Finite Group Databasebeta

← #13155758

Subgroups — #13155758

Order 1728.

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

normal subgroups only

This group has 602 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
3#11722019C211yesmaximal, characteristic, complemented
4C3×#63578C211yesmaximal, characteristic, complemented
5C3×#63570C212yesmaximal, complemented
6#10590978—32nomaximal
7#10590390—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—
24#63630—31no—
25C3×#825—32no—
26C3×#822—32no—
27C3×#805—32no—
28#63578D311yescharacteristic, complemented
29#63570D312yescomplemented
30#63554—32no—
31#63554—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))D412yes—
39C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
40C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
41C32×C3⋊C8—22no—
42#4894—91no—
43#4823—92no—
44#3828C2×D312yes—
45C3×#235—32no—
46C3×#235—32no—
47C3×#235—62no—
48C32×M16—32no—
49C32×C42—31no—
50C2×C3⋊(C3⋊C8)—32no—
51C2×C3⋊(C3⋊C8)—31no—
52C2×C3⋊(C3⋊C8)—61no—
53C2×C3⋊(C3⋊C8)—62no—
54C2×C3⋊(C3⋊C8)—121no—
55C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
56#3743—62no—
57C8×C3⋊D3—62no—
58C3×C4×Dic3—21no—
59C3×C4×Dic3C2×D312yes—
60C3×#208—22no—
61C3×#208C2×D312yes—
62C3×#208C2×D312yes—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—61no—
65#3719—32no—
66#3718—31no—
67#3717—32no—
68#3716—32no—
69#3715—32no—
70#3714—32no—
71#3713—31no—
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×C3×C3⋊D3—22no—
80C33×C4C22×C411yescharacteristic
81C33×C4C2×D411yescharacteristic
82C3×C3⋊Dic3—22no—
83C32×Dic3—41no—
84#950—92no—
85#948—92no—
86#907—92no—
87#907—92no—
88#907—182no—
89#904—91no—
90C3×#108—91no—
91C3×#104—92no—
92#825—32no—
93#822—32no—
94#815—91no—
95#805—32no—
96#803—92no—
97#803—181no—
98C2×C32×C4—21no—
99C2×C32×C4—22no—
100C2×C32×C4—41no—
101C2×C32×C4—61no—
102C2×C32×C4C22×D312yes—
103C2×C32×C4C22×D311yescharacteristic
104#638—22no—
105C4⋊(C3⋊D3)C4×D312yes—
106C4×C3⋊D3—22no—
107#634C4×D312yes—
108C3×#67—64no—
109C3×#67—62no—
110C3×#67—62no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C3×D12—32no—
114C3×D12—32no—
115C3×D12—62no—
116C3×C4×D3—64no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×Dic6—32no—
120C3×Dic6—32no—
121C3×Dic6—62no—
122C32×C8—62no—
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)C4×D311yescharacteristic
131C3⋊(C3⋊C8)C4×D311yescharacteristic
132C3×C3⋊C8—24no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—22no—
135C3×C3⋊C8—32no—
136C3×C3⋊C8—32no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—61no—
139#444—271noSylow
140C2×C33—21no—
141C2×C33C4×D411yescharacteristic
142C3×C3⋊D3—42no—
143C3×#41—92no—
144#235—32no—
145#235—32no—
146#235—62no—
147C3×M16—32no—
148C3×M16—32no—
149C3×M16—62no—
150C2×C3×C8—91no—
151C3×C42—32no—
152C3×C42—61no—
153#216—92no—
154#215—92no—
155#214—92no—
156#213—92no—
157C4×Dic3D3211yescharacteristic
158#208—182no—
159#208D3212yes—
160C2×C3⋊C8—32no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8—92no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—181no—
165C2×C3⋊C8—182no—
166C2×C3⋊C8—182no—
167C2×C3⋊C8—361no—
168Dic12—91no—
169D24—91no—
170#204—92no—
171ΓL(1,𝔽25)—182no—
172ΓL(1,𝔽25)—182no—
173ΓL(1,𝔽25)—184no—
174D3×C8—182no—
175D3×C8—182no—
176D3×C8—184no—
177C22×C32—22no—
178C22×C32—21no—
179C22×C32—41no—
180C22×C32#23712yes—
181C22×C32#23911yescharacteristic
182C2×C3⋊D3—22no—
183C2×C3×D3—64no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C32×C4—21no—
187C32×C4—22no—
188C32×C4—21no—
189C32×C4—22no—
190C32×C4—41no—
191C32×C4—41no—
192C32×C4—121no—
193C32×C4C2×C4×D312yes—
194C32×C4C2×C4×D311yescharacteristic
195C32×C4D3×D412yes—
196C32×C4D3×D411yescharacteristic
197C3⋊Dic3—22no—
198C3×Dic3—42no—
199C3×Dic3—42no—
200C3×Dic3—64no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203#135—271no—
204#131—272no—
205#108—91no—
206#104—92no—
207#97—271no—
208C33#44411yescharacteristic, complemented, Sylow
209C3×Q8—92no—
210C3×D4—92no—
211C3×D4—182no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—62no—
216C2×C3×C4—62no—
217C2×C3×C4—182no—
218C2×C3×C4C2×D3211yescharacteristic
219C2×C3×C4C2×D3212yes—
220#67—64no—
221#67—62no—
222#67—62no—
223C2×Dic3—21no—
224D12—32no—
225D12—32no—
226D12—62no—
227C4×D3—64no—
228C4×D3—62no—
229C4×D3—62no—
230Dic6—32no—
231Dic6—32no—
232Dic6—62no—
233C3×C8—64no—
234C3×C8—62no—
235C3×C8—62no—
236C3×C8—91no—
237C3×C8—91no—
238C3⋊C8—22no—
239C3⋊C8—32no—
240C3⋊C8—32no—
241C3⋊C8—61no—
242C3⋊C8—61no—
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—22no—
254C2×C32—21no—
255C2×C32—41no—
256C2×C32—42no—
257C2×C32—42no—
258C2×C32#67⋊C412yes—
259C2×C32#89311yescharacteristic
260C3⋊D3—42no—
261C3×D3—124no—
262C3×D3—122no—
263C3×D3—122no—
264#41—92no—
265Q16—271no—
266SD16—272no—
267D8—271no—
268M16—32no—
269M16—542no—
270C2×C8—91no—
271C2×C8—271no—
272C2×C8—542no—
273C42—31no—
274C22×C3—21no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—182no—
278C22×C3#379612yes—
279C22×C3#380511yescharacteristic
280C2×D3—64no—
281C2×D3—62no—
282C2×D3—62no—
283C3×C4—21no—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—122no—
290C3×C4—122no—
291C3×C4—182no—
292C3×C4C4×D3212yes—
293C3×C4C4×D3211yescharacteristic
294C3×C4#380211yescharacteristic
295C3×C4#381112yes—
296Dic3—41no—
297Dic3—64no—
298Dic3—62no—
299Dic3—62no—
300C32—22no—
301C32—21no—
302C32—41no—
303C32#482312yescomplemented
304C32#489411yescharacteristic, complemented
305Q8—92no—
306D4—92no—
307D4—182no—
308C2×C4—61no—
309C2×C4—182no—
310C2×C4D3311yescharacteristic
311C8—62no—
312C8—91no—
313C8—91no—
314C8—272no—
315C2×C3—22no—
316C2×C3—22no—
317C2×C3—21no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—42no—
321C2×C3—42no—
322C2×C3—41no—
323C2×C3—362no—
324C2×C3#6388112yes—
325C2×C3#6390211yescharacteristic
326D3—124no—
327D3—122no—
328D3—122no—
329C22—182no—
330C22#8896411yescharacteristic
331C4—121no—
332C4—182no—
333C4#8894911yescharacteristic
334C4#8895411yescharacteristic
335C3—22no—
336C3—21no—
337C3—41no—
338C3#1059039011yescharacteristic, complemented
339C3#1059097812yescomplemented
340C2—21no—
341C2—362no—
342C2#1172293811yescharacteristic
subgroupG/H
centerC4#88954
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. #11723354C2semidirect
3. #11722019C2semidirect
4. C3×#63578C2semidirect
5. C3×#63570C2semidirect
13. #88812C4semidirect
14. #88811C4semidirect
15. #88810C4semidirect
28. #63578D3semidirect
29. #63570D3semidirect
208. C33#444semidirect
303. C32#4823semidirect
304. C32#4894semidirect
338. C3#10590390semidirect
339. C3#10590978semidirect

Maximal subgroups

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

subgroupindex
#117236602
#117233542
#117220192
C3×#635782
C3×#635702
#105909783
#105903903