Finite Group Databasebeta

← #13155742

Subgroups — #13155742

Order 1728.

614 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 96 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#11723660C211yesmaximal, characteristic
2#11723354C212yesmaximal, complemented
3Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C3×#63578C211yesmaximal, characteristic, complemented
5C3×#63570C212yesmaximal, complemented
6#10590977—32nomaximal
7#10590389—31nomaximal
8C3×#3828C2212yes—
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×C4×Dic3C2211yescharacteristic
12C32×#208C2212yes—
13#88812C411yescharacteristic, complemented
14#88811C412yescomplemented
15#88810C411yescharacteristic, complemented
16#88787—22no—
17C3⋊D3×C3⋊C8—22no—
18#64116—32no—
19#63829—32no—
20#63822—31no—
21#63815—32no—
22#63815—32no—
23#63815—62no—
24C4×C3⋊(C3⋊C8)—31no—
25C3×#825—32no—
26C3×#822—32no—
27C3×#805—32no—
28#63578D311yescharacteristic, complemented
29#63570D312yescomplemented
30Dic3×C3⋊C8—32no—
31Dic3×C3⋊C8—61no—
32C2×C33×C4C2311yescharacteristic
33C3×#638—22no—
34C3×C4⋊(C3⋊D3)C2×C412yes—
35C3×C4×C3⋊D3—22no—
36C3×#634C2×C412yes—
37C2×C32×Dic3—21no—
38C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
39C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
40C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
41C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
42C32×C3⋊C8—22no—
43#4895—91no—
44#4824—92no—
45#3828C2×D312yes—
46C3×#235—32no—
47C3×#235—32no—
48C3×#235—62no—
49C32×M16—32no—
50C32×C42—31no—
51C2×C3⋊(C3⋊C8)—32no—
52C2×C3⋊(C3⋊C8)—31no—
53C2×C3⋊(C3⋊C8)—61no—
54C2×C3⋊(C3⋊C8)—62no—
55C2×C3⋊(C3⋊C8)—121no—
56C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
57#3743—62no—
58C8×C3⋊D3—62no—
59C3×C4×Dic3—21no—
60C3×C4×Dic3C2×D312yes—
61C3×#208—22no—
62C3×#208C2×D312yes—
63C3×#208C2×D312yes—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—61no—
66#3719—32no—
67#3718—31no—
68#3717—32no—
69#3716—32no—
70#3715—32no—
71#3714—32no—
72#3713—31no—
73#3711—62no—
74#3711—62no—
75#3711—64no—
76D3×C3⋊C8—62no—
77D3×C3⋊C8—62no—
78D3×C3⋊C8—64no—
79C22×C33#4111yescharacteristic
80C2×C3×C3⋊D3—22no—
81C33×C4C22×C411yescharacteristic
82C33×C4C2×D411yescharacteristic
83C3×C3⋊Dic3—22no—
84C32×Dic3—22no—
85#950—92no—
86#948—92no—
87#907—92no—
88#907—92no—
89#907—182no—
90#904—91no—
91C3×#108—91no—
92C3×#104—92no—
93#825—32no—
94#822—32no—
95C8×Dic3—91no—
96#805—32no—
97C4×C3⋊C8—92no—
98C4×C3⋊C8—181no—
99C2×C32×C4—21no—
100C2×C32×C4—22no—
101C2×C32×C4—41no—
102C2×C32×C4—61no—
103C2×C32×C4C22×D312yes—
104C2×C32×C4C22×D311yescharacteristic
105#638—22no—
106C4⋊(C3⋊D3)C4×D312yes—
107C4×C3⋊D3—22no—
108#634C4×D312yes—
109C3×#67—64no—
110C3×#67—62no—
111C3×#67—62no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—22no—
114C3×D12—32no—
115C3×D12—32no—
116C3×D12—62no—
117C3×C4×D3—64no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120C3×Dic6—32no—
121C3×Dic6—32no—
122C3×Dic6—62no—
123C32×C8—62no—
124C3⋊(C3⋊C8)—31no—
125C3⋊(C3⋊C8)—32no—
126C3⋊(C3⋊C8)—31no—
127C3⋊(C3⋊C8)—32no—
128C3⋊(C3⋊C8)—62no—
129C3⋊(C3⋊C8)—62no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—61no—
132C3⋊(C3⋊C8)—121no—
133C3⋊(C3⋊C8)—121no—
134C3⋊(C3⋊C8)C4×D311yescharacteristic
135C3⋊(C3⋊C8)C4×D311yescharacteristic
136C3×C3⋊C8—24no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—32no—
140C3×C3⋊C8—32no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143#443—271noSylow
144C2×C33—21no—
145C2×C33C4×D411yescharacteristic
146C3×C3⋊D3—42no—
147C3×#41—92no—
148#235—32no—
149#235—32no—
150#235—62no—
151C3×M16—32no—
152C3×M16—32no—
153C3×M16—62no—
154C2×C3×C8—91no—
155C3×C42—32no—
156C3×C42—61no—
157#216—92no—
158#215—92no—
159#214—92no—
160#213—92no—
161C4×Dic3D3211yescharacteristic
162#208—182no—
163#208D3212yes—
164C2×C3⋊C8—32no—
165C2×C3⋊C8—61no—
166C2×C3⋊C8—92no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—181no—
169C2×C3⋊C8—182no—
170C2×C3⋊C8—182no—
171C2×C3⋊C8—361no—
172Dic12—91no—
173D24—91no—
174#204—92no—
175ΓL(1,𝔽25)—182no—
176ΓL(1,𝔽25)—182no—
177ΓL(1,𝔽25)—184no—
178D3×C8—182no—
179D3×C8—182no—
180D3×C8—184no—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—41no—
184C22×C32#23712yes—
185C22×C32#23911yescharacteristic
186C2×C3⋊D3—22no—
187C2×C3×D3—64no—
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C32×C4—21no—
191C32×C4—22no—
192C32×C4—21no—
193C32×C4—22no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—62no—
197C32×C4C2×C4×D312yes—
198C32×C4C2×C4×D311yescharacteristic
199C32×C4D3×D412yes—
200C32×C4D3×D411yescharacteristic
201C3⋊Dic3—22no—
202C3×Dic3—24no—
203C3×Dic3—24no—
204C3×Dic3—64no—
205C3×Dic3—62no—
206C3×Dic3—62no—
207#135—271no—
208#131—272no—
209#108—91no—
210#104—92no—
211C4×C8—271no—
212C33#44311yescharacteristic, complemented, Sylow
213C3×Q8—92no—
214C3×D4—92no—
215C3×D4—182no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—182no—
222C2×C3×C4C2×D3211yescharacteristic
223C2×C3×C4C2×D3212yes—
224#67—64no—
225#67—62no—
226#67—62no—
227C2×Dic3—21no—
228D12—32no—
229D12—32no—
230D12—62no—
231C4×D3—64no—
232C4×D3—62no—
233C4×D3—62no—
234Dic6—32no—
235Dic6—32no—
236Dic6—62no—
237C3×C8—64no—
238C3×C8—62no—
239C3×C8—62no—
240C3×C8—91no—
241C3×C8—91no—
242C3⋊C8—22no—
243C3⋊C8—32no—
244C3⋊C8—32no—
245C3⋊C8—61no—
246C3⋊C8—61no—
247C3⋊C8—91no—
248C3⋊C8—91no—
249C3⋊C8—92no—
250C3⋊C8—92no—
251C3⋊C8—182no—
252C3⋊C8—182no—
253C3⋊C8—181no—
254C3⋊C8—181no—
255C3⋊C8—361no—
256C3⋊C8—361no—
257C2×C32—21no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—42no—
264C2×C32—42no—
265C2×C32#67⋊C412yes—
266C2×C32#89311yescharacteristic
267C3⋊D3—42no—
268C3×D3—124no—
269C3×D3—122no—
270C3×D3—122no—
271#41—92no—
272Q16—271no—
273SD16—272no—
274D8—271no—
275M16—32no—
276M16—542no—
277C2×C8—91no—
278C2×C8—271no—
279C2×C8—542no—
280C42—31no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3—182no—
285C22×C3#379612yes—
286C22×C3#380511yescharacteristic
287C2×D3—64no—
288C2×D3—62no—
289C2×D3—62no—
290C3×C4—21no—
291C3×C4—22no—
292C3×C4—22no—
293C3×C4—21no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—64no—
297C3×C4—64no—
298C3×C4—182no—
299C3×C4C4×D3212yes—
300C3×C4C4×D3211yescharacteristic
301C3×C4#380211yescharacteristic
302C3×C4#381112yes—
303Dic3—22no—
304Dic3—64no—
305Dic3—62no—
306Dic3—62no—
307C32—22no—
308C32—21no—
309C32—41no—
310C32#482412yescomplemented
311C32#489511yescharacteristic, complemented
312Q8—92no—
313D4—92no—
314D4—182no—
315C2×C4—61no—
316C2×C4—182no—
317C2×C4D3311yescharacteristic
318C8—62no—
319C8—91no—
320C8—91no—
321C8—271no—
322C8—271no—
323C2×C3—22no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—42no—
329C2×C3—42no—
330C2×C3—41no—
331C2×C3—362no—
332C2×C3#6388112yes—
333C2×C3#6390211yescharacteristic
334D3—124no—
335D3—122no—
336D3—122no—
337C22—182no—
338C22#8896411yescharacteristic
339C4—62no—
340C4—182no—
341C4#8894911yescharacteristic
342C4#8895411yescharacteristic
343C3—22no—
344C3—21no—
345C3—41no—
346C3#1059038911yescharacteristic, complemented
347C3#1059097712yescomplemented
348C2—21no—
349C2—362no—
350C2#1172293811yescharacteristic
subgroupG/H
centerC4#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radical#11723660C2
hypocenterC33#443
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723354C2semidirect
3. Dic3×C3⋊(C3⋊C8)C2semidirect
4. C3×#63578C2semidirect
5. C3×#63570C2semidirect
13. #88812C4semidirect
14. #88811C4semidirect
15. #88810C4semidirect
28. #63578D3semidirect
29. #63570D3semidirect
212. C33#443semidirect
310. C32#4824semidirect
311. C32#4895semidirect
346. C3#10590389semidirect
347. C3#10590977semidirect

Maximal subgroups

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

subgroupindex
#117236602
#117233542
Dic3×C3⋊(C3⋊C8)2
C3×#635782
C3×#635702
#105909773
#105903893