Finite Group Databasebeta

← #11808697

Subgroups — #11808697

Order 1152.

1221 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64135C211yesmaximal, characteristic
2C2×C4×#627C211yesmaximal, characteristic, complemented
3C3×C4×#217C212yesmaximal
4#10591715C211yesmaximal, characteristic
5#10591692C211yesmaximal, characteristic
6#10591691C212yesmaximal
7#10591397C211yesmaximal, characteristic, complemented
8#10591351C212yesmaximal, complemented
9C4×#3724C211yesmaximal, characteristic, complemented
10C4×Dic3⋊Dic3C212yesmaximal, complemented
11#10591218C211yesmaximal, characteristic, complemented
12#73480—32nomaximal
13C23×C32×C4C2211yescharacteristic
14C22×#627C2211yescharacteristic
15C2×#3757C2211yescharacteristic
16C2×#3757C2211yescharacteristic
17#64135—24no—
18C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C4×C3⋊Dic3C2211yescharacteristic
21C2×C4×C3⋊Dic3C2211yescharacteristic
22C2×C3×#217C2212yes—
23C2×C3×#217C2212yes—
24C2×C3×C4×Dic3C2212yes—
25C2×C3×C4×Dic3C2212yes—
26C2×#3724C2212yes—
27C2×#3724C2211yescharacteristic
28C2×Dic3⋊Dic3C2212yes—
29C2×Dic3⋊Dic3C2212yes—
30C2×Dic3⋊Dic3C2212yes—
31C4×#627—24no—
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34#63579C2212yes—
35#63579C2212yes—
36#63579C2211yescharacteristic
37#63579C2211yescharacteristic
38C2×#926—32no—
39C2×#926—61no—
40C2×C4×Dic6—32no—
41C3×C4×C22⋊C4—32no—
42C4×#217D312yes—
43#4949—32no—
44#4948—32no—
45#4948—32no—
46#4947—32no—
47C4×C4⋊Dic3—32no—
48C4×Dic3⋊C4—32no—
49C4×Dic3⋊C4—32no—
50#4661—32no—
51#4660—32no—
52#4657—32no—
53#4656—32no—
54C24×C32C2311yescharacteristic
55C22×C32×C4—22no—
56C22×C32×C4—22no—
57C22×C32×C4Q814yescomplemented
58C22×C32×C4C2311yescharacteristic
59C22×C32×C4C2311yescharacteristic
60C22×C3⋊Dic3C2311yescharacteristic
61C22×C3⋊Dic3C2311yescharacteristic
62C22×C3⋊Dic3C2311yescharacteristic
63C22×C3⋊Dic3C2311yescharacteristic
64C22×C3×Dic3C2312yes—
65C22×C3×Dic3C2312yes—
66C22×C3×Dic3C2312yes—
67C22×C3×Dic3C2312yes—
68C2×#627—22no—
69C2×#627—24no—
70#3757—22no—
71#3757—22no—
72C4⋊(C3⋊Dic3)—22no—
73C4⋊(C3⋊Dic3)—22no—
74C4×C3⋊Dic3—22no—
75C4×C3⋊Dic3D414yes—
76C3×#217—24no—
77C3×#217—24no—
78C3×C4×Dic3—24no—
79C3×C4×Dic3—24no—
80#3724—22no—
81#3724—24no—
82Dic3⋊Dic3—24no—
83Dic3⋊Dic3—24no—
84Dic3⋊Dic3—24no—
85#2396—91noSylow
86C23×C3×C4—21no—
87C23×C3×C4C2×D312yes—
88C22×Dic6—32no—
89C2×C3×C22⋊C4—32no—
90C2×C3×C22⋊C4—32no—
91C2×C3×C42—32no—
92C2×C3×C42—32no—
93C2×#217—32no—
94C2×#217—32no—
95C2×#217—61no—
96C2×#217—61no—
97C2×#217C2×D312yes—
98C2×#217C2×D312yes—
99#926—68no—
100#926—68no—
101C2×C4⋊Dic3—32no—
102C2×C4⋊Dic3—32no—
103C2×C4⋊Dic3—32no—
104C2×C4⋊Dic3—32no—
105C2×C4⋊Dic3—32no—
106C2×C4⋊Dic3—61no—
107C2×C4⋊Dic3—61no—
108C2×Dic3⋊C4—32no—
109C2×Dic3⋊C4—32no—
110C2×Dic3⋊C4—32no—
111C2×Dic3⋊C4—32no—
112C2×Dic3⋊C4—32no—
113C2×Dic3⋊C4—32no—
114C2×C4×Dic3—32no—
115C2×C4×Dic3—32no—
116C2×C4×Dic3—61no—
117C2×C4×Dic3—61no—
118C2×C4×Dic3C2×D312yes—
119C2×C4×Dic3C2×D312yes—
120C4×Dic6—68no—
121C3×#95—32no—
122C3×#95—32no—
123C3⋊#95—32no—
124C3⋊#95—32no—
125C3⋊#95—32no—
126C3⋊#95—32no—
127C3⋊#95—32no—
128C3⋊#95—32no—
129C3⋊#95C2×D312yes—
130C3⋊#95C2×D312yes—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—22no—
134C23×C32—22no—
135C23×C32C2×Q812yes—
136C23×C32C2411yescharacteristic
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—28no—
140C2×C32×C4C2×D412yes—
141C2×C32×C4C2×Q812yes—
142C2×C32×C4C2×Q812yes—
143C2×C32×C4#4112yes—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3—22no—
147C2×C3⋊Dic3—22no—
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3C2×D412yes—
151C2×C3⋊Dic3C2×D412yes—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3#4114yes—
163C2×C3×Dic3#4114yes—
164#627—28no—
165C2×C4×Q8—91no—
166C2×#117—91no—
167#400—91no—
168#399—91no—
169#398—92no—
170#396—92no—
171#395—91no—
172#393—91no—
173C4×C4⋊C4—91no—
174C4×C4⋊C4—92no—
175C4×C22⋊C4—32no—
176C24×C3—21no—
177C24×C3C22×D312yes—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—32no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4Dic618yescomplemented
190C22×C3×C4C22×D312yes—
191C22×C3×C4C22×D312yes—
192C22×Dic3—32no—
193C22×Dic3—32no—
194C22×Dic3—32no—
195C22×Dic3—32no—
196C22×Dic3—61no—
197C22×Dic3—61no—
198C22×Dic3—61no—
199C22×Dic3—61no—
200C22×Dic3C22×D312yes—
201C22×Dic3C22×D312yes—
202C22×Dic3C22×D312yes—
203C22×Dic3C22×D312yes—
204C2×Dic6—64no—
205C2×Dic6—64no—
206C2×Dic6—64no—
207C3×C22⋊C4—64no—
208C3×C22⋊C4—64no—
209C3×C42—64no—
210C3×C42—64no—
211#217—24no—
212#217—24no—
213#217—64no—
214#217—64no—
215#217—64no—
216#217—64no—
217C4⋊Dic3—64no—
218C4⋊Dic3—64no—
219C4⋊Dic3—64no—
220C4⋊Dic3—64no—
221C4⋊Dic3—64no—
222C4⋊Dic3—64no—
223C4⋊Dic3—64no—
224Dic3⋊C4—64no—
225Dic3⋊C4—64no—
226Dic3⋊C4—64no—
227Dic3⋊C4—64no—
228Dic3⋊C4—64no—
229Dic3⋊C4—64no—
230C4×Dic3—24no—
231C4×Dic3—24no—
232C4×Dic3—38no—
233C4×Dic3—64no—
234C4×Dic3—64no—
235C4×Dic3—64no—
236C22×C32—22no—
237C22×C32—24no—
238C22×C32—41no—
239C22×C32—42no—
240C22×C32—41no—
241C22×C32C22×D411yescharacteristic
242C22×C32C22×Q811yescharacteristic
243C22×C32C2×#4112yes—
244C22×C32C2×#4112yes—
245C22×C32C2×#4111yescharacteristic
246C32×C4—22no—
247C32×C4C22⋊Q814yes—
248C3⋊Dic3—22no—
249C3⋊Dic3—22no—
250C3⋊Dic3—41no—
251C3⋊Dic3—41no—
252C3×Dic3—24no—
253C3×Dic3—24no—
254C3×Dic3—42no—
255C3×Dic3—42no—
256C22×Q8—91no—
257C23×C4D3211yescharacteristic
258C4×Q8—184no—
259#117—184no—
260C2×C4⋊C4—92no—
261C2×C4⋊C4—91no—
262C2×C4⋊C4—92no—
263C2×C4⋊C4—92no—
264C2×C4⋊C4—92no—
265C2×C4⋊C4—91no—
266C2×C4⋊C4—91no—
267C2×C22⋊C4—32no—
268C2×C22⋊C4—32no—
269C2×C22⋊C4—91no—
270C2×C22⋊C4—91no—
271C2×C42—32no—
272C2×C42—32no—
273C2×C42—91no—
274C2×C42—91no—
275#95—32no—
276#95—32no—
277#95—92no—
278#95—91no—
279#95—91no—
280#95—92no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3C2×Dic614yes—
294C23×C3C23×D312yes—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—216no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—216no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—34no—
306C2×C3×C4—34no—
307C2×C3×C4—62no—
308C2×C3×C4—62no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4C2×Dic614yes—
318C2×C3×C4C2×Dic614yes—
319C2×C3×C4D3×D414yes—
320C2×C3×C4#23714yes—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—34no—
332C2×Dic3—34no—
333C2×Dic3—62no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347C2×Dic3—122no—
348C2×Dic3—122no—
349C2×Dic3—122no—
350C2×Dic3—122no—
351C2×Dic3#23514yes—
352C2×Dic3#23514yes—
353Dic6—616no—
354C2×C32—42no—
355C2×C32C2×C22⋊Q811yescharacteristic
356C2×C32#52511yescharacteristic
357C2×C32#52912yes—
358C2×C32#53211yescharacteristic
359C2×C32#53312yes—
360C24C2×D3211yescharacteristic
361C2×Q8—184no—
362C2×Q8—182no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—32no—
366C22×C4—32no—
367C22×C4—32no—
368C22×C4—32no—
369C22×C4—91no—
370C22×C4—91no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4#62714yescomplemented
374C22×C4C2×D3211yescharacteristic
375C22×C4C2×D3211yescharacteristic
376C4⋊C4—182no—
377C4⋊C4—182no—
378C4⋊C4—184no—
379C4⋊C4—184no—
380C4⋊C4—184no—
381C4⋊C4—182no—
382C4⋊C4—184no—
383C22⋊C4—64no—
384C22⋊C4—64no—
385C22⋊C4—182no—
386C22⋊C4—182no—
387C42—64no—
388C42—64no—
389C42—94no—
390C42—182no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—44no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3C22×Dic612yes—
409C22×C3C2×#23512yes—
410C22×C3C2×#23512yes—
411C22×C3C2×D3×D412yes—
412C22×C3C2×#23712yes—
413C22×C3D3×#4112yes—
414C22×C3D3×#4112yes—
415C3×C4—24no—
416C3×C4—24no—
417C3×C4—24no—
418C3×C4—64no—
419C3×C4—64no—
420C3×C4—122no—
421C3×C4—122no—
422C3×C4#87818yes—
423Dic3—24no—
424Dic3—24no—
425Dic3—42no—
426Dic3—42no—
427Dic3—64no—
428Dic3—64no—
429Dic3—122no—
430Dic3—122no—
431Dic3—122no—
432Dic3—122no—
433Dic3—122no—
434Dic3—122no—
435C32#239611yescharacteristic, complemented, Sylow
436C23—22no—
437C23—22no—
438C23—21no—
439C23—21no—
440C23C2×#62712yes—
441C23C22×D3211yescharacteristic
442Q8—188no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—28no—
446C2×C4—34no—
447C2×C4—34no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—92no—
459C2×C4—92no—
460C2×C4—182no—
461C2×C4—182no—
462C2×C4—182no—
463C2×C4—182no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4#380512yes—
467C2×C4C2×#62712yes—
468C2×C4C2×#62712yes—
469C2×C4#381112yes—
470C2×C3—21no—
471C2×C3—22no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—22no—
475C2×C3—44no—
476C2×C3—44no—
477C2×C3#547012yes—
478C2×C3C2×#87812yes—
479C2×C3#551712yes—
480C2×C3#553812yes—
481C2×C3#554712yes—
482C2×C3#554812yes—
483C2×C3#555712yes—
484C22—22no—
485C22—24no—
486C22—41no—
487C22—42no—
488C22—41no—
489C22D3×#23512yes—
490C22D3×#23512yes—
491C22C2×#380511yescharacteristic
492C22C22×#62711yescharacteristic
493C22C2×#381111yescharacteristic
494C4—22no—
495C4—64no—
496C4—64no—
497C4—122no—
498C4—122no—
499C4—182no—
500C4—182no—
501C4—361no—
502C4—361no—
503C4#6398214yes—
504C3—21no—
505C3#7348012yescomplemented
506C2—42no—
507C2#1059493412yes—
508C2#1059501711yescharacteristic
509C2#1059504912yes—
510C2#1059507611yescharacteristic
511C2C2×#6398211yescharacteristic
subgroupG/H
centerC22×C4C2×D32
commutator, socle, cosocleC23×C32C24
FrattiniC23C22×D32
FittingC23×C32×C4C22
prime radicalC24×C32C23
hypercenterC23×C4D32
hypocenterC32#2396
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#627C2semidirect
7. #10591397C2semidirect
8. #10591351C2semidirect
9. C4×#3724C2semidirect
10. C4×Dic3⋊Dic3C2semidirect
11. #10591218C2semidirect
57. C22×C32×C4Q8semidirect
189. C22×C3×C4Dic6semidirect
373. C22×C4#627semidirect
435. C32#2396semidirect
505. C3#73480semidirect

Maximal subgroups

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

subgroupindex
C2×#641352
C2×C4×#6272
C3×C4×#2172
#105917152
#105916922
#105916912
#105913972
#105913512
C4×#37242
C4×Dic3⋊Dic32
#105912182
#734803