Finite Group Databasebeta

← #13148536

Subgroups — #13148536

Order 1728.

1105 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88672C211yesmaximal, characteristic
2#11721733C211yesmaximal, characteristic, complemented
3#64090⋊C3C212yesmaximal, complemented
4#11721314C212yesmaximal, complemented
5C2×#88449C211yesmaximal, characteristic, complemented
6#11721138C211yesmaximal, characteristic, complemented
7C4×#6198C211yesmaximal, characteristic, complemented
8#11721120C212yesmaximal, complemented
9#11721119C212yesmaximal, complemented
10C4×#6196C211yesmaximal, characteristic, complemented
11#11721078C211yesmaximal, characteristic, complemented
12#10594049—32nomaximal
13C22×#1084C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C22×(C3⋊Dic3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×#6198C2211yescharacteristic
18#88672C418yescomplemented
19C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×#6196C2211yescharacteristic
22C22⋊C4×H(3,𝔽3)C2211yescharacteristic
23#88552C2211yescharacteristic
24#88540C2211yescharacteristic
25#88540C2211yescharacteristic
26C4×#1084C2211yescharacteristic
27C4×#1084C2211yescharacteristic
28#3752⋊C3C2212yes—
29#3750⋊C3C2212yes—
30C4×(C3⋊Dic3)⋊C3C2212yes—
31#88450C2212yes—
32#88449C2211yescharacteristic
33#88449C2211yescharacteristic
34#88449C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37C2×#3804—32no—
38#64090D312yescomplemented
39C3×#67⋊C4—32no—
40C3×#877—32no—
41C3×#877—61no—
42#67⋊Dic3—32no—
43C2×Dic3⋊Dic3—32no—
44#63952—32no—
45C4×#627—32no—
46C4×#626—32no—
47#63883—32no—
48#63880—32no—
49#63878—32no—
50D3×Dic3⋊C4—32no—
51#63837—32no—
52#63836—32no—
53C23×H(3,𝔽3)C2311yescharacteristic
54C22×(C3⋊D3)⋊C3C2312yes—
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×#1084C2×C414yescomplemented
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C2×#1084C2311yescharacteristic
61C2×#1084C2311yescharacteristic
62#638⋊C3C2×C418yescomplemented
63C2×(C3⋊Dic3)⋊C3—42no—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C4×(C3⋊D3)⋊C3—42no—
68#6198C2×C414yescomplemented
69((C3⋊D3)⋊C3)⋊C4—42no—
70((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
71#6196C2×C414yescomplemented
72#5546—92no—
73#5523—91no—
74C2×#638C2×D312yes—
75C22×C3⋊Dic3C2×D312yes—
76C2×C4×C3⋊D3C2×D312yes—
77C2×C3×#67—32no—
78C22×C3×Dic3—32no—
79C22×C3×Dic3—32no—
80C22×C3×Dic3—61no—
81C2×C3×C4×D3—32no—
82C2×#627—32no—
83C2×#626—32no—
84C2×#624—32no—
85C2×#624—32no—
86#3804—316no—
87C2×D3×Dic3—32no—
88C2×D3×Dic3—32no—
89C32×C22⋊C4—21no—
90C32×C22⋊C4C2×D312yes—
91#3752C2×D312yes—
92#3750C2×D312yes—
93C4×C3⋊Dic3C2×D312yes—
94C3×#217—32no—
95C3×#217—61no—
96C3×(C2×D3)⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—61no—
101C3×Dic3⋊C4—61no—
102C3×C4×Dic3—32no—
103C3×C4×Dic3—32no—
104C3×C4×Dic3—32no—
105C3×C4×Dic3—61no—
106C3×C4×Dic3—61no—
107#3724—32no—
108Dic3⋊Dic3—32no—
109Dic3⋊Dic3—32no—
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112Dic3⋊Dic3—32no—
113#3722—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116Dic32—32no—
117C22×H(3,𝔽3)—21no—
118C22×H(3,𝔽3)—21no—
119C22×H(3,𝔽3)C22×C412yes—
120C22×H(3,𝔽3)C2411yescharacteristic
121C2×(C3⋊D3)⋊C3—22no—
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3C22×C414yes—
124C4×H(3,𝔽3)—21no—
125C4×H(3,𝔽3)—21no—
126#1084—21no—
127#1084—21no—
128#1084C22×C412yes—
129#1084C22×C412yes—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132(C3⋊Dic3)⋊C3C22×C414yes—
133C2×#237—92no—
134C2×#235—91no—
135C3×C4×D4—92no—
136C3×#117—91no—
137D4×Dic3—92no—
138C4×#67—92no—
139C4×#67—92no—
140C2×C4⋊Dic3—91no—
141C2×Dic3⋊C4—92no—
142C2×Dic3⋊C4—92no—
143#892—92no—
144D3×C4⋊C4—92no—
145D3×C4⋊C4—92no—
146#887—92no—
147#67⋊C4—32no—
148#67⋊C4—92no—
149#67⋊C4—92no—
150#67⋊C4—92no—
151#877—32no—
152#877—61no—
153#877—92no—
154C4×D12—91no—
155C4×D12—91no—
156#872—92no—
157C4×Dic6—91no—
158C4×Dic6—92no—
159C4×Dic6—91no—
160C23×C32—21no—
161C23×C32C22×D312yes—
162C22×C3⋊D3C22×D312yes—
163C22×C3×D3—32no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D312yes—
168#638C4×D318yescomplemented
169C2×C3⋊Dic3—42no—
170C2×C3⋊Dic3C22×D312yes—
171C2×C3⋊Dic3C22×D312yes—
172C2×C3⋊Dic3C22×D312yes—
173C4×C3⋊D3—42no—
174C3×#67—38no—
175C2×C3×Dic3—38no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—64no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—122no—
189C3×C4×D3—122no—
190#627—38no—
191#626—38no—
192#624—38no—
193#624—122no—
194D3×Dic3—38no—
195D3×Dic3—122no—
196#520—271noSylow
197C2×H(3,𝔽3)—22no—
198C2×H(3,𝔽3)C23×C411yescharacteristic
199C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
200C2×H(3,𝔽3)2−1+411yescharacteristic
201(C3⋊D3)⋊C3—24no—
202C2×C3×D4—92no—
203C22×C3×C4—92no—
204C22×C3×C4—91no—
205C22×C3×C4—92no—
206C2×#67—32no—
207C2×#67—92no—
208C2×#67—92no—
209C22×Dic3—32no—
210C22×Dic3—32no—
211C22×Dic3—61no—
212C22×Dic3—92no—
213C22×Dic3—92no—
214C22×Dic3—92no—
215#237—916no—
216#235—98no—
217C2×D12—91no—
218C2×C4×D3—32no—
219C2×C4×D3—92no—
220C2×C4×D3—92no—
221C2×C4×D3—92no—
222C2×C4×D3—92no—
223C2×C4×D3—92no—
224C2×Dic6—91no—
225C2×Dic6—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—91no—
229C3×C22⋊C4—32no—
230C3×C22⋊C4—61no—
231C3×C22⋊C4—92no—
232C3×C22⋊C4—91no—
233C3×C22⋊C4D3211yescharacteristic
234C3×C42—92no—
235C3×C42—91no—
236C3×C42—91no—
237#217—32no—
238#217—61no—
239#217—92no—
240(C2×D3)⋊C4—32no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—92no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—92no—
248Dic3⋊C4—32no—
249Dic3⋊C4—32no—
250Dic3⋊C4—32no—
251Dic3⋊C4—61no—
252Dic3⋊C4—61no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260C4×Dic3—32no—
261C4×Dic3—32no—
262C4×Dic3—32no—
263C4×Dic3—61no—
264C4×Dic3—61no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C4×Dic3—92no—
268C4×Dic3—92no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—41no—
274C22×C32—41no—
275C22×C32C2×C4×D314yes—
276C22×C32C23×D312yes—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3C2×C4×D314yes—
280C2×C3×D3—34no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C32×C4—22no—
284C32×C4—22no—
285C32×C4—41no—
286C32×C4—41no—
287C3⋊Dic3—22no—
288C3⋊Dic3—22no—
289C3⋊Dic3C2×C4×D314yes—
290C3×Dic3—34no—
291C3×Dic3—34no—
292C3×Dic3—34no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C3×Dic3—62no—
296C3×Dic3—62no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C3×Dic3—121no—
300C3×Dic3—121no—
301C2×#41—271no—
302C4×Q8—271no—
303C4×Q8—271no—
304C4×D4—92no—
305C4×D4—271no—
306C4×D4—271no—
307C4×D4—272no—
308#117—91no—
309#117—272no—
310C2×C4⋊C4—272no—
311C2×C4⋊C4—271no—
312H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
313C23×C3—32no—
314C23×C3—61no—
315C23×C3—92no—
316C23×C3C2×D3211yescharacteristic
317C22×D3—32no—
318C22×D3—92no—
319C3×D4—98no—
320C2×C3×C4—32no—
321C2×C3×C4—32no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—92no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—94no—
328C2×C3×C4—92no—
329C2×C3×C4—92no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—362no—
333C2×C3×C4—362no—
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336#67—38no—
337#67—98no—
338#67—98no—
339C2×Dic3—32no—
340C2×Dic3—32no—
341C2×Dic3—38no—
342C2×Dic3—32no—
343C2×Dic3—32no—
344C2×Dic3—32no—
345C2×Dic3—32no—
346C2×Dic3—32no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—64no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—92no—
353C2×Dic3—92no—
354C2×Dic3—92no—
355C2×Dic3—98no—
356C2×Dic3—122no—
357C2×Dic3—362no—
358C2×Dic3—362no—
359D12—94no—
360C4×D3—98no—
361C4×D3—98no—
362C4×D3—122no—
363C4×D3—362no—
364C4×D3—362no—
365C4×D3—362no—
366Dic6—94no—
367Dic6—98no—
368C2×C32—24no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—42no—
373C2×C32C22×C4×D312yes—
374C2×C32#100412yes—
375C2×C32#101012yes—
376C3⋊D3—24no—
377C3×D3—64no—
378#41—278no—
379C2×Q8—271no—
380C2×D4—92no—
381C2×D4—271no—
382C22×C4—91no—
383C22×C4—92no—
384C22×C4—92no—
385C22×C4—272no—
386C22×C4—272no—
387C4⋊C4—91no—
388C4⋊C4—92no—
389C4⋊C4—91no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C4⋊C4—271no—
393C4⋊C4—272no—
394C22⋊C4—91no—
395C22⋊C4—92no—
396C22⋊C4—272no—
397C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
398C42—91no—
399C42—91no—
400C42—92no—
401C42—272no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—32no—
405C22×C3—34no—
406C22×C3—62no—
407C22×C3—62no—
408C22×C3—61no—
409C22×C3—62no—
410C22×C3—94no—
411C22×C3—121no—
412C22×C3—121no—
413C22×C3—182no—
414C22×C3—182no—
415C22×C3C4×D3212yes—
416C22×C3C22×D3211yescharacteristic
417C2×D3—34no—
418C2×D3—62no—
419C2×D3—62no—
420C2×D3—94no—
421C2×D3—182no—
422C2×D3—182no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—62no—
426C3×C4—62no—
427C3×C4—94no—
428C3×C4—92no—
429C3×C4—92no—
430C3×C4—121no—
431C3×C4—121no—
432C3×C4—182no—
433C3×C4—182no—
434C3×C4—181no—
435C3×C4—181no—
436Dic3—34no—
437Dic3—34no—
438Dic3—34no—
439Dic3—62no—
440Dic3—62no—
441Dic3—62no—
442Dic3—62no—
443Dic3—62no—
444Dic3—62no—
445Dic3—94no—
446Dic3—121no—
447Dic3—121no—
448Dic3—182no—
449Dic3—182no—
450C32—21no—
451C32#554612yescomplemented
452C23—92no—
453C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
454Q8—274no—
455D4—98no—
456D4—274no—
457C2×C4—91no—
458C2×C4—92no—
459C2×C4—91no—
460C2×C4—94no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—92no—
464C2×C4—92no—
465C2×C4—278no—
466C2×C4—362no—
467C2×C4—362no—
468C2×C4—1082no—
469C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
470C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
471C2×C3—22no—
472C2×C3—32no—
473C2×C3—32no—
474C2×C3—32no—
475C2×C3—64no—
476C2×C3—61no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—122no—
480C2×C3—184no—
481C2×C3C2×C4×D3211yescharacteristic
482C2×C3#6430911yescharacteristic
483C2×C3#6431311yescharacteristic
484D3—64no—
485D3—184no—
486C22—21no—
487C22—21no—
488C22—94no—
489C22—182no—
490C22—182no—
491C22C4×((C3⋊D3)⋊C3)⋊C212yes—
492C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
493C4—21no—
494C4—21no—
495C4—92no—
496C4—92no—
497C4—94no—
498C4—181no—
499C4—182no—
500C4—182no—
501C4—181no—
502C3—32no—
503C3—61no—
504C3#1059405111yescharacteristic
505C2—22no—
506C2—184no—
507C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
508C2#1172250211yescharacteristic
509C2#1172250611yescharacteristic
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)#520
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×#88449C2semidirect
6. #11721138C2semidirect
7. C4×#6198C2semidirect
8. #11721120C2semidirect
9. #11721119C2semidirect
10. C4×#6196C2semidirect
11. #11721078C2semidirect
18. #88672C4semidirect
38. #64090D3semidirect
57. C2×#1084C2×C4semidirect
62. #638⋊C3C2×C4semidirect
68. #6198C2×C4semidirect
70. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
71. #6196C2×C4semidirect
168. #638C4×D3semidirect
312. H(3,𝔽3)#520semidirect
451. C32#5546semidirect

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×#884492
#117211382
C4×#61982
#117211202
#117211192
C4×#61962
#117210782
#105940493