Finite Group Databasebeta

← #13155745

Subgroups — #13155745

Order 1728.

602 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 108 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#11723350C212yesmaximal, complemented
2C3×#63822C211yesmaximal, characteristic
3#11722081C211yesmaximal, characteristic, complemented
4#11722019C211yesmaximal, characteristic, complemented
5C3×#63569C212yesmaximal, complemented
6#10591676—32nomaximal
7#10590390—31nomaximal
8C32×#235C2212yes—
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C3×#3748C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4×Dic3C2211yescharacteristic
13#88786—22no—
14D3×C3⋊(C3⋊C8)—22no—
15C3×#3718C411yescharacteristic, complemented
16C3×#3715C412yescomplemented
17C3×#3713C411yescharacteristic, complemented
18#64157—32no—
19C3×#950—32no—
20#63822D311yescharacteristic
21#63815—32no—
22#63815—32no—
23#63815—62no—
24#63630—31no—
25C3×#837—32no—
26C3×#805—32no—
27#63578—31no—
28#63577—32no—
29#63569D312yescomplemented
30#63554—32no—
31#63554—61no—
32C2×C33×C4C2311yescharacteristic
33C32×#67—22no—
34C2×C32×Dic3—21no—
35C32×D12C2×C412yes—
36C32×C4×D3—22no—
37C32×Dic6C2×C412yes—
38C3⋊(C3⋊(C3⋊C8))D412yes—
39C3×C3⋊(C3⋊C8)—22no—
40C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
41C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
42#5197—91no—
43#4823—92no—
44C32×#41—32no—
45C3×#235—22no—
46C3×#235C2×D312yes—
47C3×#235C2×D312yes—
48C32×C42—31no—
49#3748—62no—
50#3748C2×D312yes—
51C2×C3⋊(C3⋊C8)—32no—
52C2×C3⋊(C3⋊C8)—31no—
53C2×C3⋊(C3⋊C8)—62no—
54C2×C3⋊(C3⋊C8)—61no—
55C2×C3⋊(C3⋊C8)—62no—
56C2×C3⋊(C3⋊C8)—121no—
57C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
58C3×#216—32no—
59C3×#215—32no—
60C3×#214—32no—
61C3×#213—32no—
62C3×C4×Dic3—21no—
63C3×C4×Dic3C2×D312yes—
64C3×#208—32no—
65C3×#208—32no—
66C3×#208—62no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—61no—
69#3718Dic311yescharacteristic, complemented
70#3715Dic312yescomplemented
71#3713Dic311yescharacteristic, complemented
72#3711—62no—
73#3711—62no—
74#3711—64no—
75D3×C3⋊C8—62no—
76D3×C3⋊C8—62no—
77D3×C3⋊C8—64no—
78C22×C33#4111yescharacteristic
79C2×C32×D3—22no—
80C33×C4C22×C411yescharacteristic
81C33×C4C2×D411yescharacteristic
82C32×Dic3—22no—
83C32×Dic3—41no—
84C3×#135—91no—
85#950—32no—
86#948—92no—
87#948—92no—
88#948—182no—
89#907—92no—
90C3×#104—92no—
91#837—32no—
92#822—92no—
93#819—91no—
94#815—91no—
95#805—32no—
96#803—92no—
97#803—181no—
98C32×Q8—32no—
99C32×D4—32no—
100C32×D4—62no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—41no—
104C2×C32×C4—62no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D311yescharacteristic
108C3×#67—24no—
109C3×#67—22no—
110C3×#67—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C3×D12—22no—
114C3×D12C4×D312yes—
115C3×D12C2×Dic312yes—
116C3×C4×D3—24no—
117C3×C4×D3—22no—
118C3×C4×D3—22no—
119C3×Dic6—22no—
120C3×Dic6C4×D312yes—
121C3×Dic6C2×Dic312yes—
122C3⋊(C3⋊C8)—22no—
123C3⋊(C3⋊C8)—32no—
124C3⋊(C3⋊C8)—32no—
125C3⋊(C3⋊C8)—32no—
126C3⋊(C3⋊C8)—62no—
127C3⋊(C3⋊C8)—62no—
128C3⋊(C3⋊C8)—62no—
129C3⋊(C3⋊C8)—122no—
130C3⋊(C3⋊C8)C2×Dic311yescharacteristic
131C3⋊(C3⋊C8)C2×Dic311yescharacteristic
132C3×C3⋊C8—32no—
133C3×C3⋊C8—32no—
134C3×C3⋊C8—61no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—64no—
137C3×C3⋊C8—62no—
138C3×C3⋊C8—62no—
139#444—271noSylow
140C2×C33—21no—
141C2×C33C4×D411yescharacteristic
142C32×D3—42no—
143C3×#41—32no—
144C3×#41—32no—
145C3×#41—62no—
146#235D3212yes—
147C3×Q16—91no—
148C3×SD16—92no—
149C3×D8—91no—
150C3×M16—92no—
151C2×C3×C8—91no—
152C3×C42—32no—
153C3×C42—61no—
154#216—32no—
155#215—32no—
156#214—32no—
157#213—32no—
158C4×Dic3D3211yescharacteristic
159#208—32no—
160#208—32no—
161#208—62no—
162#208—182no—
163#208—182no—
164#208—184no—
165C2×C3⋊C8—32no—
166C2×C3⋊C8—61no—
167C2×C3⋊C8—92no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—181no—
170C2×C3⋊C8—182no—
171C2×C3⋊C8—182no—
172C2×C3⋊C8—182no—
173C2×C3⋊C8—184no—
174C2×C3⋊C8—361no—
175ΓL(1,𝔽25)—182no—
176D3×C8—182no—
177C22×C32—21no—
178C22×C32—22no—
179C22×C32—41no—
180C22×C32—62no—
181C22×C32#23712yes—
182C22×C32#23711yescharacteristic
183C2×C3×D3—24no—
184C2×C3×D3—22no—
185C2×C3×D3—22no—
186C32×C4—21no—
187C32×C4—21no—
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—41no—
191C32×C4—41no—
192C32×C4—62no—
193C32×C4—121no—
194C32×C4C2×C4×D312yes—
195C32×C4D3×D412yes—
196C32×C4D3×D411yescharacteristic
197C32×C4C22×Dic311yescharacteristic
198C3×Dic3—24no—
199C3×Dic3—22no—
200C3×Dic3—22no—
201C3×Dic3—42no—
202C3×Dic3—42no—
203#135—91no—
204#131—272no—
205#108—271no—
206#104—92no—
207#97—271no—
208C33#44411yescharacteristic, complemented, Sylow
209C3×Q8—32no—
210C3×Q8—32no—
211C3×Q8—62no—
212C3×D4—32no—
213C3×D4—32no—
214C3×D4—62no—
215C3×D4—64no—
216C3×D4—62no—
217C3×D4—62no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—64no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4C2×D3211yescharacteristic
227C2×C3×C4C2×D3212yes—
228#67—22no—
229C2×Dic3—21no—
230D12D3×Dic312yes—
231C4×D3—22no—
232Dic6D3×Dic312yes—
233C3×C8—91no—
234C3×C8—91no—
235C3×C8—182no—
236C3⋊C8—32no—
237C3⋊C8—32no—
238C3⋊C8—61no—
239C3⋊C8—61no—
240C3⋊C8—64no—
241C3⋊C8—62no—
242C3⋊C8—62no—
243C3⋊C8—92no—
244C3⋊C8—92no—
245C3⋊C8—92no—
246C3⋊C8—182no—
247C3⋊C8—182no—
248C3⋊C8—182no—
249C3⋊C8—362no—
250C2×C32—21no—
251C2×C32—22no—
252C2×C32—22no—
253C2×C32—21no—
254C2×C32—22no—
255C2×C32—41no—
256C2×C32—42no—
257C2×C32—42no—
258C2×C32—122no—
259C2×C32#67⋊C412yes—
260C2×C32D4×Dic311yescharacteristic
261C3×D3—44no—
262C3×D3—42no—
263C3×D3—42no—
264#41—32no—
265Q16—91no—
266SD16—92no—
267D8—91no—
268M16—92no—
269M16—542no—
270C2×C8—91no—
271C2×C8—271no—
272C2×C8—542no—
273C42—31no—
274C22×C3—21no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—64no—
278C22×C3—62no—
279C22×C3—62no—
280C22×C3#380511yescharacteristic
281C22×C3#380512yes—
282C2×D3—22no—
283C3×C4—21no—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—64no—
290C3×C4—62no—
291C3×C4—62no—
292C3×C4—122no—
293C3×C4—122no—
294C3×C4C4×D3211yescharacteristic
295C3×C4#380211yescharacteristic
296C3×C4C2×D3×Dic312yes—
297C3×C4#381112yes—
298Dic3—22no—
299Dic3—41no—
300C32—21no—
301C32—22no—
302C32—41no—
303C32#482312yescomplemented
304C32#519711yescharacteristic, complemented
305Q8—32no—
306D4—32no—
307D4—62no—
308C2×C4—62no—
309C2×C4—61no—
310C2×C4D3311yescharacteristic
311C8—91no—
312C8—91no—
313C8—182no—
314C8—272no—
315C2×C3—22no—
316C2×C3—21no—
317C2×C3—22no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—42no—
321C2×C3—42no—
322C2×C3—41no—
323C2×C3—124no—
324C2×C3—122no—
325C2×C3—122no—
326C2×C3#6390211yescharacteristic
327C2×C3#67⋊Dic312yes—
328D3—42no—
329C22—62no—
330C22#8896311yescharacteristic
331C4—62no—
332C4—121no—
333C4D32×Dic311yescharacteristic
334C4#8895411yescharacteristic
335C3—21no—
336C3—22no—
337C3—41no—
338C3#1059039011yescharacteristic, complemented
339C3#1059167612yescomplemented
340C2—21no—
341C2—122no—
342C2#1172292511yescharacteristic
subgroupG/H
centerC4#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radicalC3×#63822C2
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723350C2semidirect
3. #11722081C2semidirect
4. #11722019C2semidirect
5. C3×#63569C2semidirect
15. C3×#3718C4semidirect
16. C3×#3715C4semidirect
17. C3×#3713C4semidirect
29. #63569D3semidirect
69. #3718Dic3semidirect
70. #3715Dic3semidirect
71. #3713Dic3semidirect
208. C33#444semidirect
303. C32#4823semidirect
304. C32#5197semidirect
338. C3#10590390semidirect
339. C3#10591676semidirect

Maximal subgroups

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

subgroupindex
#117233502
C3×#638222
#117220812
#117220192
C3×#635692
#105916763
#105903903