Finite Group Databasebeta

← #11803925

Subgroups — #11803925

Order 1152.

1173 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63865C211yesmaximal, characteristic
2#10592686C211yesmaximal, characteristic
3C3×#4978C212yesmaximal, complemented
4#10591375C212yesmaximal
5#10591370C211yesmaximal, characteristic
6#10591268C212yesmaximal, complemented
7#10591249C212yesmaximal, complemented
8#10591233C212yesmaximal
9#10591191C212yesmaximal, complemented
10#75299—32nomaximal
11C22×C4×C3⋊D3C2211yescharacteristic
12C22×#627C2211yescharacteristic
13C22×#624C2211yescharacteristic
14C22×#624C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×#3752C2212yes—
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×Dic3⋊C4C2212yes—
19C2×C3×C4×Dic3C2212yes—
20C2×#3724C2211yescharacteristic
21C2×Dic3⋊Dic3C2212yes—
22C2×Dic3⋊Dic3C2212yes—
23C2×#3722C2212yes—
24C2×Dic32C2212yes—
25#63865C418yescomplemented
26#63658C2212yes—
27C3×C3⋊#95C2212yes—
28C3×C3⋊#95C2212yes—
29C3×C3⋊#95C2212yes—
30C3×C3⋊#95C2212yes—
31#63579C2212yes—
32#63579C2212yes—
33C2×#896—32no—
34C3×#387—32no—
35#4987—32no—
36#4987—61no—
37#4984—32no—
38#4978D312yescomplemented
39C4×(C2×D3)⋊C4—32no—
40#4929—32no—
41#4668—32no—
42#4668—32no—
43#4666—32no—
44D3×#95—32no—
45#4655—32no—
46#4652—32no—
47#4647—32no—
48#4646—32no—
49C23×C3⋊D3C2311yescharacteristic
50C22×C32×C4C2312yes—
51C22×C32×C4C2311yescharacteristic
52C22×C3⋊Dic3C2312yes—
53C22×C3⋊Dic3C2311yescharacteristic
54C2×C4×C3⋊D3—41no—
55C2×C4×C3⋊D3—41no—
56C2×C4×C3⋊D3C2×C414yes—
57C22×C3×Dic3C2312yes—
58C22×C3×Dic3C2312yes—
59C22×C3×Dic3C2312yes—
60C22×C3×Dic3C2312yes—
61C2×#627—41no—
62C2×#627—41no—
63C2×#627C2×C414yescomplemented
64C2×#624—22no—
65C2×#624—41no—
66C2×#624—41no—
67C2×#624—41no—
68C2×#624—41no—
69C2×#624Q814yescomplemented
70C32×C4⋊C4—41no—
71#3752—24no—
72C4⋊(C3⋊Dic3)—41no—
73C3×Dic3⋊C4C2×C418yescomplemented
74C3×C4×Dic3—42no—
75#3724C2×C414yescomplemented
76Dic3⋊Dic3—42no—
77Dic3⋊Dic3—42no—
78#3722C2×C418yes—
79Dic32—42no—
80#2329—91noSylow
81C22×C4×D3—32no—
82C22×C4×D3—32no—
83C22×C4×D3—32no—
84C22×C4×D3—61no—
85C22×Dic6—32no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—32no—
88C2×C3×C4⋊C4C2×D312yes—
89C2×C3×C42—32no—
90C2×(C2×D3)⋊C4—32no—
91C2×(C2×D3)⋊C4—32no—
92C2×(C2×D3)⋊C4—32no—
93C2×(C2×D3)⋊C4—32no—
94C2×(C2×D3)⋊C4—62no—
95C2×C4⋊Dic3—32no—
96C2×C4⋊Dic3—32no—
97C2×C4⋊Dic3—32no—
98C2×C4⋊Dic3—61no—
99C2×Dic3⋊C4—32no—
100C2×Dic3⋊C4—32no—
101C2×Dic3⋊C4—32no—
102C2×Dic3⋊C4C2×D312yes—
103C2×C4×Dic3—32no—
104C2×C4×Dic3—32no—
105C2×C4×Dic3C2×D312yes—
106#896—316no—
107C3×#95—32no—
108C3×#95—32no—
109C3×#95—32no—
110C3×#95—32no—
111C3⋊#95—32no—
112C3⋊#95—32no—
113C3⋊#95—32no—
114C3⋊#95—32no—
115C3⋊#95—32no—
116C3⋊#95—32no—
117C3⋊#95—62no—
118C3⋊#95C2×D312yes—
119C3⋊#95C2×D312yes—
120C3⋊#95C2×D312yes—
121C3⋊#95C2×D312yes—
122C23×C32C2411yescharacteristic
123C22×C3⋊D3—21no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3C22×C412yes—
128C22×C3⋊D3C2×Q812yes—
129C22×C3⋊D3#4112yes—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4C22×C412yes—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3C22×C412yes—
142C4×C3⋊D3—44no—
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×Dic3C22×C414yes—
153C2×C3×Dic3C22×C414yes—
154C2×C3×Dic3C2×Q814yes—
155#627—44no—
156#624—44no—
157#624—44no—
158C2×C22⋊Q8—91no—
159#391—92no—
160#389—92no—
161#387—32no—
162#387—92no—
163#385—92no—
164#380—91no—
165C4×C22⋊C4—92no—
166C2×#95—91no—
167C23×D3—32no—
168C23×D3—61no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—32no—
172C22×C3×C4—32no—
173C22×C3×C4—32no—
174C22×C3×C4—32no—
175C22×C3×C4C22×D312yes—
176C22×C3×C4C22×D312yes—
177C22×C3×C4C22×D312yes—
178C22×Dic3—32no—
179C22×Dic3—32no—
180C22×Dic3—32no—
181C22×Dic3—61no—
182C22×Dic3—62no—
183C22×Dic3C22×D312yes—
184C22×Dic3C22×D312yes—
185C22×Dic3C22×D312yes—
186C22×Dic3C22×D312yes—
187C2×C4×D3—38no—
188C2×C4×D3—38no—
189C2×C4×D3—64no—
190C2×C4×D3—64no—
191C2×C4×D3—122no—
192C2×C4×D3—122no—
193C2×C4×D3—122no—
194C2×C4×D3—122no—
195C2×C4×D3—122no—
196C2×C4×D3—122no—
197C2×C4×D3—122no—
198C2×C4×D3—122no—
199C2×Dic6—38no—
200C2×Dic6—122no—
201C2×Dic6—122no—
202C3×C4⋊C4—38no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—42no—
205C3×C42—122no—
206(C2×D3)⋊C4—38no—
207(C2×D3)⋊C4—38no—
208(C2×D3)⋊C4—64no—
209(C2×D3)⋊C4—64no—
210(C2×D3)⋊C4—124no—
211C4⋊Dic3—122no—
212C4⋊Dic3—122no—
213C4⋊Dic3—122no—
214C4⋊Dic3—122no—
215Dic3⋊C4—38no—
216Dic3⋊C4—122no—
217Dic3⋊C4—122no—
218Dic3⋊C4C4×D318yescomplemented
219C4×Dic3—42no—
220C4×Dic3—122no—
221C4×Dic3—122no—
222C22×C32C23×C411yescharacteristic
223C22×C32C22×Q811yescharacteristic
224C22×C32C2×#4111yescharacteristic
225C22×C32H(5,𝔽2)11yescharacteristic
226C22×C32H(5,𝔽2)11yescharacteristic
227C22×C32H(5,𝔽2)11yescharacteristic
228C22×C322−1+411yescharacteristic
229C2×C3⋊D3—22no—
230C2×C3⋊D3—22no—
231C2×C3⋊D3—22no—
232C2×C3⋊D3—22no—
233C2×C3⋊D3—22no—
234C2×C3⋊D3—22no—
235C2×C3⋊D3C4×Q814yes—
236C32×C4—22no—
237C32×C4—42no—
238C3⋊Dic3—22no—
239C3⋊Dic3—42no—
240C3×Dic3—24no—
241C3×Dic3—24no—
242C3×Dic3—24no—
243C3×Dic3—42no—
244C22×Q8—91no—
245C23×C4—91no—
246C23×C4—91no—
247C23×C4—91no—
248C22⋊Q8—98no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—91no—
251C2×C4⋊C4—92no—
252C2×C4⋊C4—92no—
253C2×C4⋊C4—91no—
254C2×C4⋊C4D3211yescharacteristic
255C2×C22⋊C4—92no—
256C2×C22⋊C4—92no—
257C2×C42—32no—
258C2×C42—92no—
259#95—32no—
260#95—32no—
261#95—32no—
262#95—32no—
263#95—92no—
264#95—92no—
265#95—92no—
266C23×C3—21no—
267C23×C3C23×D312yes—
268C22×D3—34no—
269C22×D3—34no—
270C22×D3—34no—
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—62no—
274C22×D3—62no—
275C22×D3—62no—
276C22×D3—62no—
277C22×D3—62no—
278C22×D3—121no—
279C22×D3—121no—
280C22×D3—121no—
281C22×D3—121no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—34no—
294C2×C3×C4—34no—
295C2×C3×C4—34no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—62no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4C2×C4×D314yes—
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—34no—
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—121no—
329C2×Dic3—121no—
330C2×Dic3—122no—
331C2×Dic3—122no—
332C2×Dic3—122no—
333C2×Dic3C2×C4×D314yes—
334C2×Dic3C2×C4×D314yes—
335C2×Dic3D3×Q814yes—
336C4×D3—128no—
337C4×D3—128no—
338C4×D3—128no—
339C4×D3—128no—
340Dic6—128no—
341C2×C32C2×C4×Q811yescharacteristic
342C2×C32#51811yescharacteristic
343C2×C32#52011yescharacteristic
344C2×C32#53811yescharacteristic
345C2×C32#53911yescharacteristic
346C2×C32#54311yescharacteristic
347C2×C32#54411yescharacteristic
348C3⋊D3—24no—
349C24—91no—
350C2×Q8—94no—
351C2×Q8—361no—
352C2×Q8—361no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—32no—
356C22×C4—32no—
357C22×C4—94no—
358C22×C4—91no—
359C22×C4—94no—
360C22×C4—92no—
361C22×C4—182no—
362C22×C4—361no—
363C22×C4—361no—
364C22×C4—361no—
365C22×C4—361no—
366C22×C4—361no—
367C22×C4—361no—
368C22×C4C2×D3212yes—
369C22×C4C2×D3211yescharacteristic
370C4⋊C4—38no—
371C4⋊C4—41no—
372C4⋊C4—94no—
373C4⋊C4—361no—
374C4⋊C4—362no—
375C4⋊C4—362no—
376C22⋊C4—98no—
377C22⋊C4—184no—
378C42—122no—
379C42—362no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3C22×C4×D312yes—
388C22×C3#100412yes—
389C22×C3C2×D3×Q812yes—
390C22×C3D3×#4112yes—
391C22×C3#100912yes—
392C22×C3#100912yes—
393C22×C3#101012yes—
394C2×D3—38no—
395C2×D3—64no—
396C2×D3—64no—
397C2×D3—64no—
398C2×D3—64no—
399C2×D3—64no—
400C2×D3—64no—
401C2×D3—64no—
402C2×D3—122no—
403C2×D3—122no—
404C2×D3—122no—
405C2×D3—122no—
406C2×D3—122no—
407C2×D3—122no—
408C3×C4—24no—
409C3×C4—24no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—64no—
413C3×C4—64no—
414C3×C4—64no—
415C3×C4—82no—
416C3×C4—122no—
417Dic3—24no—
418Dic3—24no—
419Dic3—24no—
420Dic3—42no—
421Dic3—64no—
422Dic3—122no—
423Dic3—122no—
424Dic3—122no—
425Dic3—242no—
426C32#232911yescharacteristic, complemented, Sylow
427C23—92no—
428C23—92no—
429C23—92no—
430C23—181no—
431C23—181no—
432C23—181no—
433C23—181no—
434C23C22×D3211yescharacteristic
435Q8—364no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—34no—
442C2×C4—34no—
443C2×C4—34no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—92no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4—182no—
457C2×C4—182no—
458C2×C4—182no—
459C2×C4—364no—
460C2×C4—364no—
461C2×C4—364no—
462C2×C4C4×D3212yes—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3#552312yes—
471C2×C3#554512yes—
472C2×C3C4×D3×Q812yes—
473C2×C3#563412yes—
474C2×C3#566012yes—
475C2×C3#568912yes—
476C2×C3#570412yes—
477D3—68no—
478D3—124no—
479C22—94no—
480C22—182no—
481C22—182no—
482C22—182no—
483C22—182no—
484C22—182no—
485C22—182no—
486C22C2×C4×D3211yescharacteristic
487C22#6430811yescharacteristic
488C22#6430911yescharacteristic
489C22#6431211yescharacteristic
490C22D32×Q811yescharacteristic
491C22#6432011yescharacteristic
492C22#6433011yescharacteristic
493C4—22no—
494C4—42no—
495C4—64no—
496C4—64no—
497C4—64no—
498C4—122no—
499C4—182no—
500C4—362no—
501C3—21no—
502C3#7529912yescomplemented
503C2—184no—
504C2#1059436311yescharacteristic
505C2#1059440111yescharacteristic
506C2#1059451111yescharacteristic
507C2#1059465811yescharacteristic
508C2#1059505811yescharacteristic
509C2#1059520911yescharacteristic
510C2#1059543411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C4⋊C4C22
socle, cosocleC23×C32C24
prime radicalC23×C3⋊D3C23
hypercenterC2×C4⋊C4D32
hypocenterC32#2329
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#4978C2semidirect
6. #10591268C2semidirect
7. #10591249C2semidirect
9. #10591191C2semidirect
25. #63865C4semidirect
38. #4978D3semidirect
63. C2×#627C2×C4semidirect
69. C2×#624Q8semidirect
73. C3×Dic3⋊C4C2×C4semidirect
75. #3724C2×C4semidirect
218. Dic3⋊C4C4×D3semidirect
426. C32#2329semidirect
502. C3#75299semidirect

Maximal subgroups

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

subgroupindex
C2×#638652
#105926862
C3×#49782
#105913752
#105913702
#105912682
#105912492
#105912332
#105911912
#752993