Finite Group Databasebeta

← #13155285

Subgroups — #13155285

Order 1728.

866 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C4×#6289C211yesmaximal, characteristic
2C2×#88788C212yesmaximal
3#11722071C211yesmaximal, characteristic, complemented
4C3×#63555C211yesmaximal, characteristic, complemented
5C3×Dic3×C3⋊C8C212yesmaximal, complemented
6#10590664—32nomaximal
7C4×#3712—31nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×#6289C412yes—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4×Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14Dic3×C3⋊Dic3C412yes—
15#88788C418yescomplemented
16C2×#3743—32no—
17C42×C3⋊D3—31no—
18C4×#624—32no—
19C4×#624—61no—
20C2×#3712—32no—
21C2×#3712—32no—
22C2×#3712—62no—
23C3×#815—32no—
24C3×C8×Dic3—32no—
25C3×C4×C3⋊C8—32no—
26#63555—31no—
27#63555D311yescharacteristic, complemented
28#63554—32no—
29Dic3×C3⋊C8D312yescomplemented
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
33C4×C3⋊(C3⋊D3)C2×C414yes—
34C2×C3×C3⋊Dic3C2×C412yes—
35C2×C32×Dic3C2×C412yes—
36#6289—24no—
37C3×C3⋊(C3⋊C8)C2×C414yescomplemented
38C32×C3⋊C8C2×C414yescomplemented
39#4707—91no—
40C4×ΓL(1,𝔽25)—92no—
41C2×C4×C3⋊D3—31no—
42C2×C4×C3⋊D3—32no—
43C2×C4×C3⋊D3—32no—
44C2×C4×C3⋊D3—62no—
45C2×C4×C3⋊D3—61no—
46C2×C4×C3⋊D3—121no—
47C2×#624—34no—
48C2×#624—62no—
49C2×C32×C8—32no—
50C32×C42—31no—
51C4×C3⋊Dic3—32no—
52C4×C3⋊Dic3C2×D311yescharacteristic
53C2×C3⋊(C3⋊C8)C2×D312yes—
54#3743—38no—
55C3×C4×Dic3—21no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—61no—
58C3×C4×Dic3C2×D312yes—
59C2×C3×C3⋊C8—22no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—62no—
63C2×C3×C3⋊C8C2×D312yes—
64C2×C3×C3⋊C8C2×D312yes—
65Dic32—34no—
66Dic32—62no—
67#3712—38no—
68#3712—38no—
69#3712—68no—
70C22×C33C22×C411yescharacteristic
71C2×C3⋊(C3⋊D3)—22no—
72C2×C3⋊(C3⋊D3)C4212yes—
73C33×C4C22×C412yes—
74C3⋊(C3⋊Dic3)C4212yes—
75C3×C3⋊Dic3—22no—
76C32×Dic3M1614yescomplemented
77C2×ΓL(1,𝔽25)—92no—
78C2×ΓL(1,𝔽25)—92no—
79C2×ΓL(1,𝔽25)—92no—
80C2×ΓL(1,𝔽25)—182no—
81C42×D3—92no—
82C42×D3—91no—
83C42×D3—181no—
84C3×#97—91no—
85C3×C4×C8—92no—
86#815—32no—
87#815—92no—
88C8×Dic3—32no—
89#803—91no—
90C4×C3⋊C8—32no—
91C22×C3⋊D3—32no—
92C22×C3⋊D3—31no—
93C22×C3⋊D3—61no—
94C22×C3⋊D3—62no—
95C22×C3⋊D3—121no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—32no—
99C2×C32×C4—41no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C32×C4C22×D312yes—
102C2×C3⋊Dic3—32no—
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—61no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—121no—
107C2×C3⋊Dic3C4×D312yes—
108C4×C3⋊D3—34no—
109C4×C3⋊D3—34no—
110C4×C3⋊D3—34no—
111C4×C3⋊D3—64no—
112C4×C3⋊D3—64no—
113C4×C3⋊D3—64no—
114C4×C3⋊D3—64no—
115C4×C3⋊D3—124no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—34no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3C4×D314yes—
120#624—68no—
121#624—68no—
122C32×C8—34no—
123C3⋊(C3⋊C8)C4×D314yescomplemented
124C3×C3⋊C8—24no—
125C3×C3⋊C8—34no—
126C3×C3⋊C8—34no—
127C3×C3⋊C8—64no—
128C3×C3⋊C8C4×D314yescomplemented
129C3×C3⋊C8C4×D314yescomplemented
130C4×M16—271noSylow
131C2×C33C2×C4211yescharacteristic
132C2×C33C2×M1612yes—
133C3⋊(C3⋊D3)—22no—
134C2×C4×D3—91no—
135C2×C4×D3—92no—
136C2×C4×D3—94no—
137C2×C4×D3—92no—
138C2×C4×D3—182no—
139C2×C4×D3—181no—
140C2×C4×D3—182no—
141C2×C4×D3—361no—
142C2×C3×C8—32no—
143C2×C3×C8—32no—
144C2×C3×C8—62no—
145C2×C3×C8—92no—
146C3×C42—32no—
147C3×C42—61no—
148C3×C42—91no—
149C4×Dic3—32no—
150C4×Dic3—61no—
151C4×Dic3—94no—
152C4×Dic3—92no—
153C4×Dic3—182no—
154C4×Dic3D3211yescharacteristic
155C2×C3⋊C8—32no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—62no—
158C2×C3⋊C8D3212yes—
159ΓL(1,𝔽25)—98no—
160ΓL(1,𝔽25)—98no—
161ΓL(1,𝔽25)—98no—
162ΓL(1,𝔽25)—188no—
163C22×C32—22no—
164C22×C32—21no—
165C22×C32—41no—
166C22×C32C2×C4×D311yescharacteristic
167C22×C32C2×C4×D312yes—
168C2×C3⋊D3—32no—
169C2×C3⋊D3—34no—
170C2×C3⋊D3—62no—
171C2×C3⋊D3—64no—
172C2×C3⋊D3—62no—
173C2×C3⋊D3—64no—
174C2×C3⋊D3—64no—
175C2×C3⋊D3—122no—
176C2×C3⋊D3—124no—
177C2×C3⋊D3—124no—
178C32×C4—22no—
179C32×C4—22no—
180C32×C4—22no—
181C32×C4—34no—
182C32×C4—42no—
183C32×C4C2×C4×D312yes—
184C32×C4C2×C4×D312yes—
185C32×C4C2×C4×D312yes—
186C3⋊Dic3—22no—
187C3⋊Dic3—34no—
188C3⋊Dic3—32no—
189C3⋊Dic3—64no—
190C3⋊Dic3—62no—
191C3⋊Dic3—122no—
192C3×Dic3—24no—
193C3×Dic3—64no—
194C3×Dic3—64no—
195C3×Dic3ΓL(1,𝔽25)18yescomplemented
196C2×M16—272no—
197C2×C42—271no—
198#97—91no—
199#97—271no—
200C4×C8—92no—
201C33C4×M1611yescharacteristic, complemented, Sylow
202C22×D3—92no—
203C22×D3—91no—
204C22×D3—181no—
205C22×D3—182no—
206C22×D3—361no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—34no—
210C2×C3×C4—41no—
211C2×C3×C4—62no—
212C2×C3×C4—92no—
213C2×C3×C4C2×D3211yescharacteristic
214C2×C3×C4C2×D3212yes—
215C2×Dic3—34no—
216C2×Dic3—62no—
217C2×Dic3—92no—
218C2×Dic3—91no—
219C2×Dic3—181no—
220C2×Dic3—182no—
221C2×Dic3—361no—
222C2×Dic3#62412yes—
223C4×D3—94no—
224C4×D3—94no—
225C4×D3—94no—
226C4×D3—184no—
227C4×D3—184no—
228C4×D3—184no—
229C4×D3—188no—
230C4×D3—188no—
231C4×D3—184no—
232C4×D3—364no—
233C3×C8—34no—
234C3×C8—34no—
235C3×C8—64no—
236C3×C8—94no—
237C3⋊C8—34no—
238C3⋊C8—34no—
239C3⋊C8—64no—
240C3⋊C8#62414yescomplemented
241C2×C32—24no—
242C2×C32—22no—
243C2×C32—21no—
244C2×C32—22no—
245C2×C32—41no—
246C2×C32—42no—
247C2×C32C42×D312yes—
248C2×C32C42×D311yescharacteristic
249C2×C32C2×ΓL(1,𝔽25)14yes—
250C2×C32D3×M1612yes—
251C3⋊D3—62no—
252C3⋊D3—64no—
253C3⋊D3—64no—
254C3⋊D3—124no—
255C3⋊D3—124no—
256C22×C4—272no—
257C22×C4—271no—
258M16—278no—
259C2×C8—32no—
260C2×C8—92no—
261C42—31no—
262C42—91no—
263C42—272no—
264C22×C3—21no—
265C22×C3—22no—
266C22×C3—41no—
267C22×C3C4×D3212yes—
268C22×C3C2×#62411yescharacteristic
269C2×D3—92no—
270C2×D3—94no—
271C2×D3—182no—
272C2×D3—184no—
273C2×D3—184no—
274C2×D3—184no—
275C2×D3—182no—
276C2×D3—362no—
277C2×D3—364no—
278C2×D3—364no—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—38no—
283C3×C4—42no—
284C3×C4—64no—
285C3×C4—182no—
286C3×C4C4×D3212yes—
287C3×C4C4×D3212yes—
288C3×C4C2×#62412yes—
289Dic3—64no—
290Dic3—64no—
291Dic3—92no—
292Dic3—94no—
293Dic3—184no—
294Dic3—182no—
295Dic3—362no—
296Dic3#371214yescomplemented
297C32—21no—
298C32—22no—
299C32—41no—
300C32C4×ΓL(1,𝔽25)12yescomplemented
301C32#470711yescharacteristic, complemented
302C23—271no—
303C2×C4—32no—
304C2×C4—92no—
305C2×C4—271no—
306C2×C4—274no—
307C2×C4—544no—
308C2×C4D3311yescharacteristic
309C8—34no—
310C8—94no—
311C2×C3—22no—
312C2×C3—24no—
313C2×C3—21no—
314C2×C3—22no—
315C2×C3—42no—
316C2×C3—41no—
317C2×C3D3×ΓL(1,𝔽25)14yes—
318C2×C3C2×#371212yes—
319C2×C3C4×#62411yescharacteristic
320C2×C3C4×#62412yes—
321D3—184no—
322D3—182no—
323D3—184no—
324D3—364no—
325D3—364no—
326C22—272no—
327C22—542no—
328C22D3×#62411yescharacteristic
329C4—34no—
330C4—182no—
331C4—272no—
332C4D3×#62412yes—
333C3—21no—
334C3—22no—
335C3—41no—
336C3C4×#371211yescharacteristic, complemented
337C3#1059066412yescomplemented
338C2—542no—
339C2D3×#371212yes—
340C2#1172290011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutatorC2×C33C2×C42
Fitting, cosocleC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×C3⋊(C3⋊D3)C2×C4
hypocenterC33C4×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722071C2semidirect
4. C3×#63555C2semidirect
5. C3×Dic3×C3⋊C8C2semidirect
15. #88788C4semidirect
27. #63555D3semidirect
29. Dic3×C3⋊C8D3semidirect
37. C3×C3⋊(C3⋊C8)C2×C4semidirect
38. C32×C3⋊C8C2×C4semidirect
76. C32×Dic3M16semidirect
123. C3⋊(C3⋊C8)C4×D3semidirect
128. C3×C3⋊C8C4×D3semidirect
129. C3×C3⋊C8C4×D3semidirect
195. C3×Dic3ΓL(1,𝔽25)semidirect
201. C33C4×M16semidirect
240. C3⋊C8#624semidirect
296. Dic3#3712semidirect
300. C32C4×ΓL(1,𝔽25)semidirect
301. C32#4707semidirect
336. C3C4×#3712semidirect
337. C3#10590664semidirect

Maximal subgroups

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

subgroupindex
C4×#62892
C2×#887882
#117220712
C3×#635552
C3×Dic3×C3⋊C82
#105906643
C4×#37123