Finite Group Databasebeta

← #11807843

Subgroups — #11807843

Order 1152.

1037 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#924C211yesmaximal, characteristic
2#10592910C211yesmaximal, characteristic
3C3×#5212C211yesmaximal, characteristic
4#10591702C212yesmaximal
5#10591701C212yesmaximal
6#10591270C212yesmaximal, complemented
7#10591265C212yesmaximal, complemented
8#10591262C212yesmaximal, complemented
9#10591261C212yesmaximal, complemented
10#82750—31nomaximal
11#73886—31nomaximal
12C23×C32×C4C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C2×#3757C2212yes—
15C2×C3×#217C2211yescharacteristic
16C2×C3×#217C2211yescharacteristic
17C2×C3×#217C2212yes—
18C2×C3×C4⋊Dic3C2212yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C3×#924C418yescomplemented
21C2×C3×Dic3⋊C4C2212yes—
22C2×Dic3⋊Dic3C2212yes—
23C2×Dic3⋊Dic3C2212yes—
24C2×Dic32C2212yes—
25C2×Dic32C2212yes—
26#63658C2212yes—
27#63658C2211yescharacteristic
28#63658C2211yescharacteristic
29C3×C3⋊#95C2212yes—
30#63579C2212yes—
31#63579C2212yes—
32#63579C2212yes—
33#63579C2212yes—
34C2×C3×C22⋊Q8—31no—
35C2×#924D311yescharacteristic
36C3×#380—31no—
37#5229—32no—
38#5212D311yescharacteristic
39#5211—31no—
40#5211—31no—
41#5211—61no—
42#4984—32no—
43#4980—32no—
44#4980—32no—
45#4978—32no—
46#4976—32no—
47#4951—32no—
48#4951—32no—
49#4945—32no—
50#4944—32no—
51#4654—32no—
52#4646—32no—
53C24×C32C2311yescharacteristic
54C22×C32×C4—42no—
55C22×C32×C4C2×C414yes—
56C22×C32×C4C2311yescharacteristic
57C22×C32×C4C2311yescharacteristic
58C22×C3⋊Dic3C2312yes—
59C22×C3⋊Dic3C2312yes—
60C22×C3×Dic3C2312yes—
61C22×C3×Dic3C2312yes—
62C22×C3×Dic3C2312yes—
63C22×C3×Dic3C2312yes—
64C2×C3×Dic6—42no—
65C2×C3×Dic6C2×C414yes—
66#3757—42no—
67C3×#217—41no—
68C3×#217—41no—
69C3×#217C2×C418yes—
70C3×C4⋊Dic3—42no—
71C3×C4⋊Dic3C2×C414yes—
72C3×Dic3⋊C4C2×C418yes—
73Dic3⋊Dic3—42no—
74Dic3⋊Dic3—42no—
75Dic32—42no—
76Dic32—42no—
77#2378—91noSylow
78C22×C3×Q8—31no—
79C23×C3×C4—21no—
80C23×C3×C4C2×D311yescharacteristic
81C23×C3×C4C2×D311yescharacteristic
82C22×Dic6C2×D311yescharacteristic
83C3×C22⋊Q8—38no—
84C2×C3×C4⋊C4—32no—
85C2×C3×C4⋊C4—31no—
86C2×C3×C4⋊C4—32no—
87C2×C3×C22⋊C4—32no—
88C2×C3×C22⋊C4—31no—
89C2×C3×C22⋊C4—31no—
90C2×#217—32no—
91C2×#217—32no—
92C2×#217—62no—
93C2×#217C2×D311yescharacteristic
94C2×#217C2×D311yescharacteristic
95C2×#217C2×D312yes—
96C2×C4⋊Dic3—32no—
97C2×C4⋊Dic3—32no—
98C2×C4⋊Dic3C2×D312yes—
99C2×C4⋊Dic3C2×D311yescharacteristic
100#924Dic318yescomplemented
101C2×Dic3⋊C4—32no—
102C2×Dic3⋊C4—32no—
103C2×Dic3⋊C4C2×D312yes—
104C2×C4×Dic3—32no—
105C2×C4×Dic3—32no—
106C2×C4×Dic3—32no—
107C2×C4×Dic3—32no—
108C3×#95—32no—
109C3⋊#95—32no—
110C3⋊#95—32no—
111C3⋊#95—31no—
112C3⋊#95—31no—
113C3⋊#95—31no—
114C3⋊#95—31no—
115C3⋊#95—32no—
116C3⋊#95—32no—
117C3⋊#95—32no—
118C3⋊#95—32no—
119C3⋊#95—32no—
120C3⋊#95—32no—
121C3⋊#95—32no—
122C3⋊#95—32no—
123C3⋊#95—61no—
124C3⋊#95—62no—
125C3⋊#95—61no—
126C3⋊#95C2×D312yes—
127C23×C32—22no—
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—21no—
131C23×C32C22×C412yes—
132C23×C32C2411yescharacteristic
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—44no—
136C2×C32×C4C22×C412yes—
137C2×C32×C4C22×C412yes—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3C22×C414yes—
155C2×C3×Dic3C22×C414yes—
156C3×Dic6—44no—
157C2×C22⋊Q8—31no—
158#391—92no—
159#389—92no—
160#388—92no—
161#388—92no—
162#387—92no—
163#387—92no—
164#381—91no—
165#380—31no—
166C24×C3—21no—
167C24×C3C22×D311yescharacteristic
168C24×C3C22×D311yescharacteristic
169C2×C3×Q8—34no—
170C2×C3×Q8—122no—
171C22×C3×C4—24no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—32no—
175C22×C3×C4—32no—
176C22×C3×C4—32no—
177C22×C3×C4—32no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4—44no—
181C22×C3×C4C4×D314yes—
182C22×C3×C4C2×Dic314yes—
183C22×C3×C4C22×D311yescharacteristic
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C22×D311yescharacteristic
186C22×C3×C4C22×D311yescharacteristic
187C22×Dic3—32no—
188C22×Dic3—32no—
189C22×Dic3—32no—
190C22×Dic3—32no—
191C22×Dic3—62no—
192C22×Dic3—62no—
193C22×Dic3C22×D312yes—
194C22×Dic3C22×D312yes—
195C22×Dic3C22×D312yes—
196C22×Dic3C22×D312yes—
197C2×Dic6—42no—
198C2×Dic6C2×Dic314yes—
199C3×C4⋊C4—38no—
200C3×C4⋊C4—34no—
201C3×C4⋊C4—122no—
202C3×C22⋊C4—38no—
203C3×C22⋊C4—121no—
204C3×C22⋊C4—121no—
205#217—41no—
206#217—41no—
207#217—122no—
208#217—122no—
209#217—124no—
210#217C2×Dic318yes—
211C4⋊Dic3—42no—
212C4⋊Dic3—122no—
213C4⋊Dic3—122no—
214C4⋊Dic3C2×Dic314yes—
215Dic3⋊C4—122no—
216Dic3⋊C4—122no—
217Dic3⋊C4C2×Dic318yes—
218C4×Dic3—122no—
219C4×Dic3—122no—
220C4×Dic3—122no—
221C4×Dic3—122no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—42no—
226C22×C32—42no—
227C22×C32C23×C411yescharacteristic
228C22×C32H(5,𝔽2)12yes—
229C22×C32H(5,𝔽2)11yescharacteristic
230C22×C322−1+412yes—
231C22×C322−1+411yescharacteristic
232C32×C4—22no—
233C32×C4—22no—
234C3⋊Dic3—42no—
235C3⋊Dic3—42no—
236C3×Dic3—24no—
237C3×Dic3—24no—
238C3×Dic3—42no—
239C3×Dic3—42no—
240C22×Q8—31no—
241C23×C4D3211yescharacteristic
242C22⋊Q8—38no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—92no—
247C2×C4⋊C4—92no—
248C2×C22⋊C4—32no—
249C2×C22⋊C4—31no—
250C2×C22⋊C4—31no—
251C2×C22⋊C4—92no—
252C2×C42—92no—
253C2×C42—92no—
254#95—32no—
255#95—92no—
256#95—92no—
257#95—92no—
258#95—92no—
259#95—91no—
260#95—92no—
261#95—91no—
262C23×C3—24no—
263C23×C3—21no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—21no—
267C23×C3—22no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—22no—
273C23×C3—42no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3C2×C4×D312yes—
277C23×C3C22×Dic312yes—
278C23×C3C23×D311yescharacteristic
279C23×C3C23×D311yescharacteristic
280C3×Q8—124no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—34no—
288C2×C3×C4—34no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4—48no—
294C2×C3×C4—62no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—62no—
298C2×C3×C4—62no—
299C2×C3×C4—62no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4C2×C4×D312yes—
305C2×C3×C4C2×C4×D312yes—
306C2×C3×C4C22×Dic312yes—
307C2×C3×C4C22×Dic312yes—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—62no—
321C2×Dic3—62no—
322C2×Dic3—62no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—62no—
329C2×Dic3—62no—
330C2×Dic3—122no—
331C2×Dic3—122no—
332C2×Dic3—122no—
333C2×Dic3—122no—
334C2×Dic3—122no—
335C2×Dic3—122no—
336C2×Dic3C22×Dic314yes—
337C2×Dic3C22×Dic314yes—
338Dic6—44no—
339C2×C32—42no—
340C2×C32#51811yescharacteristic
341C2×C32#51911yescharacteristic
342C2×C32#52011yescharacteristic
343C2×C32#56212yes—
344C2×C32#56312yes—
345C24C2×D3211yescharacteristic
346C2×Q8—34no—
347C2×Q8—122no—
348C22×C4—32no—
349C22×C4—32no—
350C22×C4—32no—
351C22×C4—32no—
352C22×C4—42no—
353C22×C4—92no—
354C22×C4—92no—
355C22×C4D3×Dic314yes—
356C22×C4C2×D3211yescharacteristic
357C22×C4C2×D3211yescharacteristic
358C4⋊C4—38no—
359C4⋊C4—34no—
360C4⋊C4—122no—
361C4⋊C4—362no—
362C4⋊C4—362no—
363C22⋊C4—38no—
364C22⋊C4—121no—
365C22⋊C4—121no—
366C22⋊C4—362no—
367C42—362no—
368C42—362no—
369C22×C3—24no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—44no—
384C22×C3—44no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3C22×C4×D311yescharacteristic
390C22×C3#100411yescharacteristic
391C22×C3#100412yes—
392C22×C3#100412yes—
393C22×C3#100711yescharacteristic
394C22×C3#100712yes—
395C22×C3#100911yescharacteristic
396C22×C3#101011yescharacteristic
397C22×C3#101012yes—
398C22×C3C23×Dic311yescharacteristic
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—42no—
404C3×C4—42no—
405C3×C4—64no—
406C3×C4—64no—
407C3×C4—122no—
408C3×C4—122no—
409Dic3—24no—
410Dic3—24no—
411Dic3—42no—
412Dic3—42no—
413Dic3—122no—
414Dic3—122no—
415Dic3—122no—
416Dic3—122no—
417Dic3—242no—
418Dic3—242no—
419C32#237811yescharacteristic, complemented, Sylow
420C23—22no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23C2×D3×Dic312yes—
425C23C22×D3211yescharacteristic
426Q8—124no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—34no—
430C2×C4—34no—
431C2×C4—44no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—182no—
443C2×C4—182no—
444C2×C4—182no—
445C2×C4—182no—
446C2×C4—182no—
447C2×C4—182no—
448C2×C4C2×D3×Dic312yes—
449C2×C4C2×D3×Dic312yes—
450C2×C3—21no—
451C2×C3—22no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—22no—
455C2×C3—44no—
456C2×C3—42no—
457C2×C3—42no—
458C2×C3#548611yescharacteristic
459C2×C3#550411yescharacteristic
460C2×C3#551611yescharacteristic
461C2×C3#562112yes—
462C2×C3#564612yes—
463C2×C3#564612yes—
464C2×C3#564712yes—
465C2×C3#579811yescharacteristic
466C2×C3#581211yescharacteristic
467C2×C3#582711yescharacteristic
468C22—22no—
469C22—22no—
470C22—22no—
471C22—42no—
472C22—42no—
473C22#6429712yes—
474C22#6429711yescharacteristic
475C22#6430711yescharacteristic
476C22C22×D3×Dic311yescharacteristic
477C22#6433012yes—
478C4—22no—
479C4—22no—
480C4—64no—
481C4—64no—
482C4—122no—
483C4—122no—
484C4—362no—
485C4—362no—
486C3—21no—
487C3#7388611yescharacteristic, complemented
488C3#8275011yescharacteristic, complemented
489C2—42no—
490C2#1059481112yes—
491C2#1059481811yescharacteristic
492C2#1059497612yes—
493C2#1059498611yescharacteristic
494C2#1059550611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC23×C32×C4C22
socle, cosocleC23×C32C24
prime radicalC24×C32C23
hypercenterC23×C4D32
hypocenterC32#2378
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
6. #10591270C2semidirect
7. #10591265C2semidirect
8. #10591262C2semidirect
9. #10591261C2semidirect
20. C3×#924C4semidirect
100. #924Dic3semidirect
419. C32#2378semidirect
487. C3#73886semidirect
488. C3#82750semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9242
#105929102
C3×#52122
#105917022
#105917012
#105912702
#105912652
#105912622
#105912612
#827503
#738863