Finite Group Databasebeta

← #13162016

Subgroups — #13162016

Order 1728.

576 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 119 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×Q8×C3⋊Dic3C211yesmaximal, characteristic
2C4⋊#6296C211yesmaximal, characteristic
3C3×#63662C212yesmaximal
4#11722079C212yesmaximal
5#11722078C211yesmaximal, characteristic
6#10591630—31nomaximal
7#10591544—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×#6296—21no—
10C3×C4⋊(C3⋊Dic3)—21no—
11C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C3×C4×C3⋊Dic3—21no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88815—21no—
17Q8×C3⋊Dic3D311yescharacteristic
18C3×Q8×Dic3—32no—
19C3×Q8×Dic3—61no—
20#63919—31no—
21#63918—32no—
22#63662D312yes—
23C3×#835—32no—
24C3×#835—32no—
25C3×#835—62no—
26#63574—32no—
27#63573—32no—
28#63573—32no—
29#63572—31no—
30Dic3⋊(C3⋊C8)—32no—
31C33×Q8—21no—
32C33×Q8Q812yescomplemented
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C3×C3⋊Dic3D411yescharacteristic
39#6296—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#5020—91no—
42#4791—92no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D312yes—
47C2×C32×Q8C2×D311yescharacteristic
48C2×#627—62no—
49C2×#627—61no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)C2×D311yescharacteristic
52C4⋊(C3⋊Dic3)C2×D312yes—
53C4×C3⋊Dic3—21no—
54C4×C3⋊Dic3C2×D311yescharacteristic
55C2×C3⋊(C3⋊C8)C2×D312yes—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—62no—
60C3×C4⋊Dic3—61no—
61C3×C4⋊Dic3—62no—
62C3×C4⋊Dic3—62no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—62no—
65C3×C4×Dic3—61no—
66C3×C4×Dic3—62no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70#3724—61no—
71Dic3⋊Dic3—62no—
72C22×C33C2×D411yescharacteristic
73C33×C4—22no—
74C33×C4C2×Q811yescharacteristic
75C33×C4#4111yescharacteristic
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3—42no—
78C3×C3⋊Dic3—41no—
79C3×C3⋊Dic3SD1612yes—
80C3×C4×Q8—91no—
81Q8×Dic3—32no—
82Q8×Dic3—61no—
83#888—92no—
84#869—91no—
85C3×#103—92no—
86#835—32no—
87#835—32no—
88#835—62no—
89#817—92no—
90Dic3⋊C8—92no—
91#808—92no—
92#808—92no—
93C4⋊(C3⋊C8)—91no—
94C32×Q8—21no—
95C32×Q8—22no—
96C32×Q8—22no—
97C32×Q8—22no—
98C32×Q8—24no—
99C32×Q8—42no—
100C32×Q8—42no—
101C32×Q8—42no—
102C32×Q8Dic612yescomplemented
103C32×Q8Dic614yescomplemented
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—42no—
111C2×C32×C4—42no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C22×D312yes—
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3#6711yescharacteristic
117C2×C3⋊Dic3#6711yescharacteristic
118C2×C3×Dic3—32no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—122no—
127#627—124no—
128#627—122no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—62no—
131C3×C3⋊C8—62no—
132C3×C3⋊C8—122no—
133#475—271noSylow
134C2×C33C22⋊Q811yescharacteristic
135C2×C33C2×SD1611yescharacteristic
136C2×C33#13711yescharacteristic
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—41no—
140C2×C3×Q8D3211yescharacteristic
141C2×C3×Q8D3212yes—
142C2×Dic6—182no—
143C2×Dic6—181no—
144C2×C3×C8—92no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—92no—
147C3×C4⋊C4—181no—
148C3×C42—91no—
149C3×C42—181no—
150C4⋊Dic3—32no—
151C4⋊Dic3—32no—
152C4⋊Dic3—32no—
153C4⋊Dic3—61no—
154C4⋊Dic3—62no—
155C4⋊Dic3—62no—
156C4⋊Dic3—62no—
157C4⋊Dic3—181no—
158Dic3⋊C4—182no—
159C4×Dic3—32no—
160C4×Dic3—61no—
161C4×Dic3—62no—
162C4×Dic3—62no—
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
171C32×C4—22no—
172C32×C4—24no—
173C32×C4—24no—
174C32×C4—22no—
175C32×C4—21no—
176C32×C4—22no—
177C32×C4—21no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—44no—
181C32×C4—44no—
182C32×C4C2×Dic612yes—
183C32×C4C2×Dic611yescharacteristic
184C32×C4#23511yescharacteristic
185C32×C4#23712yes—
186C3⋊Dic3—21no—
187C3⋊Dic3—41no—
188C3⋊Dic3—42no—
189C3⋊Dic3#21512yes—
190C3×Dic3—34no—
191C3×Dic3—62no—
192C3×Dic3—62no—
193C3×Dic3—121no—
194C3×Dic3—122no—
195C3×Dic3—122no—
196C3×Dic3—122no—
197C3×Dic3—122no—
198C3×Dic3—242no—
199C4⋊Q8—271no—
200C4×Q8—91no—
201#106—272no—
202C4⋊C8—271no—
203#103—92no—
204C33#47511yescharacteristic, complemented, Sylow
205C3×Q8—22no—
206C3×Q8—22no—
207C3×Q8—21no—
208C3×Q8—22no—
209C3×Q8—24no—
210C3×Q8—42no—
211C3×Q8—42no—
212C3×Q8—42no—
213C3×Q8#62714yescomplemented
214C3×Q8#62712yescomplemented
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—182no—
226C2×C3×C4—181no—
227C2×C3×C4C2×D3212yes—
228C2×C3×C4C2×D3211yescharacteristic
229C2×Dic3—32no—
230C2×Dic3—32no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—122no—
238Dic6—364no—
239Dic6—362no—
240C3×C8—182no—
241C3⋊C8—62no—
242C3⋊C8—62no—
243C3⋊C8—122no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—22no—
249C2×C32—21no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32#87812yes—
254C2×C32D3×SD1612yes—
255C2×C32#91812yes—
256C2×C32#92411yescharacteristic
257C2×C32C2×#21511yescharacteristic
258C2×C32#95111yescharacteristic
259C2×Q8—541no—
260C2×Q8#111311yescharacteristic
261C2×C8—92no—
262C4⋊C4—92no—
263C4⋊C4—91no—
264C4⋊C4—181no—
265C4⋊C4—541no—
266C42—91no—
267C42—181no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—41no—
271C22×C3D3×#6712yes—
272C22×C3#381111yescharacteristic
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—22no—
280C3×C4—44no—
281C3×C4—44no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—92no—
285C3×C4—181no—
286C3×C4—362no—
287C3×C4—361no—
288C3×C4#380412yes—
289C3×C4#380511yescharacteristic
290C3×C4C2×#62712yes—
291C3×C4C2×#62711yescharacteristic
292Dic3—34no—
293Dic3—62no—
294Dic3—62no—
295Dic3—121no—
296Dic3—122no—
297Dic3—122no—
298Dic3—122no—
299Dic3—122no—
300Dic3—242no—
301C32—22no—
302C32—21no—
303C32—41no—
304C32#479112yescomplemented
305C32#502011yescharacteristic, complemented
306Q8—21no—
307Q8—1082no—
308Q8#629612yescomplemented
309C2×C4—21no—
310C2×C4—91no—
311C2×C4—91no—
312C2×C4—181no—
313C2×C4—182no—
314C2×C4C2×#111311yescharacteristic
315C8—182no—
316C2×C3—21no—
317C2×C3—22no—
318C2×C3—21no—
319C2×C3—22no—
320C2×C3—21no—
321C2×C3—22no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3D3×#21512yes—
326C2×C3#6394011yescharacteristic
327C2×C3#6394312yes—
328C2×C3#6394511yescharacteristic
329C2×C3#6396412yes—
330C2×C3#6398211yescharacteristic
331C22#8904911yescharacteristic
332C4—22no—
333C4—92no—
334C4—181no—
335C4—362no—
336C4—361no—
337C4#8904611yescharacteristic
338C4C2×#629611yescharacteristic
339C3—21no—
340C3—22no—
341C3—41no—
342C3#1059154412yescomplemented
343C3#1059163011yescharacteristic, complemented
344C2#1172370811yescharacteristic
345C2#1172371111yescharacteristic
346C2#1172372111yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8#1113
hypocenterC33#475
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
32. C33×Q8Q8semidirect
102. C32×Q8Dic6semidirect
103. C32×Q8Dic6semidirect
204. C33#475semidirect
213. C3×Q8#627semidirect
214. C3×Q8#627semidirect
304. C32#4791semidirect
305. C32#5020semidirect
308. Q8#6296semidirect
342. C3#10591544semidirect
343. C3#10591630semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C4⋊#62962
C3×#636622
#117220792
#117220782
#105916303
#105915443