Finite Group Databasebeta

← #13137404

Subgroups — #13137404

Order 1728.

531 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#11721718C211yesmaximal, characteristic
2C2×#88435C211yesmaximal, characteristic
3C2×#88434C211yesmaximal, characteristic
4#63630⋊C3C211yesmaximal, characteristic, complemented
5C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
6#11720303C211yesmaximal, characteristic, complemented
7#11720302C211yesmaximal, characteristic, complemented
8#10590319—31nomaximal
9#10590318—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12#88542C412yes—
13#88540C412yes—
14C4×#1084C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88435C414yescomplemented
20#88434C414yescomplemented
21C3×#872—31no—
22C3×#872—31no—
23C3×#872—61no—
24C2×#3711—31no—
25C2×#3711—31no—
26C2×D3×C3⋊C8—31no—
27C2×D3×C3⋊C8—31no—
28#63630D311yescharacteristic, complemented
29C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
30C3×#803—31no—
31C3×C4×C3⋊C8—31no—
32#63554—31no—
33#63554—31no—
34Dic3×C3⋊C8—31no—
35Dic3×C3⋊C8—31no—
36C22×#272C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2×C412yes—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084C2×C412yes—
40C2×#1084C2×C411yescharacteristic
41C4×#272C2×C412yes—
42C4×#272C2×C412yes—
43(C3⋊(C3⋊C8))⋊C3C2×C412yes—
44(C3⋊(C3⋊C8))⋊C3C2×C412yes—
45(C3⋊(C3⋊C8))⋊C3C2×C412yes—
46(C3⋊(C3⋊C8))⋊C3C2×C412yes—
47#4997—91no—
48#4708—91no—
49#4689—91no—
50C2×C3×C4×D3—31no—
51C2×C3×C4×D3—31no—
52C2×C3×C4×D3—61no—
53C32×C42—21no—
54C32×C42C2×D311yescharacteristic
55C32×C42C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
60C3×(C2×D3)⋊C4—32no—
61C3×(C2×D3)⋊C4—32no—
62C3×(C2×D3)⋊C4—62no—
63C3×Dic3⋊C4—32no—
64C3×Dic3⋊C4—32no—
65C3×Dic3⋊C4—62no—
66C3×C4×Dic3—31no—
67C3×C4×Dic3—31no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73#3711—34no—
74#3711—34no—
75D3×C3⋊C8—34no—
76D3×C3⋊C8—34no—
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C2×#272—22no—
79C2×#272C4212yes—
80C4×H(3,𝔽3)—22no—
81C4×H(3,𝔽3)C22×C411yescharacteristic
82C4×H(3,𝔽3)C22×C411yescharacteristic
83#1084—22no—
84#1084C4212yes—
85C3×#117—91no—
86C2×#208—91no—
87C22×C3⋊C8—91no—
88C2×ΓL(1,𝔽25)—91no—
89C2×ΓL(1,𝔽25)—91no—
90C2×D3×C8—91no—
91C2×D3×C8—91no—
92#872—31no—
93#872—31no—
94#872—61no—
95C3×#97—91no—
96C3×C4×C8—91no—
97#815—91no—
98#815—91no—
99C8×Dic3—91no—
100C8×Dic3—91no—
101#803—31no—
102#803—91no—
103#803—91no—
104C4×C3⋊C8—31no—
105C4×C3⋊C8—91no—
106C4×C3⋊C8—91no—
107C22×C3×D3—31no—
108C22×C3×D3—31no—
109C22×C3×D3—61no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4C4×D312yes—
113C2×C32×C4C4×D312yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C22×D311yescharacteristic
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C3×C4×D3—32no—
123C3×C4×D3—32no—
124C3×C4×D3—32no—
125C3×C4×D3—32no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3⋊(C3⋊C8)C4×D312yes—
129C3⋊(C3⋊C8)C4×D312yes—
130C3⋊(C3⋊C8)C4×D312yes—
131C3⋊(C3⋊C8)C4×D312yes—
132C3×C3⋊C8—32no—
133C3×C3⋊C8—32no—
134C3×C3⋊C8—32no—
135C3×C3⋊C8—32no—
136#405—271noSylow
137C2×H(3,𝔽3)C2×C4211yescharacteristic
138C2×H(3,𝔽3)#13112yes—
139#272—22no—
140C22×C3×C4—91no—
141C2×C4×D3—31no—
142C2×C4×D3—31no—
143C2×C4×D3—61no—
144C2×C3×C8—91no—
145C2×C3×C8—91no—
146C2×C3×C8—91no—
147C2×C3×C8—91no—
148C3×C4⋊C4—92no—
149C3×C22⋊C4—92no—
150C3×C42—31no—
151C3×C42—31no—
152C3×C42—61no—
153C3×C42—91no—
154C3×C42D3211yescharacteristic
155(C2×D3)⋊C4—32no—
156(C2×D3)⋊C4—32no—
157(C2×D3)⋊C4—62no—
158Dic3⋊C4—32no—
159Dic3⋊C4—32no—
160Dic3⋊C4—62no—
161C4×Dic3—31no—
162C4×Dic3—31no—
163C4×Dic3—61no—
164#208—94no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8—31no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—91no—
171C2×C3⋊C8—91no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—94no—
174ΓL(1,𝔽25)—94no—
175ΓL(1,𝔽25)—94no—
176D3×C8—94no—
177D3×C8—94no—
178C22×C32—21no—
179C22×C32C2×C4×D311yescharacteristic
180C22×C32C2×C4×D311yescharacteristic
181C2×C3×D3—32no—
182C2×C3×D3—32no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—122no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—21no—
190C32×C4—21no—
191C32×C4—42no—
192C32×C4C2×C4×D311yescharacteristic
193C32×C4C2×C4×D311yescharacteristic
194C32×C4C2×C4×D311yescharacteristic
195C32×C4C2×C4×D311yescharacteristic
196C3×Dic3—32no—
197C3×Dic3—32no—
198C3×Dic3—62no—
199C3×Dic3—62no—
200C3×Dic3—62no—
201C3×Dic3—122no—
202C2×M16—271no—
203C22×C8—271no—
204#117—91no—
205#97—91no—
206#97—271no—
207C4×C8—91no—
208C4×C8—271no—
209H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
210C23×C3—91no—
211C22×D3—31no—
212C22×D3—31no—
213C22×D3—61no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—62no—
219C2×C3×C4—61no—
220C2×C3×C4—91no—
221C2×C3×C4—92no—
222C2×C3×C4—92no—
223C2×C3×C4—92no—
224C2×C3×C4#62412yes—
225C2×C3×C4C2×D3211yescharacteristic
226C2×Dic3—31no—
227C2×Dic3—31no—
228C2×Dic3—32no—
229C2×Dic3—32no—
230C2×Dic3—61no—
231C2×Dic3—62no—
232C4×D3—32no—
233C4×D3—32no—
234C4×D3—32no—
235C4×D3—32no—
236C4×D3—62no—
237C4×D3—62no—
238C3×C8—92no—
239C3×C8—92no—
240C3×C8—92no—
241C3×C8—92no—
242C3⋊C8—32no—
243C3⋊C8—32no—
244C3⋊C8—32no—
245C3⋊C8—32no—
246C3⋊C8—92no—
247C3⋊C8—92no—
248C3⋊C8—92no—
249C3⋊C8—92no—
250C2×C32—22no—
251C2×C32—21no—
252C2×C32C42×D311yescharacteristic
253C2×C32C42×D311yescharacteristic
254C2×C32#90112yes—
255C2×C32#90712yes—
256C3×D3—62no—
257C3×D3—62no—
258C3×D3—122no—
259C22×C4—91no—
260M16—274no—
261C2×C8—91no—
262C2×C8—91no—
263C2×C8—91no—
264C2×C8—91no—
265C2×C8—274no—
266C4⋊C4—92no—
267C22⋊C4—92no—
268C42—91no—
269C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—61no—
273C22×C3—92no—
274C22×C3—182no—
275C22×C3C2×#62411yescharacteristic
276C2×D3—32no—
277C2×D3—32no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—62no—
281C2×D3—122no—
282C3×C4—22no—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—62no—
288C3×C4—62no—
289C3×C4—61no—
290C3×C4—61no—
291C3×C4—92no—
292C3×C4—122no—
293C3×C4—182no—
294C3×C4C4×D3211yescharacteristic
295C3×C4C4×D3211yescharacteristic
296Dic3—32no—
297Dic3—32no—
298Dic3—62no—
299Dic3—62no—
300Dic3—62no—
301Dic3—122no—
302C32—21no—
303C32#468911yescharacteristic, complemented
304C32#470811yescharacteristic, complemented
305C23—91no—
306C2×C4—91no—
307C2×C4—92no—
308C2×C4—92no—
309C2×C4—92no—
310C2×C4#619912yes—
311C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
312C8—92no—
313C8—92no—
314C8—92no—
315C8—92no—
316C2×C3—32no—
317C2×C3—32no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—62no—
321C2×C3—61no—
322C2×C3—182no—
323C2×C3#6381712yes—
324C2×C3C4×#62411yescharacteristic
325D3—62no—
326D3—62no—
327D3—122no—
328C22—92no—
329C22—182no—
330C22C2×#619911yescharacteristic
331C4—22no—
332C4—92no—
333C4—182no—
334C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
336C3—31no—
337C3—31no—
338C3—61no—
339C3#1059032111yescharacteristic
340C2—182no—
341C2#1172106412yes—
342C2C4×#619911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#405
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
4. #63630⋊C3C2semidirect
5. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
6. #11720303C2semidirect
7. #11720302C2semidirect
19. #88435C4semidirect
20. #88434C4semidirect
28. #63630D3semidirect
29. C4×C3⋊(C3⋊C8)D3semidirect
209. H(3,𝔽3)#405semidirect
303. C32#4689semidirect
304. C32#4708semidirect

Maximal subgroups

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

subgroupindex
#117217182
C2×#884352
C2×#884342
#63630⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#117203032
#117203022
#105903193
#105903183