Finite Group Databasebeta

← #13147777

Subgroups — #13147777

Order 1728.

1083 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 1083 conjugacy classes of subgroups — too many to draw. Only its 215 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
1C2×((C3⋊D3)⋊C3)⋊Q8C212yesmaximal
2#11721715C211yesmaximal, characteristic, complemented
3C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
4#64078⋊C3C211yesmaximal, characteristic, complemented
5#11721129C214yesmaximal
6#11721116C214yesmaximal, complemented
7#11721090C212yesmaximal, complemented
8#10593909—31nomaximal
9D3×#869—31nomaximal
10C2×#6229C2212yes—
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2212yes—
13C2×#634⋊C3C2212yes—
14C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
15C2×#6196C2214yes—
16((C3⋊D3)⋊C3)⋊Q8—28no—
17C42×H(3,𝔽3)C2211yescharacteristic
18C4⋊#1084C2214yes—
19C4⋊((C3⋊Dic3)⋊C3)C2214yes—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C4×(C3⋊Dic3)⋊C3C2212yes—
22#88448C2218yes—
23C2×#3797—32no—
24C2×D3×Dic6—32no—
25C42×C3⋊D3D311yescharacteristic
26#64078D311yescharacteristic, complemented
27C3×C42×D3—31no—
28C3×#869—31no—
29C3×#869—31no—
30C3×#869—31no—
31C3×#869—61no—
32#63919—34no—
33#63918—34no—
34#63900—34no—
35D3×C4⋊Dic3—34no—
36#63873—32no—
37#63866—32no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2312yes—
41C2×#1084C2314yes—
42#6229—24no—
43C2×(C3⋊Dic3)⋊C3C2314yes—
44C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C4×(C3⋊D3)⋊C3D414yes—
47C4×(C3⋊D3)⋊C3Q818yescomplemented
48#634⋊C3—24no—
49((C3⋊D3)⋊C3)⋊C4—28no—
50#6196—28no—
51D3×C4⋊Q8—91no—
52D3×C4⋊Q8—91no—
53C2×#869—91no—
54C2×C4×C3⋊D3C2×D312yes—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×#634C2×D312yes—
57C2×C3×C4×D3—31no—
58C2×C3×C4×D3—32no—
59C2×C3×Dic6—32no—
60C2×C3×Dic6—32no—
61C2×C3×Dic6—32no—
62C2×C3×Dic6—62no—
63C2×#627—34no—
64C2×#627—34no—
65C2×#624—34no—
66C2×D3×Dic3—34no—
67#3797—68no—
68D3×Dic6—68no—
69C32×C42—21no—
70C32×C42C2×D311yescharacteristic
71C32×C42C2×D311yescharacteristic
72C4⋊(C3⋊Dic3)C2×D314yes—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C4×C3⋊Dic3C2×D312yes—
75C3×C4⋊Dic3—34no—
76C3×C4⋊Dic3—34no—
77C3×C4⋊Dic3—34no—
78C3×C4⋊Dic3—64no—
79C3×C4×Dic3—31no—
80C3×C4×Dic3—32no—
81#3724—38no—
82Dic3⋊Dic3—38no—
83C22×H(3,𝔽3)C2411yescharacteristic
84C2×(C3⋊D3)⋊C3C2×D412yes—
85C2×(C3⋊D3)⋊C3C2×Q814yes—
86C4×H(3,𝔽3)C2×D412yes—
87C4×H(3,𝔽3)C2×Q814yescomplemented
88#1084—24no—
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3C2×D412yes—
91(C3⋊Dic3)⋊C3C2×Q814yescomplemented
92C2×D3×Q8—92no—
93C2×D3×Q8—92no—
94C22×Dic6—92no—
95C3×C4⋊Q8—91no—
96C3×C4⋊Q8—91no—
97C2×C3×C42—91no—
98#944—92no—
99#944—92no—
100C2×C4⋊Dic3—94no—
101D3×C4⋊C4—94no—
102D3×C4⋊C4—94no—
103#888—94no—
104#888—94no—
105C42×D3—31no—
106C42×D3—91no—
107#869—31no—
108#869—31no—
109#869—31no—
110#869—61no—
111#869—91no—
112#869—94no—
113#869—92no—
114C22×C3⋊D3C22×D311yescharacteristic
115C22×C3×D3—31no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D312yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D312yes—
122C2×C3⋊Dic3C22×D314yes—
123C2×C3⋊Dic3C22×D312yes—
124C2×C3⋊Dic3C22×D311yescharacteristic
125C4×C3⋊D3Dic618yescomplemented
126C4×C3⋊D3D1214yes—
127#634—24no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—64no—
134C3×C4×D3—34no—
135C3×C4×D3—38no—
136C3×Dic6—64no—
137C3×Dic6—64no—
138C3×Dic6—64no—
139C3×Dic6—124no—
140#627—68no—
141#627—68no—
142#624—68no—
143D3×Dic3—68no—
144C2×C4⋊Q8—271noSylow
145C2×H(3,𝔽3)C22×D411yescharacteristic
146C2×H(3,𝔽3)C22×Q812yes—
147(C3⋊D3)⋊C3C4⋊Q814yescomplemented
148C2×C3×Q8—92no—
149C2×C3×Q8—92no—
150C22×C3×C4—92no—
151C22×C3×C4—91no—
152C22×Dic3—94no—
153D3×Q8—188no—
154D3×Q8—188no—
155C2×C4×D3—32no—
156C2×C4×D3—31no—
157C2×C4×D3—92no—
158C2×C4×D3—91no—
159C2×C4×D3—94no—
160C2×C4×D3—94no—
161C2×Dic6—32no—
162C2×Dic6—32no—
163C2×Dic6—32no—
164C2×Dic6—62no—
165C2×Dic6—92no—
166C2×Dic6—94no—
167C2×Dic6—94no—
168C2×Dic6—94no—
169C2×Dic6—188no—
170C3×C4⋊C4—94no—
171C3×C4⋊C4—94no—
172C3×C42—31no—
173C3×C42—31no—
174C3×C42—61no—
175C3×C42—92no—
176C3×C42—91no—
177C3×C42D3211yescharacteristic
178C4⋊Dic3—34no—
179C4⋊Dic3—34no—
180C4⋊Dic3—34no—
181C4⋊Dic3—64no—
182C4⋊Dic3—94no—
183C4⋊Dic3—98no—
184Dic3⋊C4—98no—
185Dic3⋊C4—98no—
186C4×Dic3—32no—
187C4×Dic3—31no—
188C4×Dic3—92no—
189C4×Dic3—91no—
190C22×C32—21no—
191C22×C32C23×D311yescharacteristic
192C22×C32C23×D311yescharacteristic
193C2×C3⋊D3C2×Dic614yes—
194C2×C3⋊D3C2×D1212yes—
195C2×C3×D3—32no—
196C2×C3×D3—34no—
197C32×C4—24no—
198C32×C4—22no—
199C32×C4C2×Dic614yescomplemented
200C32×C4C2×D1212yes—
201C32×C4D3×D412yes—
202C32×C4D3×Q814yescomplemented
203C3⋊Dic3—24no—
204C3⋊Dic3C2×Dic614yescomplemented
205C3⋊Dic3C2×D1212yes—
206C3×Dic3—32no—
207C3×Dic3—34no—
208C3×Dic3—64no—
209C3×Dic3—64no—
210C3×Dic3—64no—
211C3×Dic3—124no—
212C22×Q8—272no—
213C4⋊Q8—91no—
214C4⋊Q8—91no—
215C4⋊Q8—274no—
216C4⋊Q8—272no—
217C2×C4⋊C4—274no—
218C2×C42—91no—
219H(3,𝔽3)C2×C4⋊Q811yescharacteristic, complemented, Sylow
220C23×C3—91no—
221C22×D3—31no—
222C22×D3—91no—
223C3×Q8—184no—
224C3×Q8—184no—
225C2×C3×C4—32no—
226C2×C3×C4—31no—
227C2×C3×C4—32no—
228C2×C3×C4—31no—
229C2×C3×C4—62no—
230C2×C3×C4—61no—
231C2×C3×C4—94no—
232C2×C3×C4—98no—
233C2×C3×C4—92no—
234C2×C3×C4—91no—
235C2×C3×C4—94no—
236C2×C3×C4—94no—
237C2×C3×C4C2×D3212yes—
238C2×C3×C4C2×D3211yescharacteristic
239C2×Dic3—32no—
240C2×Dic3—31no—
241C2×Dic3—34no—
242C2×Dic3—34no—
243C2×Dic3—34no—
244C2×Dic3—64no—
245C2×Dic3—92no—
246C2×Dic3—91no—
247C2×Dic3—94no—
248C2×Dic3—188no—
249C4×D3—38no—
250C4×D3—34no—
251C4×D3—98no—
252C4×D3—94no—
253C4×D3—188no—
254C4×D3—188no—
255Dic6—64no—
256Dic6—64no—
257Dic6—64no—
258Dic6—124no—
259Dic6—184no—
260Dic6—188no—
261Dic6—188no—
262Dic6—188no—
263C2×C32—22no—
264C2×C32—21no—
265C2×C32C22×Dic612yes—
266C2×C32C22×D1211yescharacteristic
267C2×C32C2×D3×D411yescharacteristic
268C2×C32C2×D3×Q812yes—
269C3⋊D3#86914yescomplemented
270C3×D3—34no—
271C2×Q8—92no—
272C2×Q8—92no—
273C2×Q8—274no—
274C2×Q8—548no—
275C22×C4—92no—
276C22×C4—91no—
277C22×C4—274no—
278C4⋊C4—94no—
279C4⋊C4—94no—
280C4⋊C4—278no—
281C42—91no—
282C42—92no—
283C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—61no—
287C22×C3—92no—
288C22×C3—94no—
289C22×C3C22×D3211yescharacteristic
290C2×D3—34no—
291C2×D3—32no—
292C2×D3—92no—
293C2×D3—94no—
294C3×C4—34no—
295C3×C4—34no—
296C3×C4—32no—
297C3×C4—32no—
298C3×C4—64no—
299C3×C4—62no—
300C3×C4—92no—
301C3×C4—94no—
302C3×C4—184no—
303C3×C4—184no—
304C3×C4D3×Dic614yes—
305C3×C4D3×D1212yes—
306Dic3—34no—
307Dic3—32no—
308Dic3—64no—
309Dic3—64no—
310Dic3—64no—
311Dic3—92no—
312Dic3—94no—
313Dic3—124no—
314Dic3—184no—
315C32—21no—
316C32C2×#86911yescharacteristic, complemented
317C32D3×C4⋊Q811yescharacteristic, complemented
318C23—91no—
319Q8—184no—
320Q8—184no—
321Q8—548no—
322C2×C4—92no—
323C2×C4—91no—
324C2×C4—98no—
325C2×C4—94no—
326C2×C4—94no—
327C2×C4—94no—
328C2×C4—548no—
329C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
330C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
331C2×C3—32no—
332C2×C3—32no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—62no—
336C2×C3—61no—
337C2×C3—94no—
338C2×C3C2×D3×Dic612yes—
339C2×C3C2×D3×D1211yescharacteristic
340D3—34no—
341D3—94no—
342C22—94no—
343C22—92no—
344C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
345C4—94no—
346C4—92no—
347C4—184no—
348C4—184no—
349C4((C3⋊D3)⋊C3)⋊Q814yescomplemented
350C4#8865412yes—
351C3—31no—
352C3—31no—
353C3—61no—
354C3D3×#86911yescharacteristic
355C2—94no—
356C2C2×((C3⋊D3)⋊C3)⋊Q812yes—
357C2C2×#8865411yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721715C2semidirect
4. #64078⋊C3C2semidirect
6. #11721116C2semidirect
7. #11721090C2semidirect
26. #64078D3semidirect
47. C4×(C3⋊D3)⋊C3Q8semidirect
87. C4×H(3,𝔽3)C2×Q8semidirect
91. (C3⋊Dic3)⋊C3C2×Q8semidirect
125. C4×C3⋊D3Dic6semidirect
147. (C3⋊D3)⋊C3C4⋊Q8semidirect
199. C32×C4C2×Dic6semidirect
202. C32×C4D3×Q8semidirect
204. C3⋊Dic3C2×Dic6semidirect
219. H(3,𝔽3)C2×C4⋊Q8semidirect
269. C3⋊D3#869semidirect
316. C32C2×#869semidirect
317. C32D3×C4⋊Q8semidirect
349. C4((C3⋊D3)⋊C3)⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊Q82
#117217152
C42×(C3⋊D3)⋊C32
#64078⋊C32
#117211292
#117211162
#117210902
#105939093
D3×#8693