Finite Group Databasebeta

← #13155487

Subgroups — #13155487

Order 1728.

646 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 138 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#11723689C211yesmaximal, characteristic
2#11723404C211yesmaximal, characteristic
3C2×C3×#3712C211yesmaximal, characteristic
4#11722052C212yesmaximal, complemented
5C3×#63575C212yesmaximal, complemented
6#10590683—32nomaximal
7#10590582—31nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6294—21no—
10C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C32×C4⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15#88803—21no—
16Dic3×C3⋊Dic3—21no—
17C3×#3712C414yescomplemented
18#64101—31no—
19C2×C3×ΓL(1,𝔽25)—32no—
20#63900—31no—
21D3×C4⋊Dic3—32no—
22#63841—32no—
23#63841—61no—
24C2×#3712D311yescharacteristic
25#63644—32no—
26C3×#818—32no—
27C3×#807—32no—
28#63575D312yescomplemented
29#63573—32no—
30#63573—32no—
31#63573—62no—
32C22×C3×C3⋊D3D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×C4×C3⋊D3C2×C412yes—
38C3×C4×C3⋊D3Q812yes—
39C2×C32×Dic3—21no—
40#6294—42no—
41C32×C3⋊C8C2×C414yescomplemented
42#5121—91no—
43#4881—92no—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×C4×D3—32no—
46C2×C3×C4×D3—61no—
47C2×#624—61no—
48C2×D3×Dic3—62no—
49C2×C32×C8—32no—
50C32×C4⋊C4—31no—
51#3752—61no—
52C4⋊(C3⋊Dic3)—31no—
53C4⋊(C3⋊Dic3)—32no—
54C4⋊(C3⋊Dic3)—61no—
55C4⋊(C3⋊Dic3)—62no—
56C4⋊(C3⋊Dic3)—121no—
57C4⋊(C3⋊Dic3)C2×D312yes—
58C4×C3⋊Dic3—61no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—32no—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—62no—
63C3×C4⋊Dic3C2×D312yes—
64C2×C3×C3⋊C8—22no—
65C2×C3×C3⋊C8C2×D312yes—
66C2×C3×C3⋊C8C2×D312yes—
67C3×ΓL(1,𝔽25)—38no—
68#3724—61no—
69Dic3⋊Dic3—62no—
70(C2×D3)⋊Dic3—62no—
71(C2×D3)⋊Dic3—62no—
72Dic32—62no—
73Dic32—62no—
74#3712Dic314yescomplemented
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—21no—
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3C4⋊C412yes—
79C33×C4C22×C411yescharacteristic
80C33×C4C2×Q811yescharacteristic
81C3⋊(C3⋊Dic3)—41no—
82C3×C3⋊Dic3—42no—
83C3×C3⋊Dic3C4⋊C412yes—
84C32×Dic3—41no—
85C2×C3×M16—91no—
86#926—91no—
87C2×C4⋊Dic3—91no—
88C2×ΓL(1,𝔽25)—32no—
89#892—92no—
90#892—181no—
91D3×C4⋊C4—92no—
92C3×#107—92no—
93#818—32no—
94#817—92no—
95#817—92no—
96#817—182no—
97#808—92no—
98#807—32no—
99C22×C3⋊D3D1211yescharacteristic
100C22×C3×D3—32no—
101C22×C3×D3—61no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—41no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C32×C4C22×D312yes—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—62no—
111C2×C3⋊Dic3—62no—
112C2×C3⋊Dic3—122no—
113C2×C3⋊Dic3—122no—
114C2×C3⋊Dic3D1211yescharacteristic
115C4×C3⋊D3Dic612yes—
116C4×C3⋊D3C2×Dic312yes—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—122no—
124C3×C4×D3—34no—
125C3×C4×D3—34no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128#624—122no—
129D3×Dic3—124no—
130C32×C8—34no—
131C3×C3⋊C8—24no—
132C3×C3⋊C8C4×D314yescomplemented
133C3×C3⋊C8C2×Dic314yescomplemented
134#428—271noSylow
135C2×C33C2×C4⋊C411yescharacteristic
136C2×C33AGL(1,ℤ/8)11yescharacteristic
137C2×C33#13711yescharacteristic
138C3×C3⋊D3—22no—
139C22×C3×C4—91no—
140C22×Dic3—181no—
141C2×C4×D3—32no—
142C2×C4×D3—61no—
143C2×C4×D3—182no—
144C3×M16—94no—
145C2×C3×C8—32no—
146C2×C3×C8—32no—
147C2×C3×C8—62no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—92no—
151#217—181no—
152(C2×D3)⋊C4—182no—
153(C2×D3)⋊C4—182no—
154C4⋊Dic3—32no—
155C4⋊Dic3—32no—
156C4⋊Dic3—62no—
157C4⋊Dic3—92no—
158C4⋊Dic3—91no—
159C4⋊Dic3—181no—
160C4⋊Dic3—182no—
161C4⋊Dic3—181no—
162C4⋊Dic3—361no—
163C4⋊Dic3D3211yescharacteristic
164Dic3⋊C4—182no—
165C4×Dic3—181no—
166C4×Dic3—182no—
167C4×Dic3—182no—
168C2×C3⋊C8D3212yes—
169ΓL(1,𝔽25)—38no—
170C22×C32—22no—
171C22×C32—21no—
172C22×C32—41no—
173C22×C32C2×D1211yescharacteristic
174C22×C32D3×D412yes—
175C2×C3⋊D3—21no—
176C2×C3⋊D3—21no—
177C2×C3⋊D3C4⋊Dic312yes—
178C2×C3×D3—34no—
179C2×C3×D3—62no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—121no—
183C2×C3×D3—121no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—21no—
187C32×C4—22no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—121no—
191C32×C4C2×Dic611yescharacteristic
192C32×C4C2×C4×D312yes—
193C32×C4D3×Q812yes—
194C32×C4C22×Dic311yescharacteristic
195C3⋊Dic3—42no—
196C3⋊Dic3—121no—
197C3⋊Dic3—122no—
198C3⋊Dic3—122no—
199C3⋊Dic3—242no—
200C3⋊Dic3—242no—
201C3⋊Dic3C4⋊Dic312yes—
202C3×Dic3—34no—
203C3×Dic3—42no—
204C3×Dic3—42no—
205C3×Dic3—62no—
206C3×Dic3—122no—
207C3×Dic3—122no—
208C3×Dic3—242no—
209C2×M16—91no—
210#117—271no—
211C2×C4⋊C4—271no—
212#107—92no—
213#106—272no—
214C33#42811yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—32no—
217C22×D3—61no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—41no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—91no—
224C2×C3×C4—92no—
225C2×C3×C4—92no—
226C2×C3×C4—182no—
227C2×C3×C4C2×D3212yes—
228C2×C3×C4C2×D3211yescharacteristic
229C2×Dic3—21no—
230C2×Dic3—32no—
231C2×Dic3—61no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—122no—
235C2×Dic3—182no—
236C2×Dic3—182no—
237C2×Dic3—181no—
238C2×Dic3—362no—
239C2×Dic3—362no—
240C2×Dic3—362no—
241C4×D3—34no—
242C4×D3—34no—
243C4×D3—62no—
244C4×D3—62no—
245C4×D3—364no—
246C3×C8—34no—
247C3×C8—34no—
248C3×C8—64no—
249C3⋊C8D3×Dic314yescomplemented
250C2×C32—22no—
251C2×C32—22no—
252C2×C32—21no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32D3×C4⋊C412yes—
260C2×C32#90811yescharacteristic
261C2×C32#90911yescharacteristic
262C2×C32#91112yes—
263C2×C32#91812yes—
264C2×C32C2×C4⋊Dic311yescharacteristic
265C3⋊D3—22no—
266C3×D3—64no—
267C3×D3—122no—
268C22×C4—91no—
269C22×C4—541no—
270M16—94no—
271C2×C8—32no—
272C4⋊C4—31no—
273C4⋊C4—92no—
274C4⋊C4—271no—
275C4⋊C4—541no—
276C22⋊C4—541no—
277C42—541no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3—92no—
282C22×C3—181no—
283C22×C3—181no—
284C22×C3D3×D1212yes—
285C22×C3#381111yescharacteristic
286C2×D3—34no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—62no—
290C2×D3—121no—
291C2×D3—121no—
292C3×C4—22no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—22no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—92no—
299C3×C4—122no—
300C3×C4—122no—
301C3×C4—362no—
302C3×C4D3×Dic612yes—
303C3×C4#379711yescharacteristic
304C3×C4C4×D3211yescharacteristic
305C3×C4C2×D3×Dic312yes—
306Dic3—34no—
307Dic3—41no—
308Dic3—62no—
309Dic3—122no—
310Dic3—122no—
311Dic3—242no—
312Dic3—362no—
313Dic3—362no—
314Dic3—361no—
315Dic3—722no—
316Dic3—722no—
317C32—21no—
318C32—22no—
319C32—41no—
320C32#488112yescomplemented
321C32#512111yescharacteristic, complemented
322C23—91no—
323C2×C4—61no—
324C2×C4—91no—
325C2×C4—92no—
326C2×C4—92no—
327C2×C4—182no—
328C2×C4—541no—
329C2×C4—1082no—
330C2×C4D3311yescharacteristic
331C8—34no—
332C2×C3—22no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—41no—
341C2×C3—182no—
342C2×C3#6379512yes—
343C2×C3#6380112yes—
344C2×C3#6388311yescharacteristic
345C2×C3D3×C4⋊Dic312yes—
346C2×C3#6392711yescharacteristic
347C2×C3#6394511yescharacteristic
348D3—64no—
349D3—122no—
350C22—92no—
351C22—181no—
352C22—181no—
353C22#8895511yescharacteristic
354C4—92no—
355C4—121no—
356C4—362no—
357C4—1081no—
358C4D32×Dic311yescharacteristic
359C4#8895711yescharacteristic
360C3—21no—
361C3—22no—
362C3—41no—
363C3#1059058211yescharacteristic, complemented
364C3#1059068312yescomplemented
365C2—182no—
366C2#1172288011yescharacteristic
367C2#1172289711yescharacteristic
368C2#1172290811yescharacteristic
subgroupG/H
centerC22#88955
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#428
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722052C2semidirect
5. C3×#63575C2semidirect
17. C3×#3712C4semidirect
28. #63575D3semidirect
41. C32×C3⋊C8C2×C4semidirect
74. #3712Dic3semidirect
132. C3×C3⋊C8C4×D3semidirect
133. C3×C3⋊C8C2×Dic3semidirect
214. C33#428semidirect
249. C3⋊C8D3×Dic3semidirect
320. C32#4881semidirect
321. C32#5121semidirect
363. C3#10590582semidirect
364. C3#10590683semidirect

Maximal subgroups

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

subgroupindex
#117236892
#117234042
C2×C3×#37122
#117220522
C3×#635752
#105906833
#105905823