Finite Group Databasebeta

← #13155727

Subgroups — #13155727

Order 1728.

586 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C3×#63822C211yesmaximal, characteristic
2C3×#63814C212yesmaximal, complemented
3#11722081C211yesmaximal, characteristic, complemented
4#11722043C212yesmaximal, complemented
5#11722018C211yesmaximal, characteristic, complemented
6#10591673—32nomaximal
7#10590921—31nomaximal
8C32×#235C2212yes—
9C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C3×#3748C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C2×C32×C3⋊C8C2211yescharacteristic
13C3×#3718C411yescharacteristic, complemented
14C3×#3715C412yescomplemented
15C3×#3713C411yescharacteristic, complemented
16C3×#3711—22no—
17C3×D3×C3⋊C8—22no—
18C3×#950—32no—
19C3×#948—32no—
20C3×#901—32no—
21#63822D311yescharacteristic
22#63814D312yescomplemented
23#63664—32no—
24C8×C3⋊Dic3—31no—
25#63578—31no—
26#63577—32no—
27#63568—32no—
28#63568—32no—
29#63568—62no—
30#63553—32no—
31#63553—61no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C32×#67—22no—
35C32×D12C2×C412yes—
36C32×C4×D3—22no—
37C32×Dic6C2×C412yes—
38C3×C3⋊(C3⋊C8)—22no—
39C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
40C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
41C32×C3⋊C8D411yescharacteristic
42C32×C3⋊C8D411yescharacteristic
43#5196—91no—
44#5144—92no—
45C32×#41—32no—
46C3×#235—22no—
47C3×#235C2×D312yes—
48C3×#235C2×D312yes—
49C2×C32×C8—31no—
50C4×C3⋊Dic3—32no—
51C4×C3⋊Dic3—31no—
52C4×C3⋊Dic3—61no—
53C4×C3⋊Dic3—62no—
54C4×C3⋊Dic3—121no—
55#3748C2×D312yes—
56C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
57C3×#216—32no—
58C3×#215—32no—
59C3×#214—32no—
60C3×#213—32no—
61C3×#208—32no—
62C3×#208—32no—
63C3×#208—62no—
64C3×#208—62no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—61no—
68C2×C3×C3⋊C8—62no—
69C2×C3×C3⋊C8C2×D312yes—
70C3×ΓL(1,𝔽25)—62no—
71C3×D3×C8—62no—
72#3718Dic311yescharacteristic, complemented
73#3715Dic312yescomplemented
74#3713Dic311yescharacteristic, complemented
75#3711—22no—
76D3×C3⋊C8—22no—
77C22×C33#4111yescharacteristic
78C2×C32×D3—22no—
79C33×C4C22×C411yescharacteristic
80C33×C4C2×D411yescharacteristic
81C3⋊(C3⋊Dic3)—22no—
82C32×Dic3—22no—
83C3×#135—91no—
84C3×#131—92no—
85#950—32no—
86#948—32no—
87#901—32no—
88#837—92no—
89#837—92no—
90#837—182no—
91#825—92no—
92#822—92no—
93#819—91no—
94C8×Dic3—92no—
95C8×Dic3—91no—
96C8×Dic3—181no—
97C32×Q8—32no—
98C32×D4—32no—
99C32×D4—62no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—41no—
103C2×C32×C4—62no—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—61no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—62no—
109C2×C3⋊Dic3—122no—
110C2×C3⋊Dic3—122no—
111C3×#67—22no—
112C3×#67—22no—
113C3×#67—42no—
114C3×D12—22no—
115C3×D12C4×D312yes—
116C3×D12C2×Dic312yes—
117C3×C4×D3—22no—
118C3×C4×D3—22no—
119C3×C4×D3—42no—
120C3×Dic6—22no—
121C3×Dic6C4×D312yes—
122C3×Dic6C2×Dic312yes—
123C32×C8—31no—
124C32×C8—31no—
125C3⋊(C3⋊C8)—22no—
126C3⋊(C3⋊C8)C2×Dic311yescharacteristic
127C3⋊(C3⋊C8)C2×Dic311yescharacteristic
128C3×C3⋊C8—21no—
129C3×C3⋊C8—21no—
130C3×C3⋊C8—32no—
131C3×C3⋊C8—32no—
132C3×C3⋊C8—61no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8—62no—
135C3×C3⋊C8—62no—
136C3×C3⋊C8—122no—
137C3×C3⋊C8#6712yes—
138C3×C3⋊C8#6712yes—
139#443—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—
151C3×M16—182no—
152C2×C3×C8—32no—
153C2×C3×C8—61no—
154C2×C3×C8—91no—
155C2×C3×C8—182no—
156#216—32no—
157#215—32no—
158#214—32no—
159#213—32no—
160C4×Dic3—92no—
161C4×Dic3—91no—
162C4×Dic3—181no—
163C4×Dic3—182no—
164C4×Dic3—361no—
165#208—32no—
166#208—32no—
167#208—62no—
168#208—62no—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—61no—
171C2×C3⋊C8—62no—
172C2×C3⋊C8D3211yescharacteristic
173ΓL(1,𝔽25)—62no—
174D3×C8—62no—
175C22×C32—21no—
176C22×C32—22no—
177C22×C32—41no—
178C22×C32—62no—
179C22×C32#23512yes—
180C22×C32#23711yescharacteristic
181C2×C3×D3—22no—
182C2×C3×D3—22no—
183C2×C3×D3—42no—
184C32×C4—21no—
185C32×C4—22no—
186C32×C4—21no—
187C32×C4—22no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—62no—
191C32×C4C2×C4×D312yes—
192C32×C4D3×D411yescharacteristic
193C32×C4C22×Dic311yescharacteristic
194C32×C4C2×#6712yes—
195C3⋊Dic3—62no—
196C3⋊Dic3—64no—
197C3⋊Dic3—64no—
198C3⋊Dic3—124no—
199C3⋊Dic3—124no—
200C3×Dic3—22no—
201C3×Dic3—22no—
202C3×Dic3—42no—
203#135—91no—
204#131—92no—
205#108—271no—
206#104—272no—
207C4×C8—271no—
208C33#44311yescharacteristic, complemented, Sylow
209C3×Q8—32no—
210C3×Q8—32no—
211C3×Q8—62no—
212C3×D4—32no—
213C3×D4—32no—
214C3×D4—62no—
215C3×D4—62no—
216C3×D4—62no—
217C3×D4—122no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—122no—
224C2×C3×C4C2×D3211yescharacteristic
225C2×C3×C4C2×D3212yes—
226#67—22no—
227C2×Dic3—182no—
228C2×Dic3—181no—
229C2×Dic3—182no—
230C2×Dic3—362no—
231C2×Dic3—362no—
232D12D3×Dic312yes—
233C4×D3—22no—
234Dic6D3×Dic312yes—
235C3×C8—32no—
236C3×C8—32no—
237C3×C8—61no—
238C3×C8—61no—
239C3×C8—91no—
240C3×C8—91no—
241C3×C8—182no—
242C3⋊C8—32no—
243C3⋊C8—32no—
244C3⋊C8—61no—
245C3⋊C8—61no—
246C3⋊C8—62no—
247C3⋊C8—62no—
248C3⋊C8—122no—
249C3⋊C8#62511yescharacteristic
250C3⋊C8#62511yescharacteristic
251C2×C32—21no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—22no—
256C2×C32—41no—
257C2×C32—42no—
258C2×C32—42no—
259C2×C32—122no—
260C2×C32C4×#6712yes—
261C2×C32D4×Dic311yescharacteristic
262C3×D3—42no—
263C3×D3—42no—
264C3×D3—82no—
265#41—32no—
266Q16—91no—
267SD16—92no—
268D8—91no—
269M16—92no—
270M16—182no—
271C2×C8—31no—
272C2×C8—91no—
273C2×C8—182no—
274C42—271no—
275C22×C3—21no—
276C22×C3—22no—
277C22×C3—41no—
278C22×C3—62no—
279C22×C3—62no—
280C22×C3—122no—
281C22×C3#379811yescharacteristic
282C22×C3#380412yes—
283C2×D3—22no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—62no—
291C3×C4—62no—
292C3×C4—122no—
293C3×C4C4×D3211yescharacteristic
294C3×C4C2×D3×Dic312yes—
295C3×C4C2×#62511yescharacteristic
296C3×C4D3×#6712yes—
297Dic3—22no—
298Dic3—182no—
299Dic3—184no—
300Dic3—184no—
301Dic3—364no—
302Dic3—364no—
303C32—22no—
304C32—21no—
305C32—41no—
306C32#514412yescomplemented
307C32#519611yescharacteristic, complemented
308Q8—32no—
309D4—32no—
310D4—62no—
311C2×C4—62no—
312C2×C4—541no—
313C2×C4D3311yescharacteristic
314C8—31no—
315C8—31no—
316C8—91no—
317C8—91no—
318C8—182no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—21no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—42no—
325C2×C3—42no—
326C2×C3—41no—
327C2×C3—122no—
328C2×C3—122no—
329C2×C3—242no—
330C2×C3C4×#62511yescharacteristic
331C2×C3Dic3×#6712yes—
332D3—42no—
333C22—62no—
334C22#8895911yescharacteristic
335C4—62no—
336C4—542no—
337C4D32×Dic311yescharacteristic
338C4D3×#62511yescharacteristic
339C3—21no—
340C3—22no—
341C3—41no—
342C3#1059092111yescharacteristic, complemented
343C3#1059167312yescomplemented
344C2—21no—
345C2—122no—
346C2Dic3×#62511yescharacteristic
subgroupG/H
centerC4D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radicalC3×#63822C2
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. C3×#63814C2semidirect
3. #11722081C2semidirect
4. #11722043C2semidirect
5. #11722018C2semidirect
13. C3×#3718C4semidirect
14. C3×#3715C4semidirect
15. C3×#3713C4semidirect
22. #63814D3semidirect
72. #3718Dic3semidirect
73. #3715Dic3semidirect
74. #3713Dic3semidirect
208. C33#443semidirect
306. C32#5144semidirect
307. C32#5196semidirect
342. C3#10590921semidirect
343. C3#10591673semidirect

Maximal subgroups

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

subgroupindex
C3×#638222
C3×#638142
#117220812
#117220432
#117220182
#105916733
#105909213