Finite Group Databasebeta

← #10596198⋊C3 #13153317

Subgroups — #10596198⋊C3 #13153317

Order 1728.

1048 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C212yesmaximal, complemented
2H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
3#64155⋊C3C212yesmaximal, complemented
4#64105⋊C3C214yesmaximal, complemented
5#64102⋊C3C212yesmaximal
6#64097⋊C3C212yesmaximal, complemented
7C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
8C4×#634⋊C3C211yesmaximal, characteristic, complemented
9#10596198C311yesmaximal, characteristic, complemented
10C3×#5727—31nomaximal
11C2×Q8×H(3,𝔽3)C2212yes—
12Q8×(C3⋊D3)⋊C3—28no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2214yes—
16C2×#634⋊C3C2214yes—
17C2×#634⋊C3C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2212yes—
19C4⋊C4×H(3,𝔽3)C2212yes—
20C42×H(3,𝔽3)C2211yescharacteristic
21#3752⋊C3C2212yes—
22#3752⋊C3C2214yes—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26#3750⋊C3C2214yes—
27C4×(C3⋊Dic3)⋊C3C2212yes—
28C2×Q8×C3⋊D3C2×C312yescomplemented
29C2×C3×D3×Q8—32no—
30C32×C4⋊Q8—21no—
31C32×C4⋊Q8D311yescharacteristic, complemented
32C32×C4⋊Q8C2×C311yescharacteristic, complemented
33#64155C2×C312yescomplemented
34#64105C2×C314yescomplemented
35#64102C2×C312yes—
36#64097C2×C312yescomplemented
37C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
38C4×#634C2×C311yescharacteristic, complemented
39C3×#946—32no—
40C3×#896—34no—
41C3×#893—32no—
42C3×#888—32no—
43C3×C4×D12—31no—
44C3×C4×Dic6—31no—
45C22×(C3⋊D3)⋊C3C2312yes—
46Q8×H(3,𝔽3)—24no—
47C2×C4×H(3,𝔽3)C2314yes—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C4⋊((C3⋊D3)⋊C3)Q814yescomplemented
55C4×(C3⋊D3)⋊C3—24no—
56C4×(C3⋊D3)⋊C3—28no—
57#634⋊C3—28no—
58#634⋊C3D414yescomplemented
59C3×D4×Q8—91no—
60#5727—31no—
61#5727—91no—
62C2×C32×Q8—22no—
63C2×C32×Q8C2×D312yes—
64C2×C32×Q8C22×C312yes—
65Q8×C3⋊D3—28no—
66C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
67C2×C4×C3⋊D3C22×C312yes—
68C2×C4×C3⋊D3C22×C314yes—
69C2×#634C22×C314yes—
70C2×#634C22×C311yescharacteristic
71C3×D3×Q8—68no—
72C2×C3×D12—31no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—32no—
75C2×C3×Dic6—31no—
76C2×C3×Dic6—34no—
77C32×C4⋊C4—22no—
78C32×C4⋊C4—22no—
79C32×C4⋊C4C2×D312yes—
80C32×C4⋊C4C2×D312yes—
81C32×C4⋊C4C22×C312yes—
82C32×C4⋊C4C22×C312yes—
83C32×C42—21no—
84C32×C42C2×D311yescharacteristic
85C32×C42C22×C311yescharacteristic
86#3752C22×C312yes—
87#3752C22×C314yes—
88C4⋊(C3⋊Dic3)C22×C311yescharacteristic
89C4⋊(C3⋊Dic3)C22×C311yescharacteristic
90#3750C22×C312yes—
91#3750C22×C314yes—
92C4×C3⋊Dic3C22×C312yes—
93C3×(C2×D3)⋊C4—34no—
94C3×(C2×D3)⋊C4—32no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×Dic3⋊C4—34no—
98C3×Dic3⋊C4—32no—
99C3×C4×Dic3—32no—
100C22×H(3,𝔽3)C2411yescharacteristic
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3C2×Q814yescomplemented
104C4×H(3,𝔽3)—24no—
105C4×H(3,𝔽3)—21no—
106C4×H(3,𝔽3)C2×D412yes—
107C4×H(3,𝔽3)C2×Q812yescomplemented
108(C3⋊Dic3)⋊C3—22no—
109(C3⋊Dic3)⋊C3—22no—
110(C3⋊Dic3)⋊C3C2×D414yes—
111C22×C3×Q8—92no—
112C2×D3×Q8—32no—
113C2×D3×Q8—92no—
114C3×C4⋊Q8—31no—
115C3×C4⋊Q8—31no—
116C3×C4⋊Q8—61no—
117C3×C4⋊Q8—92no—
118C3×C4⋊Q8C3×D311yescharacteristic
119C3×C22⋊Q8—94no—
120C3×C22⋊Q8—92no—
121C3×C4×Q8—91no—
122C3×C4×D4—91no—
123C3×C4×D4—92no—
124#946—32no—
125#946—92no—
126#896—34no—
127#896—94no—
128#893—32no—
129#893—92no—
130#888—32no—
131#888—92no—
132C4×D12—31no—
133C4×D12—91no—
134C4×Dic6—31no—
135C4×Dic6—91no—
136C22×C3⋊D3C23×C312yes—
137C22×C3×D3—32no—
138C32×Q8—24no—
139C32×Q8—24no—
140C32×Q8—44no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—24no—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D314yes—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C23×C311yescharacteristic
150C2×C32×C4C23×C314yes—
151C2×C32×C4C23×C311yescharacteristic
152C2×C32×C4C23×C311yescharacteristic
153C2×C3⋊Dic3C23×C312yes—
154C2×C3⋊Dic3C23×C312yes—
155C2×C3⋊Dic3C23×C312yes—
156C4⋊(C3⋊D3)C3×Q814yescomplemented
157C4×C3⋊D3—24no—
158C4×C3⋊D3—28no—
159#634—28no—
160#634C3×D414yescomplemented
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C3×D12—34no—
165C3×C4×D3—68no—
166C3×C4×D3—64no—
167C3×Dic6—34no—
168C3×Dic6—68no—
169D4×Q8—91noSylow
170C2×H(3,𝔽3)C22×D411yescharacteristic
171C2×H(3,𝔽3)C22×Q811yescharacteristic
172C2×H(3,𝔽3)2−1+411yescharacteristic
173(C3⋊D3)⋊C3—24no—
174C2×C3×Q8—32no—
175C2×C3×Q8—32no—
176C2×C3×Q8—62no—
177C2×C3×Q8—91no—
178C2×C3×Q8—94no—
179C2×C3×Q8—188no—
180C2×C3×Q8C2×C3×D312yes—
181C2×C3×D4—91no—
182C22×C3×C4—92no—
183C22×C3×C4—94no—
184D3×Q8—68no—
185D3×Q8—188no—
186C2×D12—31no—
187C2×D12—91no—
188C2×C4×D3—32no—
189C2×C4×D3—34no—
190C2×C4×D3—92no—
191C2×C4×D3—94no—
192C2×Dic6—31no—
193C2×Dic6—34no—
194C2×Dic6—91no—
195C2×Dic6—94no—
196C3×C4⋊C4—32no—
197C3×C4⋊C4—32no—
198C3×C4⋊C4—32no—
199C3×C4⋊C4—32no—
200C3×C4⋊C4—62no—
201C3×C4⋊C4—62no—
202C3×C4⋊C4—92no—
203C3×C4⋊C4—94no—
204C3×C4⋊C4—91no—
205C3×C4⋊C4—91no—
206C3×C4⋊C4C2×C3×D312yes—
207C3×C4⋊C4C2×C3×D312yes—
208C3×C22⋊C4—92no—
209C3×C22⋊C4—94no—
210C3×C42—31no—
211C3×C42—31no—
212C3×C42—61no—
213C3×C42—92no—
214C3×C42C2×C3×D311yescharacteristic
215(C2×D3)⋊C4—32no—
216(C2×D3)⋊C4—34no—
217(C2×D3)⋊C4—92no—
218(C2×D3)⋊C4—94no—
219C4⋊Dic3—31no—
220C4⋊Dic3—31no—
221C4⋊Dic3—91no—
222C4⋊Dic3—91no—
223Dic3⋊C4—32no—
224Dic3⋊C4—34no—
225Dic3⋊C4—92no—
226Dic3⋊C4—94no—
227C4×Dic3—32no—
228C4×Dic3—92no—
229C22×C32—21no—
230C22×C32C23×D311yescharacteristic
231C22×C32C24×C311yescharacteristic
232C2×C3⋊D3—22no—
233C2×C3⋊D3—22no—
234C2×C3⋊D3C2×C3×Q814yescomplemented
235C2×C3×D3—34no—
236C2×C3×D3—62no—
237C2×C3×D3—62no—
238C32×C4—22no—
239C32×C4—24no—
240C32×C4—21no—
241C32×C4—24no—
242C32×C4—21no—
243C32×C4—22no—
244C32×C4—22no—
245C32×C4—44no—
246C32×C4D3×D412yes—
247C32×C4D3×Q812yescomplemented
248C32×C4C2×C3×D412yes—
249C32×C4C2×C3×Q812yescomplemented
250C3⋊Dic3—22no—
251C3⋊Dic3—22no—
252C3⋊Dic3C2×C3×D414yes—
253C3×Dic3—34no—
254C3×Dic3—62no—
255C3×Dic3—62no—
256C22×Q8—92no—
257C4⋊Q8—92no—
258C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
259C22⋊Q8—92no—
260C22⋊Q8—94no—
261C4×Q8—91no—
262C4×D4—92no—
263C4×D4—91no—
264H(3,𝔽3)D4×Q811yescharacteristic, complemented, Sylow
265C23×C3—92no—
266C22×D3—32no—
267C22×D3—92no—
268C3×Q8—24no—
269C3×Q8—64no—
270C3×Q8—64no—
271C3×Q8—94no—
272C3×Q8—124no—
273C3×Q8—188no—
274C3×D4—94no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—34no—
281C2×C3×C4—31no—
282C2×C3×C4—34no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—64no—
287C2×C3×C4—92no—
288C2×C3×C4—92no—
289C2×C3×C4—92no—
290C2×C3×C4—184no—
291C2×C3×C4—188no—
292C2×C3×C4C22×C3×D311yescharacteristic
293C2×C3×C4C22×C3×D314yes—
294C2×C3×C4C22×C3×D311yescharacteristic
295C2×C3×C4C22×C3×D311yescharacteristic
296C2×Dic3—32no—
297C2×Dic3—32no—
298C2×Dic3—32no—
299C2×Dic3—92no—
300C2×Dic3—92no—
301C2×Dic3—92no—
302D12—34no—
303D12—94no—
304C4×D3—68no—
305C4×D3—64no—
306C4×D3—188no—
307C4×D3—184no—
308Dic6—34no—
309Dic6—68no—
310Dic6—94no—
311Dic6—188no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32C2×D3×D411yescharacteristic
316C2×C32C2×D3×Q811yescharacteristic
317C2×C32#100711yescharacteristic
318C2×C32C22×C3×D411yescharacteristic
319C2×C32C22×C3×Q811yescharacteristic
320C2×C32C3×2−1+411yescharacteristic
321C3⋊D3—24no—
322C3×D3—64no—
323C2×Q8—91no—
324C2×Q8—94no—
325C2×Q8—188no—
326C2×Q8C2×(C3⋊D3)⋊C312yes—
327C2×D4—91no—
328C22×C4—94no—
329C22×C4—92no—
330C4⋊C4—91no—
331C4⋊C4—92no—
332C4⋊C4—91no—
333C4⋊C4—94no—
334C4⋊C4C2×(C3⋊D3)⋊C312yes—
335C4⋊C4C2×(C3⋊D3)⋊C312yes—
336C22⋊C4—94no—
337C22⋊C4—92no—
338C42—92no—
339C42C2×(C3⋊D3)⋊C311yescharacteristic
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—61no—
343C22×C3—94no—
344C22×C3—182no—
345C22×C3—182no—
346C22×C3C23×C3×D311yescharacteristic
347C2×D3—34no—
348C2×D3—62no—
349C2×D3—62no—
350C2×D3—94no—
351C2×D3—182no—
352C2×D3—182no—
353C3×C4—24no—
354C3×C4—21no—
355C3×C4—32no—
356C3×C4—32no—
357C3×C4—32no—
358C3×C4—32no—
359C3×C4—64no—
360C3×C4—64no—
361C3×C4—61no—
362C3×C4—61no—
363C3×C4—62no—
364C3×C4—62no—
365C3×C4—62no—
366C3×C4—94no—
367C3×C4—124no—
368C3×C4—182no—
369C3×C4—182no—
370C3×C4C3×D3×D412yes—
371C3×C4C3×D3×Q812yes—
372Dic3—34no—
373Dic3—62no—
374Dic3—62no—
375Dic3—94no—
376Dic3—182no—
377Dic3—182no—
378C32—21no—
379C32#572711yescharacteristic, complemented
380C32C3×D4×Q811yescharacteristic, complemented
381C23—92no—
382Q8—24no—
383Q8—94no—
384Q8—188no—
385D4—94no—
386C2×C4—92no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—188no—
390C2×C4—184no—
391C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
392C2×C4C22×(C3⋊D3)⋊C314yes—
393C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
394C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—61no—
404C2×C3—184no—
405C2×C3C2×C3×D3×D411yescharacteristic
406C2×C3C2×C3×D3×Q811yescharacteristic
407C2×C3C3×#100711yescharacteristic
408D3—64no—
409D3—184no—
410C22—94no—
411C22—182no—
412C22—182no—
413C22C23×(C3⋊D3)⋊C311yescharacteristic
414C4—24no—
415C4—21no—
416C4—94no—
417C4—182no—
418C4—182no—
419C4D4×(C3⋊D3)⋊C312yes—
420C4Q8×(C3⋊D3)⋊C312yescomplemented
421C3—31no—
422C3—31no—
423C3—61no—
424C3C3×#572711yescharacteristic
425C2—184no—
426C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
427C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
428C2#1172262711yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×(C3⋊D3)⋊C3C2semidirect
2. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #64155⋊C3C2semidirect
4. #64105⋊C3C2semidirect
6. #64097⋊C3C2semidirect
8. C4×#634⋊C3C2semidirect
9. #10596198C3semidirect
28. C2×Q8×C3⋊D3C2×C3semidirect
31. C32×C4⋊Q8D3semidirect
32. C32×C4⋊Q8C2×C3semidirect
33. #64155C2×C3semidirect
34. #64105C2×C3semidirect
36. #64097C2×C3semidirect
38. C4×#634C2×C3semidirect
54. C4⋊((C3⋊D3)⋊C3)Q8semidirect
58. #634⋊C3D4semidirect
103. C2×(C3⋊D3)⋊C3C2×Q8semidirect
107. C4×H(3,𝔽3)C2×Q8semidirect
156. C4⋊(C3⋊D3)C3×Q8semidirect
160. #634C3×D4semidirect
234. C2×C3⋊D3C2×C3×Q8semidirect
247. C32×C4D3×Q8semidirect
249. C32×C4C2×C3×Q8semidirect
258. C4⋊Q8(C3⋊D3)⋊C3semidirect
264. H(3,𝔽3)D4×Q8semidirect
379. C32#5727semidirect
380. C32C3×D4×Q8semidirect
420. C4Q8×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
H(3,𝔽3)×C4⋊Q82
#64155⋊C32
#64105⋊C32
#64102⋊C32
#64097⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#105961983
C3×#57273