Finite Group Databasebeta

← #13155751

Subgroups — #13155751

Order 1728.

574 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 102 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
2C3×#63817C212yesmaximal, complemented
3#11722045C212yesmaximal, complemented
4#11722021C211yesmaximal, characteristic, complemented
5C3×#63578C211yesmaximal, characteristic, complemented
6#10590994—32nomaximal
7#10590922—31nomaximal
8C3×#3828C2212yes—
9C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×#208C2212yes—
12C2×C32×C3⋊C8C2211yescharacteristic
13#88812C411yescharacteristic, complemented
14#88811C412yescomplemented
15#88810C411yescharacteristic, complemented
16C3×#3712—22no—
17C3×#3710—22no—
18C3×#907—32no—
19C3×#901—32no—
20#63829—32no—
21#63822—31no—
22#63817D312yescomplemented
23#63652—32no—
24#63642—31no—
25C3×#822—32no—
26#63578D311yescharacteristic, complemented
27#63568—32no—
28#63568—32no—
29#63568—62no—
30#63555—32no—
31#63555—61no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C3×#638—22no—
35C3×C4⋊(C3⋊D3)C2×C412yes—
36C3×C4×C3⋊D3—22no—
37C3×#634C2×C412yes—
38C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
39C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
40C32×C3⋊C8—22no—
41C32×C3⋊C8D412yes—
42#5145—92no—
43#4894—91no—
44#3828C2×D312yes—
45C3×#235—32no—
46C3×#235—32no—
47C3×#235—62no—
48C32×M16—32no—
49C2×C32×C8—31no—
50C4×C3⋊Dic3—32no—
51C4×C3⋊Dic3—31no—
52C4×C3⋊Dic3—61no—
53C4×C3⋊Dic3—62no—
54C4×C3⋊Dic3—121no—
55C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
56C3×#208—22no—
57C3×#208C2×D312yes—
58C3×#208C2×D312yes—
59C2×C3×C3⋊C8—21no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—61no—
62C2×C3×C3⋊C8C2×D312yes—
63C3×ΓL(1,𝔽25)—62no—
64C3×ΓL(1,𝔽25)—62no—
65C3×D3×C8—62no—
66C3×D3×C8—62no—
67#3719—32no—
68#3718—31no—
69#3717—32no—
70#3716—32no—
71#3715—32no—
72#3714—32no—
73#3713—31no—
74#3712—22no—
75#3710—22no—
76C22×C33#4111yescharacteristic
77C2×C3×C3⋊D3—22no—
78C33×C4C22×C411yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3⋊(C3⋊Dic3)—41no—
81C3×C3⋊Dic3—22no—
82C3×#131—92no—
83#950—92no—
84#907—32no—
85#904—91no—
86#901—32no—
87C3×#108—91no—
88#837—92no—
89#825—92no—
90#825—92no—
91#825—182no—
92#822—32no—
93#815—92no—
94#815—91no—
95#815—181no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—41no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C32×C4C22×D312yes—
101#638—22no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—62no—
105C2×C3⋊Dic3—122no—
106C2×C3⋊Dic3—122no—
107C4⋊(C3⋊D3)C4×D312yes—
108C4×C3⋊D3—22no—
109#634C4×D312yes—
110C3×#67—62no—
111C3×#67—62no—
112C3×#67—122no—
113C3×D12—32no—
114C3×D12—32no—
115C3×D12—62no—
116C3×C4×D3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—122no—
119C3×Dic6—32no—
120C3×Dic6—32no—
121C3×Dic6—62no—
122C32×C8—32no—
123C32×C8—62no—
124C3⋊(C3⋊C8)C4×D311yescharacteristic
125C3⋊(C3⋊C8)C4×D311yescharacteristic
126C3×C3⋊C8—22no—
127C3×C3⋊C8—22no—
128C3×C3⋊C8—22no—
129C3×C3⋊C8—32no—
130C3×C3⋊C8—32no—
131C3×C3⋊C8—42no—
132C3×C3⋊C8—61no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8#6712yes—
135C3×C3⋊C8#6712yes—
136#444—271noSylow
137C2×C33—21no—
138C2×C33C4×D411yescharacteristic
139C3×C3⋊D3—42no—
140C3×#41—92no—
141#235—32no—
142#235—32no—
143#235—62no—
144C3×M16—32no—
145C3×M16—32no—
146C3×M16—62no—
147C3×M16—182no—
148C2×C3×C8—32no—
149C2×C3×C8—61no—
150C2×C3×C8—91no—
151C2×C3×C8—182no—
152#216—92no—
153#215—92no—
154#214—92no—
155#213—92no—
156C4×Dic3—91no—
157C4×Dic3—92no—
158C4×Dic3—181no—
159C4×Dic3—182no—
160C4×Dic3—361no—
161#208D3212yes—
162C2×C3⋊C8—32no—
163C2×C3⋊C8—61no—
164C2×C3⋊C8D3211yescharacteristic
165Dic12—91no—
166D24—91no—
167#204—92no—
168ΓL(1,𝔽25)—62no—
169ΓL(1,𝔽25)—62no—
170D3×C8—62no—
171D3×C8—62no—
172C22×C32—21no—
173C22×C32—22no—
174C22×C32—41no—
175C22×C32#23512yes—
176C22×C32#23911yescharacteristic
177C2×C3⋊D3—22no—
178C2×C3×D3—62no—
179C2×C3×D3—62no—
180C2×C3×D3—122no—
181C32×C4—21no—
182C32×C4—22no—
183C32×C4—21no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4C2×C4×D311yescharacteristic
188C32×C4C2×C4×D312yes—
189C32×C4D3×D411yescharacteristic
190C32×C4C2×#6712yes—
191C3⋊Dic3—22no—
192C3⋊Dic3—121no—
193C3⋊Dic3—122no—
194C3⋊Dic3—122no—
195C3⋊Dic3—124no—
196C3⋊Dic3—124no—
197C3×Dic3—62no—
198C3×Dic3—62no—
199C3×Dic3—122no—
200#135—271no—
201#131—92no—
202#108—91no—
203#104—272no—
204#97—271no—
205C33#44411yescharacteristic, complemented, Sylow
206C3×Q8—92no—
207C3×D4—92no—
208C3×D4—182no—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—182no—
213C2×C3×C4C2×D3211yescharacteristic
214C2×C3×C4C2×D3212yes—
215#67—62no—
216#67—62no—
217#67—122no—
218C2×Dic3—182no—
219C2×Dic3—181no—
220C2×Dic3—182no—
221C2×Dic3—362no—
222C2×Dic3—362no—
223D12—32no—
224D12—32no—
225D12—62no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—122no—
229Dic6—32no—
230Dic6—32no—
231Dic6—62no—
232C3×C8—32no—
233C3×C8—32no—
234C3×C8—62no—
235C3×C8—62no—
236C3×C8—62no—
237C3×C8—91no—
238C3×C8—91no—
239C3×C8—122no—
240C3⋊C8—22no—
241C3⋊C8—32no—
242C3⋊C8—32no—
243C3⋊C8—61no—
244C3⋊C8—61no—
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#89311yescharacteristic
255C2×C32C4×#6712yes—
256C3⋊D3—42no—
257C3×D3—122no—
258C3×D3—122no—
259C3×D3—242no—
260#41—92no—
261Q16—271no—
262SD16—272no—
263D8—271no—
264M16—32no—
265M16—182no—
266C2×C8—31no—
267C2×C8—91no—
268C2×C8—182no—
269C42—271no—
270C22×C3—21no—
271C22×C3—22no—
272C22×C3—41no—
273C22×C3—182no—
274C22×C3#379811yescharacteristic
275C22×C3#379912yes—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—122no—
279C3×C4—21no—
280C3×C4—22no—
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—182no—
286C3×C4C4×D3211yescharacteristic
287C3×C4C4×D3212yes—
288C3×C4C2×#62511yescharacteristic
289C3×C4D3×#6712yes—
290Dic3—62no—
291Dic3—62no—
292Dic3—122no—
293Dic3—362no—
294Dic3—362no—
295Dic3—361no—
296Dic3—364no—
297Dic3—364no—
298C32—22no—
299C32—21no—
300C32—41no—
301C32#489411yescharacteristic, complemented
302C32#514512yescomplemented
303Q8—92no—
304D4—92no—
305D4—182no—
306C2×C4—182no—
307C2×C4—541no—
308C2×C4D3311yescharacteristic
309C8—32no—
310C8—62no—
311C8—91no—
312C8—91no—
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—362no—
322C2×C3C4×#62511yescharacteristic
323C2×C3#6390412yes—
324D3—122no—
325D3—122no—
326D3—242no—
327C22—182no—
328C22#8896511yescharacteristic
329C4—182no—
330C4—1081no—
331C4#8894911yescharacteristic
332C4D3×#62511yescharacteristic
333C3—22no—
334C3—21no—
335C3—41no—
336C3#1059092211yescharacteristic, complemented
337C3#1059099412yescomplemented
338C2—21no—
339C2—362no—
340C2#1172293111yescharacteristic
subgroupG/H
centerC4D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radical#11723660C2
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×#63817C2semidirect
3. #11722045C2semidirect
4. #11722021C2semidirect
5. C3×#63578C2semidirect
13. #88812C4semidirect
14. #88811C4semidirect
15. #88810C4semidirect
22. #63817D3semidirect
26. #63578D3semidirect
205. C33#444semidirect
301. C32#4894semidirect
302. C32#5145semidirect
336. C3#10590922semidirect
337. C3#10590994semidirect

Maximal subgroups

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

subgroupindex
#117236602
C3×#638172
#117220452
#117220212
C3×#635782
#105909943
#105909223