Finite Group Databasebeta

← #13156452

Subgroups — #13156452

Order 1728.

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

normal subgroups only

This group has 588 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#11723435C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic
3#11722072C211yesmaximal, characteristic
4#11722054C212yesmaximal, complemented
5C3×#63565C212yesmaximal, complemented
6#10591553—31nomaximal
7#10590516—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×Dic6C2211yescharacteristic
10C2×C3×#624—21no—
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3C2212yes—
14#88806—21no—
15#88803—21no—
16C3×#3724—21no—
17#64155—31no—
18C3×D3×C4⋊C4—32no—
19#63900D311yescharacteristic
20#63856—32no—
21#63856—61no—
22#63635—32no—
23C3×#835—32no—
24C3×#810—32no—
25#63573—32no—
26#63573—32no—
27#63573—62no—
28#63565D312yescomplemented
29#63557—31no—
30(C2×D3)⋊(C3⋊C8)—32no—
31C22×C3×C3⋊D3D411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C3×C4×C3⋊D3—22no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3—22no—
38C32×Dic6—22no—
39C3×#624—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#4971—91no—
42#4805—92no—
43C2×C32×Q8—31no—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×C4×D3—32no—
46C2×C3×C4×D3—62no—
47C2×C3×C4×D3—61no—
48C2×C3×Dic6—21no—
49C2×C3×Dic6C2×D312yes—
50C2×#624—21no—
51C32×C4⋊C4—32no—
52#3752—61no—
53C4⋊(C3⋊Dic3)—31no—
54C4⋊(C3⋊Dic3)—32no—
55C4⋊(C3⋊Dic3)—61no—
56C4⋊(C3⋊Dic3)—62no—
57C4⋊(C3⋊Dic3)—121no—
58#3750—61no—
59C2×C3⋊(C3⋊C8)C2×D312yes—
60C3×C4⋊Dic3—22no—
61C3×C4⋊Dic3C2×D312yes—
62C3×C4⋊Dic3C2×D312yes—
63C3×Dic3⋊C4—62no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—62no—
67#3724—21no—
68#3724—62no—
69#3724—62no—
70(C2×D3)⋊Dic3—62no—
71(C2×D3)⋊Dic3—62no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3—21no—
75C2×C3×C3⋊D3SD1612yes—
76C33×C4#4112yes—
77C3⋊(C3⋊Dic3)—41no—
78C3×C3⋊Dic3—21no—
79C32×Dic3—41no—
80C32×Dic3—42no—
81C2×C3×C4⋊C4—91no—
82#946—92no—
83#946—181no—
84#924—91no—
85D3×C4⋊C4—32no—
86C3×#103—92no—
87#835—32no—
88#830—91no—
89(C2×D3)⋊C8—92no—
90#817—92no—
91#810—32no—
92#808—92no—
93#808—92no—
94#808—182no—
95C22×C3⋊D3#6711yescharacteristic
96C22×C3×D3—32no—
97C22×C3×D3—61no—
98C32×Q8—62no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—41no—
102C2×C32×C4—61no—
103C2×C32×C4—62no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C22×D312yes—
106C2×C3⋊Dic3—61no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—62no—
109C2×C3⋊Dic3—122no—
110C2×C3⋊Dic3—122no—
111C2×C3⋊Dic3#6711yescharacteristic
112C4×C3⋊D3—22no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—42no—
119C2×C3×Dic3—61no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—122no—
123C3×C4×D3—124no—
124C3×Dic6—22no—
125C3×Dic6—22no—
126C3×Dic6—42no—
127#624—42no—
128C3⋊(C3⋊C8)—22no—
129C3×C3⋊C8—64no—
130C3×C3⋊C8—62no—
131C3×C3⋊C8—62no—
132#483—271noSylow
133C2×C33#12311yescharacteristic
134C2×C33C2×SD1611yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—22no—
137C2×C3×Q8—32no—
138C2×C3×Q8—61no—
139C22×C3×C4—91no—
140C22×C3×C4—181no—
141C2×C4×D3—32no—
142C2×C4×D3—61no—
143C2×C4×D3—62no—
144C2×Dic6D3211yescharacteristic
145C2×C3×C8—92no—
146C3×C4⋊C4—32no—
147C3×C4⋊C4—32no—
148C3×C4⋊C4—62no—
149C3×C4⋊C4—181no—
150#217—181no—
151(C2×D3)⋊C4—182no—
152(C2×D3)⋊C4—182no—
153C4⋊Dic3—91no—
154C4⋊Dic3—92no—
155C4⋊Dic3—181no—
156C4⋊Dic3—182no—
157C4⋊Dic3—361no—
158C4⋊Dic3D3212yes—
159Dic3⋊C4—62no—
160Dic3⋊C4—181no—
161Dic3⋊C4—182no—
162Dic3⋊C4—182no—
163C2×C3⋊C8—32no—
164C2×C3⋊C8—32no—
165C2×C3⋊C8—62no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32D3×D412yes—
170C22×C32C2×#6711yescharacteristic
171C2×C3⋊D3—21no—
172C2×C3⋊D3—21no—
173C2×C3⋊D3#21412yes—
174C2×C3×D3—34no—
175C2×C3×D3—62no—
176C2×C3×D3—62no—
177C2×C3×D3—62no—
178C2×C3×D3—121no—
179C2×C3×D3—121no—
180C32×C4—22no—
181C32×C4—22no—
182C32×C4—22no—
183C32×C4—42no—
184C32×C4—121no—
185C32×C4—122no—
186C32×C4#23512yes—
187C32×C4#23512yes—
188C32×C4#23712yes—
189C3⋊Dic3—21no—
190C3⋊Dic3—121no—
191C3⋊Dic3—122no—
192C3⋊Dic3—122no—
193C3⋊Dic3—242no—
194C3⋊Dic3—242no—
195C3×Dic3—42no—
196C3×Dic3—42no—
197C3×Dic3—42no—
198C3×Dic3—42no—
199C3×Dic3—62no—
200C3×Dic3—82no—
201C3×Dic3—121no—
202C22⋊Q8—271no—
203C2×C4⋊C4—91no—
204#106—272no—
205#103—92no—
206C22⋊C8—271no—
207C33#48311yescharacteristic, complemented, Sylow
208C23×C3—91no—
209C22×D3—32no—
210C22×D3—61no—
211C3×Q8—62no—
212C3×Q8—62no—
213C3×Q8—122no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—91no—
222C2×C3×C4—122no—
223C2×C3×C4—182no—
224C2×C3×C4—362no—
225C2×C3×C4C2×D3212yes—
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—22no—
228C2×Dic3—21no—
229C2×Dic3—32no—
230C2×Dic3—61no—
231C2×Dic3—182no—
232C2×Dic3—182no—
233C2×Dic3—181no—
234C2×Dic3—362no—
235C2×Dic3—362no—
236C4×D3—62no—
237C4×D3—62no—
238C4×D3—122no—
239C4×D3—124no—
240Dic6—22no—
241C3×C8—182no—
242C3⋊C8—64no—
243C3⋊C8—62no—
244C3⋊C8—62no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—22no—
250C2×C32—21no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32#88312yes—
255C2×C32D3×SD1612yes—
256C2×C32#91812yes—
257C2×C32#92911yescharacteristic
258C2×C32C2×#21411yescharacteristic
259C2×C32#94211yescharacteristic
260C3⋊D3—22no—
261C3×D3—64no—
262C3×D3—122no—
263C2×Q8—31no—
264C22×C4—91no—
265C22×C4—181no—
266C2×C8—92no—
267C4⋊C4—32no—
268C4⋊C4—181no—
269C4⋊C4—271no—
270C4⋊C4—541no—
271C22⋊C4—541no—
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—41no—
275C22×C3—92no—
276C22×C3—181no—
277C22×C3—181no—
278C22×C3D3×#6712yes—
279C22×C3#381111yescharacteristic
280C2×D3—34no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—62no—
284C2×D3—121no—
285C2×D3—121no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—42no—
290C3×C4—122no—
291C3×C4—122no—
292C3×C4—122no—
293C3×C4—122no—
294C3×C4—181no—
295C3×C4—242no—
296C3×C4#379512yes—
297C3×C4#379812yes—
298C3×C4#380412yes—
299Dic3—42no—
300Dic3—41no—
301Dic3—62no—
302Dic3—121no—
303Dic3—362no—
304Dic3—361no—
305Dic3—362no—
306Dic3—722no—
307Dic3—722no—
308C32—22no—
309C32—21no—
310C32—41no—
311C32#480512yescomplemented
312C32#497111yescharacteristic, complemented
313C23—91no—
314Q8—62no—
315C2×C4—61no—
316C2×C4—62no—
317C2×C4—91no—
318C2×C4—182no—
319C2×C4—362no—
320C2×C4—541no—
321C2×C4D3311yescharacteristic
322C8—182no—
323C2×C3—22no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—22no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—182no—
333C2×C3#6390512yes—
334C2×C3D3×#21412yes—
335C2×C3#6394011yescharacteristic
336C2×C3#6394511yescharacteristic
337C2×C3#6394812yes—
338C2×C3#6397511yescharacteristic
339D3—64no—
340D3—122no—
341C22—92no—
342C22—181no—
343C22—181no—
344C22#8895211yescharacteristic
345C4—122no—
346C4—121no—
347C4—181no—
348C4—1081no—
349C4#8896012yes—
350C3—21no—
351C3—22no—
352C3—41no—
353C3#1059051612yescomplemented
354C3#1059155311yescharacteristic, complemented
355C2—182no—
356C2#1172283711yescharacteristic
357C2#1172289011yescharacteristic
358C2#1172299211yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#483
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722054C2semidirect
5. C3×#63565C2semidirect
28. #63565D3semidirect
207. C33#483semidirect
311. C32#4805semidirect
312. C32#4971semidirect
353. C3#10590516semidirect
354. C3#10591553semidirect

Maximal subgroups

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

subgroupindex
#117234352
C3×#639002
#117220722
#117220542
C3×#635652
#105915533
#105905163