Finite Group Databasebeta

← #11803888

Subgroups — #11803888

Order 1152.

1584 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64087C211yesmaximal, characteristic
2C32×C4×C22⋊C4C211yesmaximal, characteristic
3C4×#3757C211yesmaximal, characteristic
4C2×#63658C211yesmaximal, characteristic
5#10592649C211yesmaximal, characteristic
6#10592645C211yesmaximal, characteristic
7#10592641C211yesmaximal, characteristic
8#10592632C211yesmaximal, characteristic, complemented
9#10592632C211yesmaximal, characteristic, complemented
10#10592628C211yesmaximal, characteristic, complemented
11#10592627C211yesmaximal, characteristic, complemented
12#10592354C211yesmaximal, characteristic, complemented
13#10592353C211yesmaximal, characteristic, complemented
14#10592346C211yesmaximal, characteristic, complemented
15#10592345C211yesmaximal, characteristic, complemented
16#73447—34nomaximal
17C23×C32×C4C2211yescharacteristic
18C23×C3⋊Dic3C2211yescharacteristic
19C23×C3⋊Dic3C2211yescharacteristic
20C22×#634C2211yescharacteristic
21C2×C32×C22⋊C4C2211yescharacteristic
22C2×C32×C22⋊C4C2211yescharacteristic
23C2×C32×C42C2211yescharacteristic
24C2×C32×C42C2211yescharacteristic
25C2×#3757C2211yescharacteristic
26C2×#3757C2211yescharacteristic
27C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C2×#3750C2211yescharacteristic
31C2×#3750C2211yescharacteristic
32C2×#3750C2211yescharacteristic
33C2×#3750C2211yescharacteristic
34C2×#3750C2211yescharacteristic
35C2×#3750C2211yescharacteristic
36C2×C4×C3⋊Dic3C2211yescharacteristic
37C2×C4×C3⋊Dic3C2211yescharacteristic
38#64087C418yescomplemented
39C32×#95C2211yescharacteristic
40C32×#95C2211yescharacteristic
41#63658C2211yescharacteristic
42#63658C2211yescharacteristic
43#63658C2211yescharacteristic
44#63658C2211yescharacteristic
45#63658C2211yescharacteristic
46#63658C2211yescharacteristic
47#63658C2211yescharacteristic
48#63658C2211yescharacteristic
49#63658C2211yescharacteristic
50#63658C2211yescharacteristic
51#63658C2211yescharacteristic
52#63658C2211yescharacteristic
53C2×#878—34no—
54C3×C4×C22⋊C4D314yes—
55C4×#217—34no—
56C2×C3⋊#95—34no—
57#4950—34no—
58#4946—34no—
59#4942—34no—
60#4933—34no—
61#4933—34no—
62#4929—34no—
63#4928—34no—
64#4655—34no—
65#4654—34no—
66#4647—34no—
67#4646—34no—
68C24×C32C2311yescharacteristic
69C22×C32×C4—41no—
70C22×C32×C4—41no—
71C22×C32×C4Q814yescomplemented
72C22×C32×C4C2311yescharacteristic
73C22×C32×C4C2311yescharacteristic
74C22×C32×C4C2311yescharacteristic
75C22×C32×C4C2311yescharacteristic
76C22×C32×C4C2311yescharacteristic
77C22×C32×C4C2311yescharacteristic
78C22×C3⋊Dic3—22no—
79C22×C3⋊Dic3—41no—
80C22×C3⋊Dic3—41no—
81C22×C3⋊Dic3—41no—
82C22×C3⋊Dic3—41no—
83C22×C3⋊Dic3C2×C414yes—
84C22×C3⋊Dic3C2311yescharacteristic
85C22×C3⋊Dic3C2311yescharacteristic
86C22×C3⋊Dic3C2311yescharacteristic
87C22×C3⋊Dic3C2311yescharacteristic
88C22×C3⋊Dic3C2311yescharacteristic
89C22×C3⋊Dic3C2311yescharacteristic
90C22×C3⋊Dic3C2311yescharacteristic
91C22×C3⋊Dic3C2311yescharacteristic
92C2×#634—41no—
93C2×#634—41no—
94C2×#634C2×C414yescomplemented
95C32×C22⋊C4—22no—
96C32×C22⋊C4C2×C414yes—
97C32×C42—41no—
98C32×C42—41no—
99#3757—22no—
100#3757C2×C414yes—
101C4⋊(C3⋊Dic3)—41no—
102C4⋊(C3⋊Dic3)—41no—
103C4⋊(C3⋊Dic3)C2×C414yescomplemented
104#3750—41no—
105#3750—41no—
106#3750—41no—
107#3750—41no—
108#3750C2×C414yescomplemented
109#3750C2×C414yescomplemented
110C4×C3⋊Dic3—41no—
111C4×C3⋊Dic3—41no—
112#2329—91noSylow
113C23×C3×C4C2×D314yes—
114C23×Dic3—34no—
115C23×Dic3—34no—
116C22×Dic6—34no—
117C2×C3×C22⋊C4C2×D314yes—
118C2×C3×C22⋊C4C2×D314yes—
119C2×C3×C42C2×D314yes—
120C2×C3×C42C2×D314yes—
121C2×#217—34no—
122C2×#217—34no—
123C2×C4⋊Dic3—34no—
124C2×C4⋊Dic3—34no—
125C2×C4⋊Dic3—34no—
126C2×Dic3⋊C4—34no—
127C2×Dic3⋊C4—34no—
128C2×Dic3⋊C4—34no—
129C2×Dic3⋊C4—34no—
130C2×Dic3⋊C4—34no—
131C2×Dic3⋊C4—34no—
132C2×C4×Dic3—34no—
133C2×C4×Dic3—34no—
134#878—332no—
135C3×#95C2×D314yes—
136C3×#95C2×D314yes—
137C3⋊#95—34no—
138C3⋊#95—34no—
139C3⋊#95—34no—
140C3⋊#95—34no—
141C3⋊#95—34no—
142C3⋊#95—34no—
143C3⋊#95—34no—
144C3⋊#95—34no—
145C3⋊#95—34no—
146C3⋊#95—34no—
147C3⋊#95—34no—
148C3⋊#95—34no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32—21no—
152C23×C32—21no—
153C23×C32C22×C412yes—
154C23×C32C2×Q812yes—
155C23×C32#4112yes—
156C23×C32C2411yescharacteristic
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—44no—
172C2×C32×C4C22×C412yes—
173C2×C32×C4C22×C412yes—
174C2×C32×C4C2×Q812yes—
175C2×C32×C4C2×Q812yes—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3—21no—
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3—21no—
191C2×C3⋊Dic3—21no—
192C2×C3⋊Dic3—21no—
193C2×C3⋊Dic3—21no—
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—21no—
196C2×C3⋊Dic3—44no—
197C2×C3⋊Dic3—44no—
198C2×C3⋊Dic3C22×C412yes—
199C2×C3⋊Dic3C22×C412yes—
200C2×C3⋊Dic3C22×C412yes—
201C2×C3⋊Dic3C22×C412yes—
202#634—44no—
203C2×C22⋊Q8—91no—
204#391—91no—
205#391—91no—
206#389—91no—
207#389—91no—
208#387—91no—
209#387—91no—
210#387—91no—
211#387—91no—
212#385—91no—
213#385—91no—
214#380—91no—
215C4×C22⋊C4—91no—
216C4×C22⋊C4C3⋊D311yescharacteristic
217C2×#95—91no—
218C24×C3C22×D314yes—
219C22×C3×C4—44no—
220C22×C3×C4—44no—
221C22×C3×C4Dic6116yescomplemented
222C22×C3×C4C22×D314yes—
223C22×C3×C4C22×D314yes—
224C22×C3×C4C22×D314yes—
225C22×C3×C4C22×D314yes—
226C22×C3×C4C22×D314yes—
227C22×C3×C4C22×D314yes—
228C22×Dic3—34no—
229C22×Dic3—34no—
230C22×Dic3—34no—
231C22×Dic3—34no—
232C22×Dic3—316no—
233C22×Dic3—34no—
234C22×Dic3—34no—
235C22×Dic3—34no—
236C22×Dic3—34no—
237C22×Dic3—68no—
238C22×Dic3—124no—
239C22×Dic3—124no—
240C22×Dic3—124no—
241C22×Dic3—124no—
242C2×Dic6—316no—
243C2×Dic6—124no—
244C2×Dic6—124no—
245C3×C22⋊C4—28no—
246C3×C22⋊C4C4×D3116yes—
247C3×C42—44no—
248C3×C42—44no—
249#217—316no—
250#217—68no—
251C4⋊Dic3—316no—
252C4⋊Dic3—124no—
253C4⋊Dic3—124no—
254Dic3⋊C4—316no—
255Dic3⋊C4—316no—
256Dic3⋊C4—124no—
257Dic3⋊C4—124no—
258Dic3⋊C4—124no—
259Dic3⋊C4—124no—
260C4×Dic3—124no—
261C4×Dic3—124no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32C4×Q814yes—
269C22×C32C23×C411yescharacteristic
270C22×C32C22×Q811yescharacteristic
271C22×C32C2×#4111yescharacteristic
272C22×C32H(5,𝔽2)11yescharacteristic
273C22×C32H(5,𝔽2)11yescharacteristic
274C22×C32H(5,𝔽2)11yescharacteristic
275C22×C322−1+411yescharacteristic
276C32×C4—22no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—22no—
280C32×C4—41no—
281C32×C4—41no—
282C3⋊Dic3—22no—
283C3⋊Dic3—22no—
284C3⋊Dic3—22no—
285C3⋊Dic3—22no—
286C3⋊Dic3—41no—
287C3⋊Dic3—41no—
288C3⋊Dic3—41no—
289C3⋊Dic3—41no—
290C22×Q8—91no—
291C23×C4—91no—
292C23×C4—91no—
293C23×C4C2×C3⋊D311yescharacteristic
294C22⋊Q8—98no—
295C2×C4⋊C4—91no—
296C2×C4⋊C4—91no—
297C2×C4⋊C4—91no—
298C2×C4⋊C4—91no—
299C2×C4⋊C4—91no—
300C2×C4⋊C4—91no—
301C2×C4⋊C4—91no—
302C2×C4⋊C4—91no—
303C2×C4⋊C4—91no—
304C2×C22⋊C4—91no—
305C2×C22⋊C4—91no—
306C2×C22⋊C4C2×C3⋊D311yescharacteristic
307C2×C22⋊C4C2×C3⋊D311yescharacteristic
308C2×C42—91no—
309C2×C42—91no—
310C2×C42C2×C3⋊D311yescharacteristic
311C2×C42C2×C3⋊D311yescharacteristic
312#95—91no—
313#95—91no—
314#95—91no—
315#95—91no—
316#95—91no—
317#95—91no—
318#95—91no—
319#95—91no—
320#95—91no—
321#95—91no—
322#95—91no—
323#95—91no—
324#95C2×C3⋊D311yescharacteristic
325#95C2×C3⋊D311yescharacteristic
326C23×C3—24no—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3C2×Dic618yes—
331C23×C3C2×C4×D318yes—
332C23×C3#23518yes—
333C23×C3C23×D314yes—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—416no—
349C2×C3×C4C2×Dic618yes—
350C2×C3×C4C2×Dic618yes—
351C2×C3×C4C2×C4×D318yes—
352C2×C3×C4C2×C4×D318yes—
353C2×Dic3—38no—
354C2×Dic3—38no—
355C2×Dic3—38no—
356C2×Dic3—38no—
357C2×Dic3—64no—
358C2×Dic3—64no—
359C2×Dic3—64no—
360C2×Dic3—64no—
361C2×Dic3—64no—
362C2×Dic3—64no—
363C2×Dic3—64no—
364C2×Dic3—64no—
365C2×Dic3—64no—
366C2×Dic3—64no—
367C2×Dic3—64no—
368C2×Dic3—64no—
369C2×Dic3—64no—
370C2×Dic3—64no—
371C2×Dic3—64no—
372C2×Dic3—64no—
373C2×Dic3—64no—
374C2×Dic3—64no—
375C2×Dic3—64no—
376C2×Dic3—64no—
377C2×Dic3—1216no—
378C2×Dic3—1216no—
379Dic6—1216no—
380C2×C32—24no—
381C2×C32C2×C4×Q811yescharacteristic
382C2×C32#51811yescharacteristic
383C2×C32#52011yescharacteristic
384C2×C32#53811yescharacteristic
385C2×C32#53911yescharacteristic
386C2×C32#54311yescharacteristic
387C2×C32#54411yescharacteristic
388C24C22×C3⋊D311yescharacteristic
389C2×Q8—94no—
390C2×Q8—361no—
391C2×Q8—361no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—91no—
395C22×C4—91no—
396C22×C4—91no—
397C22×C4—91no—
398C22×C4—94no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—91no—
402C22×C4—91no—
403C22×C4—182no—
404C22×C4—361no—
405C22×C4—361no—
406C22×C4—361no—
407C22×C4—361no—
408C22×C4#63414yescomplemented
409C22×C4C22×C3⋊D311yescharacteristic
410C22×C4C22×C3⋊D311yescharacteristic
411C22×C4C22×C3⋊D311yescharacteristic
412C22×C4C22×C3⋊D311yescharacteristic
413C22×C4C22×C3⋊D311yescharacteristic
414C22×C4C22×C3⋊D311yescharacteristic
415C4⋊C4—94no—
416C4⋊C4—94no—
417C4⋊C4—94no—
418C4⋊C4—361no—
419C4⋊C4—361no—
420C4⋊C4—361no—
421C4⋊C4—361no—
422C4⋊C4—361no—
423C4⋊C4—361no—
424C22⋊C4—22no—
425C22⋊C4—94no—
426C22⋊C4—182no—
427C22⋊C4C4×C3⋊D314yes—
428C42—41no—
429C42—41no—
430C42—361no—
431C42—361no—
432C22×C3—28no—
433C22×C3—28no—
434C22×C3—28no—
435C22×C3—28no—
436C22×C3—28no—
437C22×C3—28no—
438C22×C3C4×Dic6116yes—
439C22×C3C22×Dic614yes—
440C22×C3C22×C4×D314yes—
441C22×C3C2×#23514yes—
442C22×C3#100414yes—
443C22×C3#100414yes—
444C22×C3#100914yes—
445C22×C3#101014yes—
446C3×C4—28no—
447C3×C4—28no—
448C3×C4—28no—
449C3×C4—28no—
450C3×C4—44no—
451C3×C4—44no—
452Dic3—68no—
453Dic3—68no—
454Dic3—68no—
455Dic3—68no—
456Dic3—124no—
457Dic3—124no—
458Dic3—124no—
459Dic3—124no—
460C32#232911yescharacteristic, complemented, Sylow
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23C2×#63412yes—
466C23C2×C4×C3⋊D312yes—
467C23#382812yes—
468C23C23×C3⋊D311yescharacteristic
469Q8—364no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—44no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4—92no—
488C2×C4—92no—
489C2×C4—181no—
490C2×C4—181no—
491C2×C4—181no—
492C2×C4—181no—
493C2×C4—181no—
494C2×C4—181no—
495C2×C4—181no—
496C2×C4—181no—
497C2×C4—181no—
498C2×C4—181no—
499C2×C4—181no—
500C2×C4—181no—
501C2×C4—181no—
502C2×C4—181no—
503C2×C4—181no—
504C2×C4—181no—
505C2×C4—181no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4—181no—
509C2×C4—364no—
510C2×C4—364no—
511C2×C4C2×#63412yes—
512C2×C4C2×#63412yes—
513C2×C4C2×C4×C3⋊D312yes—
514C2×C4C2×C4×C3⋊D312yes—
515C2×C3—216no—
516C2×C3C2×C4×Dic614yes—
517C2×C3#548314yes—
518C2×C3#551914yes—
519C2×C3#554514yes—
520C2×C3#554614yes—
521C2×C3#556014yes—
522C2×C3#556414yes—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22C4×#63414yes—
530C22C22×#63411yescharacteristic
531C22C22×C4×C3⋊D311yescharacteristic
532C22C2×#382811yescharacteristic
533C22#6436111yescharacteristic
534C22#6436111yescharacteristic
535C22#6436611yescharacteristic
536C22#6436711yescharacteristic
537C4—22no—
538C4—22no—
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—182no—
544C4—182no—
545C4—182no—
546C4—182no—
547C4—361no—
548C4—361no—
549C4—361no—
550C4—361no—
551C3#7344714yescomplemented
552C2—24no—
553C2C2×C4×#63411yescharacteristic
554C2#1059595411yescharacteristic
555C2#1059599011yescharacteristic
556C2#1059601611yescharacteristic
557C2#1059601711yescharacteristic
558C2#1059603111yescharacteristic
559C2#1059603511yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC32×C4×C22⋊C4C2
socle, cosocleC23×C32C24
prime radicalC24×C32C23
hypercenterC4×C22⋊C4C3⋊D3
hypocenterC32#2329
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
8. #10592632C2semidirect
9. #10592632C2semidirect
10. #10592628C2semidirect
11. #10592627C2semidirect
12. #10592354C2semidirect
13. #10592353C2semidirect
14. #10592346C2semidirect
15. #10592345C2semidirect
38. #64087C4semidirect
71. C22×C32×C4Q8semidirect
94. C2×#634C2×C4semidirect
103. C4⋊(C3⋊Dic3)C2×C4semidirect
108. #3750C2×C4semidirect
109. #3750C2×C4semidirect
221. C22×C3×C4Dic6semidirect
408. C22×C4#634semidirect
460. C32#2329semidirect
551. C3#73447semidirect

Maximal subgroups

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

subgroupindex
C2×#640872
C32×C4×C22⋊C42
C4×#37572
C2×#636582
#105926492
#105926452
#105926412
#105926322
#105926282
#105926272
#105923542
#105923532
#105923462
#105923452
#734473