Finite Group Databasebeta

← #13148535

Subgroups — #13148535

Order 1728.

1161 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 1161 conjugacy classes of subgroups — too many to draw. Only its 211 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×#88672C211yesmaximal, characteristic
2#11721733C211yesmaximal, characteristic, complemented
3#64090⋊C3C212yesmaximal, complemented
4#11721314C212yesmaximal, complemented
5C2×#88447C211yesmaximal, characteristic, complemented
6#11721122C212yesmaximal, complemented
7#11721121C212yesmaximal, complemented
8C4×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
9#11721077C212yesmaximal, complemented
10#10594048—32nomaximal
11C22×#1084C2211yescharacteristic
12C2×#638⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C2×C4×(C3⋊D3)⋊C3C2212yes—
15C2×#6198C2211yescharacteristic
16#88672C418yescomplemented
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×#6196C2211yescharacteristic
20C22⋊C4×H(3,𝔽3)C2211yescharacteristic
21#88552C2211yescharacteristic
22#88540C2212yes—
23C4×#1084C2212yes—
24#3752⋊C3C2212yes—
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27#88450C2212yes—
28#88448C2212yes—
29#88447C2212yes—
30#88447C2211yescharacteristic
31#88447C2211yescharacteristic
32#88447C2212yes—
33C2×#3804—32no—
34#64090D312yescomplemented
35C3×#67⋊C4—32no—
36C3×#877—32no—
37C3×#877—61no—
38#67⋊Dic3—32no—
39C2×Dic32—32no—
40#63952—32no—
41#63904—32no—
42C4×#624—32no—
43#63881—32no—
44#63877—32no—
45C4×D3×Dic3—32no—
46#63838—32no—
47#63837—32no—
48#63836—32no—
49C23×H(3,𝔽3)C2311yescharacteristic
50C22×(C3⋊D3)⋊C3C2312yes—
51C2×C4×H(3,𝔽3)C2312yes—
52C2×#1084C2×C414yescomplemented
53C2×#1084C2312yes—
54C2×#1084C2311yescharacteristic
55C2×#1084C2311yescharacteristic
56#638⋊C3C2×C418yescomplemented
57C2×(C3⋊Dic3)⋊C3—24no—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C4×(C3⋊D3)⋊C3—24no—
62#6198C2×C414yescomplemented
63((C3⋊D3)⋊C3)⋊C4—24no—
64((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
65#6196C2×C414yescomplemented
66C4×#237—92no—
67#5522—91no—
68C2×#638C2×D312yes—
69C22×C3⋊Dic3C2×D312yes—
70C2×C4×C3⋊D3C2×D312yes—
71C2×C3×#67—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—61no—
75C2×C3×C4×D3—32no—
76C2×#627—32no—
77C2×#626—32no—
78C2×#624—32no—
79C2×#624—32no—
80#3804—316no—
81C2×D3×Dic3—32no—
82C2×D3×Dic3—32no—
83C32×C22⋊C4—21no—
84C32×C22⋊C4C2×D312yes—
85#3752C2×D312yes—
86#3750C2×D312yes—
87C4×C3⋊Dic3C2×D312yes—
88C3×#217—32no—
89C3×#217—61no—
90C3×(C2×D3)⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—62no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—32no—
97C3×C4×Dic3—32no—
98C3×C4×Dic3—62no—
99#3724—32no—
100Dic3⋊Dic3—32no—
101#3722—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104Dic32—32no—
105Dic32—32no—
106Dic32—32no—
107Dic32—32no—
108Dic32—32no—
109C22×H(3,𝔽3)—22no—
110C22×H(3,𝔽3)C22×C412yes—
111C22×H(3,𝔽3)C2411yescharacteristic
112C2×(C3⋊D3)⋊C3—22no—
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3C22×C414yes—
115C4×H(3,𝔽3)—22no—
116#1084—22no—
117#1084C22×C412yes—
118#1084C22×C412yes—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3C22×C414yes—
121(C3⋊Dic3)⋊C3#4114yescomplemented
122C2×#237—92no—
123C2×#235—91no—
124C3×C4×D4—92no—
125C3×#117—91no—
126D4×Dic3—92no—
127C4×#67—92no—
128C4×#67—92no—
129C2×C4×Dic3—92no—
130C2×C4×Dic3—91no—
131C2×C4×Dic3—92no—
132#893—92no—
133#892—92no—
134#887—92no—
135#887—92no—
136#67⋊C4—32no—
137#67⋊C4—92no—
138#67⋊C4—92no—
139#67⋊C4—92no—
140#877—32no—
141#877—61no—
142#877—92no—
143#872—92no—
144C42×D3—92no—
145C42×D3—92no—
146C4×Dic6—92no—
147C23×C32—21no—
148C23×C32C22×D312yes—
149C22×C3⋊D3C22×D312yes—
150C22×C3×D3—32no—
151C2×C32×C4—22no—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D312yes—
154#638C4×D318yescomplemented
155C2×C3⋊Dic3—24no—
156C2×C3⋊Dic3C22×D312yes—
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D312yes—
159C4×C3⋊D3—24no—
160C3×#67—38no—
161C2×C3×Dic3—38no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—64no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—64no—
174C3×C4×D3—64no—
175#627—38no—
176#626—38no—
177#624—38no—
178#624—64no—
179D3×Dic3—38no—
180D3×Dic3—64no—
181C4×#41—271noSylow
182C2×H(3,𝔽3)—22no—
183C2×H(3,𝔽3)C23×C411yescharacteristic
184C2×H(3,𝔽3)C2×#4112yes—
185(C3⋊D3)⋊C3—24no—
186C2×C3×D4—92no—
187C22×C3×C4—92no—
188C22×C3×C4—91no—
189C22×C3×C4—92no—
190C2×#67—32no—
191C2×#67—92no—
192C2×#67—92no—
193C22×Dic3—32no—
194C22×Dic3—32no—
195C22×Dic3—61no—
196C22×Dic3—92no—
197C22×Dic3—92no—
198C22×Dic3—92no—
199#237—916no—
200#235—98no—
201C2×D12—91no—
202C2×C4×D3—32no—
203C2×C4×D3—92no—
204C2×C4×D3—92no—
205C2×C4×D3—92no—
206C2×C4×D3—92no—
207C2×C4×D3—92no—
208C2×Dic6—91no—
209C2×Dic6—92no—
210C3×C4⋊C4—92no—
211C3×C4⋊C4—92no—
212C3×C22⋊C4—32no—
213C3×C22⋊C4—61no—
214C3×C22⋊C4—92no—
215C3×C22⋊C4—91no—
216C3×C22⋊C4D3211yescharacteristic
217C3×C42—92no—
218C3×C42—92no—
219#217—32no—
220#217—61no—
221#217—92no—
222(C2×D3)⋊C4—32no—
223(C2×D3)⋊C4—92no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4—92no—
226C4⋊Dic3—92no—
227Dic3⋊C4—32no—
228Dic3⋊C4—32no—
229Dic3⋊C4—32no—
230Dic3⋊C4—62no—
231Dic3⋊C4—92no—
232Dic3⋊C4—92no—
233Dic3⋊C4—92no—
234C4×Dic3—32no—
235C4×Dic3—32no—
236C4×Dic3—32no—
237C4×Dic3—62no—
238C4×Dic3—92no—
239C4×Dic3—91no—
240C4×Dic3—92no—
241C4×Dic3—91no—
242C4×Dic3—92no—
243C4×Dic3—92no—
244C4×Dic3—92no—
245C4×Dic3—92no—
246C4×Dic3—92no—
247C4×Dic3—92no—
248C4×Dic3—92no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—21no—
252C22×C32—22no—
253C22×C32—42no—
254C22×C32C2×C4×D314yes—
255C22×C32C23×D312yes—
256C2×C3⋊D3—22no—
257C2×C3⋊D3—22no—
258C2×C3⋊D3C2×C4×D314yes—
259C2×C3×D3—34no—
260C2×C3×D3—62no—
261C2×C3×D3—62no—
262C32×C4—22no—
263C32×C4—22no—
264C32×C4—42no—
265C3⋊Dic3—22no—
266C3⋊Dic3C2×C4×D314yes—
267C3⋊Dic3#23714yescomplemented
268C3×Dic3—34no—
269C3×Dic3—34no—
270C3×Dic3—34no—
271C3×Dic3—34no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C3×Dic3—62no—
276C3×Dic3—62no—
277C3×Dic3—122no—
278C2×#41—271no—
279C4×Q8—272no—
280C4×D4—92no—
281C4×D4—272no—
282C4×D4—272no—
283#117—91no—
284#117—272no—
285C2×C42—272no—
286C2×C42—271no—
287H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
288C23×C3—32no—
289C23×C3—61no—
290C23×C3—92no—
291C23×C3C2×D3211yescharacteristic
292C22×D3—32no—
293C22×D3—92no—
294C3×D4—98no—
295C2×C3×C4—32no—
296C2×C3×C4—32no—
297C2×C3×C4—62no—
298C2×C3×C4—92no—
299C2×C3×C4—92no—
300C2×C3×C4—91no—
301C2×C3×C4—91no—
302C2×C3×C4—94no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—184no—
306C2×C3×C4—184no—
307C2×C3×C4C2×D3212yes—
308#67—38no—
309#67—98no—
310#67—98no—
311C2×Dic3—32no—
312C2×Dic3—32no—
313C2×Dic3—38no—
314C2×Dic3—32no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—64no—
322C2×Dic3—62no—
323C2×Dic3—64no—
324C2×Dic3—92no—
325C2×Dic3—92no—
326C2×Dic3—92no—
327C2×Dic3—98no—
328C2×Dic3—184no—
329C2×Dic3—184no—
330D12—94no—
331C4×D3—64no—
332C4×D3—98no—
333C4×D3—98no—
334C4×D3—184no—
335C4×D3—184no—
336C4×D3—184no—
337Dic6—94no—
338Dic6—98no—
339C2×C32—24no—
340C2×C32—21no—
341C2×C32—22no—
342C2×C32—42no—
343C2×C32C22×C4×D312yes—
344C2×C32C2×#23712yes—
345C2×C32D3×#4112yes—
346C3⋊D3—24no—
347C3×D3—64no—
348#41—278no—
349C2×Q8—271no—
350C2×D4—92no—
351C2×D4—271no—
352C22×C4—91no—
353C22×C4—92no—
354C22×C4—92no—
355C22×C4—272no—
356C22×C4—272no—
357C4⋊C4—92no—
358C4⋊C4—92no—
359C4⋊C4—272no—
360C22⋊C4—91no—
361C22⋊C4—92no—
362C22⋊C4—272no—
363C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
364C42—92no—
365C42—92no—
366C42—271no—
367C42—271no—
368C42—272no—
369C42—272no—
370C22×C3—22no—
371C22×C3—32no—
372C22×C3—34no—
373C22×C3—62no—
374C22×C3—62no—
375C22×C3—61no—
376C22×C3—62no—
377C22×C3—94no—
378C22×C3—122no—
379C22×C3—182no—
380C22×C3—182no—
381C22×C3C4×D3212yes—
382C22×C3C22×D3211yescharacteristic
383C2×D3—34no—
384C2×D3—62no—
385C2×D3—62no—
386C2×D3—94no—
387C2×D3—182no—
388C2×D3—182no—
389C3×C4—22no—
390C3×C4—62no—
391C3×C4—62no—
392C3×C4—94no—
393C3×C4—94no—
394C3×C4—92no—
395C3×C4—92no—
396C3×C4—122no—
397C3×C4—182no—
398C3×C4—182no—
399Dic3—34no—
400Dic3—34no—
401Dic3—34no—
402Dic3—34no—
403Dic3—62no—
404Dic3—62no—
405Dic3—62no—
406Dic3—62no—
407Dic3—62no—
408Dic3—94no—
409Dic3—94no—
410Dic3—122no—
411Dic3—182no—
412C32—21no—
413C32C4×#23712yescomplemented
414C23—92no—
415C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416Q8—274no—
417D4—98no—
418D4—274no—
419C2×C4—91no—
420C2×C4—92no—
421C2×C4—92no—
422C2×C4—91no—
423C2×C4—94no—
424C2×C4—92no—
425C2×C4—92no—
426C2×C4—184no—
427C2×C4—184no—
428C2×C4—278no—
429C2×C4—544no—
430C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
431C2×C3—22no—
432C2×C3—32no—
433C2×C3—32no—
434C2×C3—32no—
435C2×C3—64no—
436C2×C3—62no—
437C2×C3—61no—
438C2×C3—122no—
439C2×C3—184no—
440C2×C3C2×C4×D3211yescharacteristic
441C2×C3D3×#23712yes—
442D3—64no—
443D3—184no—
444C22—22no—
445C22—94no—
446C22—182no—
447C22—182no—
448C22C4×((C3⋊D3)⋊C3)⋊C212yes—
449C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
450C4—22no—
451C4—94no—
452C4—92no—
453C4—92no—
454C4—94no—
455C4—182no—
456C4—182no—
457C3—32no—
458C3—61no—
459C3#1059405011yescharacteristic
460C2—22no—
461C2—184no—
462C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
463C2#1172250412yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC2×#88672C2
cosocleC22×H(3,𝔽3)C24
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721733C2semidirect
3. #64090⋊C3C2semidirect
4. #11721314C2semidirect
5. C2×#88447C2semidirect
6. #11721122C2semidirect
7. #11721121C2semidirect
8. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
9. #11721077C2semidirect
16. #88672C4semidirect
34. #64090D3semidirect
52. C2×#1084C2×C4semidirect
56. #638⋊C3C2×C4semidirect
62. #6198C2×C4semidirect
64. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
65. #6196C2×C4semidirect
121. (C3⋊Dic3)⋊C3#41semidirect
154. #638C4×D3semidirect
267. C3⋊Dic3#237semidirect
287. H(3,𝔽3)C4×#41semidirect
413. C32C4×#237semidirect

Maximal subgroups

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

subgroupindex
C2×#886722
#117217332
#64090⋊C32
#117213142
C2×#884472
#117211222
#117211212
C4×((C3⋊D3)⋊C3)⋊C42
#117210772
#105940483