
294C8—272no—
295C8—541no—
296C2×C3—22no—
297C2×C3—22no—
298C2×C3—22no—
299C2×C3—42no—
300C2×C3—42no—
301C2×C3—42no—
302C2×C3—124no—
303C2×C3#6428311yescharacteristic
304C2×C3#6428711yescharacteristic
305C2×C3D4×#15711yescharacteristic
306D3—41no—
307D3—364no—
308D3—361no—
309D3—724no—
310C22—61no—
311C22—61no—
312C22—61no—
313C22—541no—
314C22—541no—
315C22—541no—
316C22C2×D3×#15711yescharacteristic
317C4—21no—
318C4—61no—
319C4—92no—
320C4—91no—
321C4—91no—
322C4—541no—
323C3—22no—
324C3—42no—
325C3#1059374211yescharacteristic, complemented
326C2—121no—
327C2—1081no—
328C2#1172425411yescharacteristic
329C2#1172425911yescharacteristic
330C2#1172426711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×#6291C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88926C2semidirect
2. C2×#88925C2semidirect
4. C4×C3⋊#622C2semidirect
5. C3×#63779C2semidirect
6. #11722726C2semidirect
7. (C2×D3)⋊#622C2semidirect
27. #63779D3semidirect
43. C3⋊#622D4semidirect
77. C3×C3⋊Dic3M16semidirect
172. C3⋊Dic3ΓL(1,𝔽25)semidirect
186. C33#436semidirect
279. C32#4838semidirect
325. C3#10593742semidirect

Maximal subgroups

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

subgroupindex
C2×#889262
C2×#889252
#117234612
C4×C3⋊#6222
C3×#637792
#117227262
(C2×D3)⋊#6222
#105937423
#48389