Finite Group Databasebeta

← #11801703

Subgroups — #11801703

Order 1152.

1517 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64090C211yesmaximal, characteristic
2C2×#63957C211yesmaximal, characteristic, complemented
3C2×#63849C211yesmaximal, characteristic, complemented
4C3×#4951C212yesmaximal, complemented
5#10591705C211yesmaximal, characteristic, complemented
6#10591704C212yesmaximal
7#10591264C212yesmaximal, complemented
8#10591260C211yesmaximal, characteristic, complemented
9#10591257C211yesmaximal, characteristic, complemented
10C4×Dic32C211yesmaximal, characteristic, complemented
11#10591202C212yesmaximal, complemented
12#75450—32nomaximal
13C22×#638C2211yescharacteristic
14C23×C3⋊Dic3C2211yescharacteristic
15C22×C4×C3⋊D3C2211yescharacteristic
16C2×C32×C22⋊C4C2211yescharacteristic
17C2×#3752C2211yescharacteristic
18C2×#3750C2211yescharacteristic
19C2×C4×C3⋊Dic3C2211yescharacteristic
20#64090C418yescomplemented
21C2×C3×#217C2212yes—
22C2×C3×#217C2212yes—
23C2×C3×C4⋊Dic3C2212yes—
24C2×C3×C4×Dic3C2212yes—
25C2×#3724C2212yes—
26C2×#3724C2212yes—
27C2×#3722C2212yes—
28C2×#3722C2212yes—
29#63957—24no—
30C2×Dic32C2211yescharacteristic
31C2×Dic32C2212yes—
32C2×Dic32C2211yescharacteristic
33C2×Dic32C2211yescharacteristic
34C2×Dic32C2211yescharacteristic
35#63849—24no—
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2212yes—
38#63579C2212yes—
39C2×#892—32no—
40C2×#67⋊C4—32no—
41C2×#67⋊C4—61no—
42C2×#877—32no—
43C3×#388—32no—
44#4976—32no—
45#4957—32no—
46#4957—32no—
47#4951D312yescomplemented
48Dic3×C22⋊C4—32no—
49#4940—32no—
50C42×Dic3—32no—
51#4669—32no—
52#4666—32no—
53#4649—32no—
54#4647—32no—
55C24×C32C2311yescharacteristic
56C23×C3⋊D3C2311yescharacteristic
57C22×C32×C4C2311yescharacteristic
58C22×C32×C4C2311yescharacteristic
59C2×#638—24no—
60C2×#638C2×C414yes—
61C22×C3⋊Dic3—22no—
62C22×C3⋊Dic3—22no—
63C22×C3⋊Dic3C2×C414yes—
64C22×C3⋊Dic3C2311yescharacteristic
65C22×C3⋊Dic3C2311yescharacteristic
66C22×C3⋊Dic3C2311yescharacteristic
67C2×C4×C3⋊D3—22no—
68C2×C4×C3⋊D3—22no—
69C2×C4×C3⋊D3C2×C414yes—
70C22×C3×Dic3C2312yes—
71C22×C3×Dic3C2312yes—
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2312yes—
74C32×C22⋊C4C2×C414yes—
75#3752C2×C414yes—
76#3750C2×C414yes—
77C4×C3⋊Dic3C2×C414yes—
78C3×#217—24no—
79C3×#217—24no—
80C3×C4⋊Dic3—24no—
81C3×C4×Dic3—24no—
82#3724—24no—
83#3724—24no—
84#3722—24no—
85#3722—24no—
86Dic32—22no—
87Dic32—22no—
88Dic32—24no—
89Dic32—22no—
90Dic32D414yescomplemented
91#2293—91noSylow
92C22×#67—32no—
93C22×#67—61no—
94C23×Dic3—32no—
95C23×Dic3—61no—
96C22×C4×D3—32no—
97C22×C4×D3—61no—
98C2×C3×C4⋊C4—32no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—32no—
101C2×C3×C22⋊C4—32no—
102C2×C3×C22⋊C4C2×D312yes—
103C2×C3×C42—32no—
104C2×#217C2×D312yes—
105C2×#217C2×D312yes—
106C2×(C2×D3)⋊C4—32no—
107C2×(C2×D3)⋊C4—32no—
108C2×(C2×D3)⋊C4—32no—
109C2×(C2×D3)⋊C4—32no—
110C2×(C2×D3)⋊C4—32no—
111C2×(C2×D3)⋊C4—61no—
112C2×C4⋊Dic3C2×D312yes—
113C2×Dic3⋊C4—32no—
114C2×Dic3⋊C4—32no—
115C2×Dic3⋊C4—32no—
116C2×Dic3⋊C4—32no—
117C2×Dic3⋊C4—32no—
118C2×Dic3⋊C4—61no—
119C2×C4×Dic3—32no—
120C2×C4×Dic3—32no—
121C2×C4×Dic3—32no—
122C2×C4×Dic3—32no—
123C2×C4×Dic3—32no—
124C2×C4×Dic3—32no—
125C2×C4×Dic3—32no—
126C2×C4×Dic3—61no—
127C2×C4×Dic3C2×D312yes—
128#892—68no—
129#67⋊C4—316no—
130#67⋊C4—68no—
131#877—68no—
132C3×#95—32no—
133C3×#95—32no—
134C3⋊#95—32no—
135C3⋊#95—32no—
136C3⋊#95C2×D312yes—
137C3⋊#95C2×D312yes—
138C23×C32—22no—
139C23×C32—21no—
140C23×C32—22no—
141C23×C32—21no—
142C23×C32C22×C412yes—
143C23×C32C2411yescharacteristic
144C22×C3⋊D3—22no—
145C22×C3⋊D3—21no—
146C22×C3⋊D3—22no—
147C22×C3⋊D3—21no—
148C22×C3⋊D3C22×C412yes—
149C2×C32×C4—21no—
150C2×C32×C4—22no—
151C2×C32×C4—21no—
152C2×C32×C4C22×C412yes—
153C2×C32×C4C22×C412yes—
154#638—28no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—28no—
158C2×C3⋊Dic3C22×C412yes—
159C2×C3⋊Dic3C22×C412yes—
160C2×C3⋊Dic3C22×C412yes—
161C2×C3⋊Dic3C2×D412yes—
162C2×C3⋊Dic3#4114yes—
163C2×C3⋊Dic3#4112yes—
164C4×C3⋊D3—28no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3C2×D414yes—
176C2×C3×Dic3#4114yes—
177C2×C4×D4—91no—
178C2×#117—91no—
179C2×#117—91no—
180#390—92no—
181#389—92no—
182#388—32no—
183#388—92no—
184C4×C4⋊C4—91no—
185C4×C22⋊C4—91no—
186C4×C22⋊C4—91no—
187C43—91no—
188C24×C3—21no—
189C24×C3C22×D312yes—
190C23×D3—32no—
191C23×D3—61no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—32no—
195C22×C3×C4—32no—
196C22×C3×C4—32no—
197C22×C3×C4—32no—
198C22×C3×C4C22×D312yes—
199C22×C3×C4C22×D312yes—
200C2×#67—38no—
201C2×#67—64no—
202C2×#67—64no—
203C2×#67—64no—
204C2×#67—68no—
205C22×Dic3—32no—
206C22×Dic3—32no—
207C22×Dic3—38no—
208C22×Dic3—32no—
209C22×Dic3—61no—
210C22×Dic3—61no—
211C22×Dic3—64no—
212C22×Dic3—61no—
213C22×Dic3—64no—
214C22×Dic3—64no—
215C22×Dic3—122no—
216C22×Dic3—122no—
217C22×Dic3C22×D312yes—
218C22×Dic3C22×D312yes—
219C22×Dic3C22×D312yes—
220C22×Dic3C22×D312yes—
221C2×C4×D3—38no—
222C2×C4×D3—64no—
223C2×C4×D3—64no—
224C2×C4×D3—64no—
225C2×C4×D3—122no—
226C2×C4×D3—122no—
227C3×C4⋊C4—64no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—64no—
230C3×C22⋊C4—64no—
231C3×C22⋊C4C4×D318yes—
232C3×C42—64no—
233#217—24no—
234#217—24no—
235(C2×D3)⋊C4—38no—
236(C2×D3)⋊C4—64no—
237(C2×D3)⋊C4—64no—
238(C2×D3)⋊C4—64no—
239(C2×D3)⋊C4—64no—
240(C2×D3)⋊C4—64no—
241C4⋊Dic3—24no—
242Dic3⋊C4—38no—
243Dic3⋊C4—64no—
244Dic3⋊C4—64no—
245Dic3⋊C4—64no—
246Dic3⋊C4—64no—
247Dic3⋊C4—64no—
248C4×Dic3—24no—
249C4×Dic3—38no—
250C4×Dic3—38no—
251C4×Dic3—64no—
252C4×Dic3—64no—
253C4×Dic3—64no—
254C4×Dic3—64no—
255C4×Dic3—64no—
256C4×Dic3—64no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32C23×C411yescharacteristic
263C22×C32C22×D411yescharacteristic
264C22×C32C2×#4112yes—
265C22×C32C2×#4112yes—
266C22×C32C2×#4111yescharacteristic
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—22no—
270C2×C3⋊D3—42no—
271C2×C3⋊D3—42no—
272C32×C4—22no—
273C32×C4—22no—
274C3⋊Dic3—22no—
275C3⋊Dic3#11714yescomplemented
276C3⋊Dic3C4×D414yes—
277C3×Dic3—24no—
278C3×Dic3—24no—
279C3×Dic3—42no—
280C3×Dic3—42no—
281C22×D4—91no—
282C23×C4—91no—
283C23×C4—91no—
284C4×D4—98no—
285#117—184no—
286#117—184no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—91no—
289C2×C4⋊C4—92no—
290C2×C4⋊C4—92no—
291C2×C22⋊C4—32no—
292C2×C22⋊C4—32no—
293C2×C22⋊C4—91no—
294C2×C22⋊C4—92no—
295C2×C22⋊C4—92no—
296C2×C22⋊C4D3211yescharacteristic
297C2×C42—32no—
298C2×C42—91no—
299C2×C42—91no—
300C2×C42—91no—
301C2×C42—91no—
302C2×C42—91no—
303C2×C42—92no—
304#95—32no—
305#95—32no—
306#95—92no—
307C23×C3—24no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—42no—
319C23×C3C2×C4×D314yes—
320C23×C3C23×D312yes—
321C22×D3—34no—
322C22×D3—62no—
323C22×D3—64no—
324C22×D3—62no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—62no—
328C22×D3—62no—
329C22×D3—62no—
330C22×D3—121no—
331C22×D3—121no—
332C22×D3—122no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—34no—
340C2×C3×C4—34no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—62no—
352C2×C3×C4—62no—
353C2×C3×C4—62no—
354C2×C3×C4C2×C4×D314yes—
355C2×C3×C4C2×C4×D314yes—
356#67—616no—
357#67—616no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—34no—
369C2×Dic3—34no—
370C2×Dic3—34no—
371C2×Dic3—34no—
372C2×Dic3—34no—
373C2×Dic3—34no—
374C2×Dic3—34no—
375C2×Dic3—62no—
376C2×Dic3—62no—
377C2×Dic3—62no—
378C2×Dic3—62no—
379C2×Dic3—64no—
380C2×Dic3—62no—
381C2×Dic3—616no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—128no—
385C2×Dic3—121no—
386C2×Dic3—121no—
387C2×Dic3D3×D414yes—
388C2×Dic3#23714yes—
389C4×D3—616no—
390C4×D3—128no—
391C2×C32—42no—
392C2×C32C2×#11711yescharacteristic
393C2×C32C2×C4×D411yescharacteristic
394C2×C32C4×#4111yescharacteristic
395C2×C32#52912yes—
396C2×C32#53212yes—
397C3⋊D3—42no—
398C24—91no—
399C24C2×D3211yescharacteristic
400C2×D4—94no—
401C2×D4—184no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—91no—
407C22×C4—91no—
408C22×C4—91no—
409C22×C4—94no—
410C22×C4—94no—
411C22×C4—182no—
412C22×C4—182no—
413C22×C4—182no—
414C22×C4—182no—
415C22×C4C2×D3211yescharacteristic
416C22×C4C2×D3211yescharacteristic
417C4⋊C4—64no—
418C4⋊C4—94no—
419C4⋊C4—184no—
420C4⋊C4—184no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—94no—
424C22⋊C4—184no—
425C22⋊C4—184no—
426C22⋊C4#62414yes—
427C42—64no—
428C42—94no—
429C42—94no—
430C42—182no—
431C42—182no—
432C42—184no—
433C42—182no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—24no—
437C22×C3—24no—
438C22×C3—22no—
439C22×C3—21no—
440C22×C3—22no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—44no—
446C22×C3—44no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3C22×C4×D312yes—
452C22×C3C2×D3×D412yes—
453C22×C3C2×#23712yes—
454C22×C3D3×#4112yes—
455C22×C3D3×#4112yes—
456C22×C3D3×#4112yes—
457C22×C3D3×#4112yes—
458C2×D3—64no—
459C2×D3—64no—
460C2×D3—64no—
461C2×D3—64no—
462C2×D3—122no—
463C2×D3—122no—
464C2×D3—124no—
465C2×D3—124no—
466C2×D3—122no—
467C2×D3—122no—
468C2×D3—122no—
469C2×D3—122no—
470C3×C4—24no—
471C3×C4—24no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—64no—
475C3×C4—64no—
476C3×C4—122no—
477C3×C4—122no—
478Dic3—24no—
479Dic3—24no—
480Dic3—38no—
481Dic3—38no—
482Dic3—42no—
483Dic3—42no—
484Dic3—64no—
485Dic3—64no—
486Dic3—64no—
487Dic3—122no—
488C32#229311yescharacteristic, complemented, Sylow
489C23—22no—
490C23—21no—
491C23—22no—
492C23—21no—
493C23—92no—
494C23—182no—
495C23—181no—
496C23—182no—
497C23—181no—
498C23C2×#62412yes—
499C23C22×D3211yescharacteristic
500D4—188no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—34no—
505C2×C4—34no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—94no—
517C2×C4—92no—
518C2×C4—92no—
519C2×C4—92no—
520C2×C4—92no—
521C2×C4—92no—
522C2×C4—188no—
523C2×C4—188no—
524C2×C4—181no—
525C2×C4—181no—
526C2×C4C2×#62412yes—
527C2×C4C2×#62412yes—
528C2×C3—22no—
529C2×C3—21no—
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—44no—
534C2×C3—44no—
535C2×C3D3×#11712yes—
536C2×C3C4×#23712yes—
537C2×C3C4×D3×D412yes—
538C2×C3#559612yes—
539C2×C3#564912yes—
540C2×C3#566812yes—
541C2×C3#570612yes—
542D3—124no—
543D3—124no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—42no—
549C22—182no—
550C22—182no—
551C22—182no—
552C22—362no—
553C22—362no—
554C22#6430612yes—
555C22D32×D411yescharacteristic
556C22D3×#23712yes—
557C22#6431211yescharacteristic
558C22C22×#62411yescharacteristic
559C4—22no—
560C4—22no—
561C4—64no—
562C4—64no—
563C4—94no—
564C4—94no—
565C4—122no—
566C4—122no—
567C4—182no—
568C3—21no—
569C3#7545012yescomplemented
570C2—42no—
571C2—362no—
572C2#1059406112yes—
573C2#1059413012yes—
574C2#1059496911yescharacteristic
575C2D4×#62411yescharacteristic
576C2#1059519011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C22⋊C4C22
socle, cosocleC23×C32C24
prime radicalC22×#638C22
hypercenterC2×C22⋊C4D32
hypocenterC32#2293
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#63957C2semidirect
3. C2×#63849C2semidirect
4. C3×#4951C2semidirect
5. #10591705C2semidirect
7. #10591264C2semidirect
8. #10591260C2semidirect
9. #10591257C2semidirect
10. C4×Dic32C2semidirect
11. #10591202C2semidirect
20. #64090C4semidirect
47. #4951D3semidirect
90. Dic32D4semidirect
275. C3⋊Dic3#117semidirect
488. C32#2293semidirect
569. C3#75450semidirect

Maximal subgroups

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

subgroupindex
C2×#640902
C2×#639572
C2×#638492
C3×#49512
#105917052
#105917042
#105912642
#105912602
#105912572
C4×Dic322
#105912022
#754503