Finite Group Databasebeta

← #13155743

Subgroups — #13155743

Order 1728.

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

normal subgroups only

This group has 574 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#11722021C211yesmaximal, characteristic, complemented
6#10591674—32nomaximal
7#10590922—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—
24#63642—31no—
25#63578—31no—
26#63577—32no—
27#63568—32no—
28#63568—32no—
29#63568—62no—
30#63555—32no—
31#63555—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⋊C8D412yes—
42#5197—91no—
43#5145—92no—
44C32×#41—32no—
45C3×#235—22no—
46C3×#235C2×D312yes—
47C3×#235C2×D312yes—
48C2×C32×C8—31no—
49C4×C3⋊Dic3—32no—
50C4×C3⋊Dic3—31no—
51C4×C3⋊Dic3—61no—
52C4×C3⋊Dic3—62no—
53C4×C3⋊Dic3—121no—
54#3748C2×D312yes—
55C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
56C3×#216—32no—
57C3×#215—32no—
58C3×#214—32no—
59C3×#213—32no—
60C3×#208—32no—
61C3×#208—32no—
62C3×#208—62no—
63C3×#208—62no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—61no—
67C2×C3×C3⋊C8—62no—
68C2×C3×C3⋊C8C2×D312yes—
69C3×ΓL(1,𝔽25)—62no—
70C3×D3×C8—62no—
71#3718Dic311yescharacteristic, complemented
72#3715Dic312yescomplemented
73#3713Dic311yescharacteristic, complemented
74#3711—22no—
75D3×C3⋊C8—22no—
76C22×C33#4111yescharacteristic
77C2×C32×D3—22no—
78C33×C4C22×C411yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3⋊(C3⋊Dic3)—41no—
81C32×Dic3—22no—
82C3×#135—91no—
83C3×#131—92no—
84#950—32no—
85#948—32no—
86#901—32no—
87#837—92no—
88#837—92no—
89#837—182no—
90#825—92no—
91#822—92no—
92#819—91no—
93#815—92no—
94#815—91no—
95#815—181no—
96C32×Q8—32no—
97C32×D4—32no—
98C32×D4—62no—
99C2×C32×C4—21no—
100C2×C32×C4—22no—
101C2×C32×C4—41no—
102C2×C32×C4—62no—
103C2×C32×C4C22×D312yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—62no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—122no—
109C2×C3⋊Dic3—122no—
110C3×#67—22no—
111C3×#67—22no—
112C3×#67—42no—
113C3×D12—22no—
114C3×D12C4×D312yes—
115C3×D12C2×Dic312yes—
116C3×C4×D3—22no—
117C3×C4×D3—22no—
118C3×C4×D3—42no—
119C3×Dic6—22no—
120C3×Dic6C4×D312yes—
121C3×Dic6C2×Dic312yes—
122C32×C8—32no—
123C3⋊(C3⋊C8)—22no—
124C3⋊(C3⋊C8)C2×Dic311yescharacteristic
125C3⋊(C3⋊C8)C2×Dic311yescharacteristic
126C3×C3⋊C8—22no—
127C3×C3⋊C8—32no—
128C3×C3⋊C8—32no—
129C3×C3⋊C8—61no—
130C3×C3⋊C8—61no—
131C3×C3⋊C8—62no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—122no—
134C3×C3⋊C8#6712yes—
135C3×C3⋊C8#6712yes—
136#444—271noSylow
137C2×C33—21no—
138C2×C33C4×D411yescharacteristic
139C32×D3—42no—
140C3×#41—32no—
141C3×#41—32no—
142C3×#41—62no—
143#235D3212yes—
144C3×Q16—91no—
145C3×SD16—92no—
146C3×D8—91no—
147C3×M16—92no—
148C3×M16—182no—
149C2×C3×C8—32no—
150C2×C3×C8—61no—
151C2×C3×C8—91no—
152C2×C3×C8—182no—
153#216—32no—
154#215—32no—
155#214—32no—
156#213—32no—
157C4×Dic3—92no—
158C4×Dic3—91no—
159C4×Dic3—181no—
160C4×Dic3—182no—
161C4×Dic3—361no—
162#208—32no—
163#208—32no—
164#208—62no—
165#208—62no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—62no—
169C2×C3⋊C8D3211yescharacteristic
170ΓL(1,𝔽25)—62no—
171D3×C8—62no—
172C22×C32—21no—
173C22×C32—22no—
174C22×C32—41no—
175C22×C32—62no—
176C22×C32#23512yes—
177C22×C32#23711yescharacteristic
178C2×C3×D3—22no—
179C2×C3×D3—22no—
180C2×C3×D3—42no—
181C32×C4—21no—
182C32×C4—22no—
183C32×C4—21no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—62no—
188C32×C4C2×C4×D312yes—
189C32×C4D3×D411yescharacteristic
190C32×C4C22×Dic311yescharacteristic
191C32×C4C2×#6712yes—
192C3⋊Dic3—121no—
193C3⋊Dic3—122no—
194C3⋊Dic3—122no—
195C3⋊Dic3—124no—
196C3⋊Dic3—124no—
197C3×Dic3—22no—
198C3×Dic3—22no—
199C3×Dic3—42no—
200#135—91no—
201#131—92no—
202#108—271no—
203#104—272no—
204#97—271no—
205C33#44411yescharacteristic, complemented, Sylow
206C3×Q8—32no—
207C3×Q8—32no—
208C3×Q8—62no—
209C3×D4—32no—
210C3×D4—32no—
211C3×D4—62no—
212C3×D4—62no—
213C3×D4—62no—
214C3×D4—122no—
215C2×C3×C4—21no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—122no—
221C2×C3×C4C2×D3211yescharacteristic
222C2×C3×C4C2×D3212yes—
223#67—22no—
224C2×Dic3—182no—
225C2×Dic3—181no—
226C2×Dic3—182no—
227C2×Dic3—362no—
228C2×Dic3—362no—
229D12D3×Dic312yes—
230C4×D3—22no—
231Dic6D3×Dic312yes—
232C3×C8—32no—
233C3×C8—32no—
234C3×C8—62no—
235C3×C8—91no—
236C3×C8—91no—
237C3×C8—182no—
238C3⋊C8—32no—
239C3⋊C8—32no—
240C3⋊C8—61no—
241C3⋊C8—61no—
242C3⋊C8—62no—
243C3⋊C8—62no—
244C3⋊C8—122no—
245C3⋊C8#62512yes—
246C2×C32—21no—
247C2×C32—22no—
248C2×C32—22no—
249C2×C32—21no—
250C2×C32—22no—
251C2×C32—41no—
252C2×C32—42no—
253C2×C32—42no—
254C2×C32—122no—
255C2×C32C4×#6712yes—
256C2×C32D4×Dic311yescharacteristic
257C3×D3—42no—
258C3×D3—42no—
259C3×D3—82no—
260#41—32no—
261Q16—91no—
262SD16—92no—
263D8—91no—
264M16—92no—
265M16—182no—
266C2×C8—31no—
267C2×C8—91no—
268C2×C8—182no—
269C42—271no—
270C22×C3—21no—
271C22×C3—22no—
272C22×C3—41no—
273C22×C3—62no—
274C22×C3—62no—
275C22×C3—122no—
276C22×C3#379811yescharacteristic
277C22×C3#380412yes—
278C2×D3—22no—
279C3×C4—21no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—21no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—62no—
286C3×C4—62no—
287C3×C4—122no—
288C3×C4C4×D3211yescharacteristic
289C3×C4C2×D3×Dic312yes—
290C3×C4C2×#62511yescharacteristic
291C3×C4D3×#6712yes—
292Dic3—22no—
293Dic3—362no—
294Dic3—362no—
295Dic3—361no—
296Dic3—364no—
297Dic3—364no—
298C32—22no—
299C32—21no—
300C32—41no—
301C32#514512yescomplemented
302C32#519711yescharacteristic, complemented
303Q8—32no—
304D4—32no—
305D4—62no—
306C2×C4—62no—
307C2×C4—541no—
308C2×C4D3311yescharacteristic
309C8—32no—
310C8—91no—
311C8—91no—
312C8—182no—
313C2×C3—22no—
314C2×C3—22no—
315C2×C3—21no—
316C2×C3—22no—
317C2×C3—21no—
318C2×C3—42no—
319C2×C3—42no—
320C2×C3—41no—
321C2×C3—122no—
322C2×C3—122no—
323C2×C3—242no—
324C2×C3C4×#62511yescharacteristic
325C2×C3Dic3×#6712yes—
326D3—42no—
327C22—62no—
328C22#8895911yescharacteristic
329C4—62no—
330C4—1081no—
331C4D32×Dic311yescharacteristic
332C4D3×#62511yescharacteristic
333C3—21no—
334C3—22no—
335C3—41no—
336C3#1059092211yescharacteristic, complemented
337C3#1059167412yescomplemented
338C2—21no—
339C2—122no—
340C2Dic3×#62511yescharacteristic
subgroupG/H
centerC4D3×#625
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
2. C3×#63814C2semidirect
3. #11722081C2semidirect
4. #11722043C2semidirect
5. #11722021C2semidirect
13. C3×#3718C4semidirect
14. C3×#3715C4semidirect
15. C3×#3713C4semidirect
22. #63814D3semidirect
71. #3718Dic3semidirect
72. #3715Dic3semidirect
73. #3713Dic3semidirect
205. C33#444semidirect
301. C32#5145semidirect
302. C32#5197semidirect
336. C3#10590922semidirect
337. C3#10591674semidirect

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
#117220212
#105916743
#105909223