Finite Group Databasebeta

← #11870364

Subgroups — #11870364

Order 1152.

943 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10595981C211yesmaximal, characteristic
2#10594990C211yesmaximal, characteristic, complemented
3C3×#4964C211yesmaximal, characteristic, complemented
4C2×C3×#808C211yesmaximal, characteristic, complemented
5#10591106C211yesmaximal, characteristic, complemented
6#10591105C211yesmaximal, characteristic, complemented
7#10591073C211yesmaximal, characteristic, complemented
8#10590451C212yesmaximal, complemented
9#10590450C212yesmaximal, complemented
10#10590445C212yesmaximal, complemented
11#10590443C212yesmaximal, complemented
12#79222—31nomaximal
13#79112—31nomaximal
14C2×#3828C2211yescharacteristic
15C2×#3828C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17#64139—22no—
18#64106—21no—
19#64106C2212yes—
20#64104—21no—
21#64104C2212yes—
22C3×#926C2211yescharacteristic
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26#63952—22no—
27#63900C2212yes—
28#63894—21no—
29#63849C2212yes—
30#63844—21no—
31C3×#808C2212yes—
32C3×#808C2212yes—
33C3×#808C2212yes—
34C3×#807C2212yes—
35#63567C2211yescharacteristic
36#63567C2211yescharacteristic
37#63567C2212yes—
38#63564C2211yescharacteristic
39#63564C2211yescharacteristic
40#63564C2212yes—
41#63557C2212yes—
42#63557C2212yes—
43#5523—31no—
44#5510—31no—
45#5510—31no—
46#5510—61no—
47#5504—31no—
48C3×#428—31no—
49C2×C3×#106—31no—
50#5133—31no—
51#5129—31no—
52#5113—31no—
53#5110—31no—
54#5002—31no—
55#4966—31no—
56#4964D311yescharacteristic, complemented
57C2×#808D311yescharacteristic, complemented
58#4888—32no—
59#4885—32no—
60#4868—32no—
61#4866—32no—
62#4863—32no—
63#4863—32no—
64#4862—32no—
65#4862—32no—
66C22×C32×C4—21no—
67C22×C32×C4C2311yescharacteristic
68C2×#638—21no—
69C2×#638D412yes—
70#3828—22no—
71#3828—22no—
72#3828—22no—
73#3828—22no—
74C2×C4⋊(C3⋊D3)—22no—
75C2×C4⋊(C3⋊D3)D411yescharacteristic
76C2×C4⋊(C3⋊D3)C2311yescharacteristic
77C2×C4×C3⋊D3—21no—
78C2×C4×C3⋊D3C2312yes—
79C2×#634D411yescharacteristic
80C2×#634C2311yescharacteristic
81C22×C3×Dic3—21no—
82C2×#624—22no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4C2312yes—
85#3752—22no—
86#3752—22no—
87C4⋊(C3⋊Dic3)—22no—
88C3×#217—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3C2312yes—
91C3×C4⋊Dic3C2311yescharacteristic
92C3×C4⋊Dic3C2311yescharacteristic
93C3×C4×Dic3—21no—
94C3×#208—22no—
95C2×C3×C3⋊C8—22no—
96C2×C3×C3⋊C8C2311yescharacteristic
97C2×C3×C3⋊C8C2311yescharacteristic
98C2×C3×C3⋊C8C2312yes—
99#3724—22no—
100#3722—22no—
101Dic32—22no—
102#3099—91noSylow
103C2×#235—31no—
104C2×#235—31no—
105C2×#235—31no—
106C2×#235—31no—
107C2×#235—61no—
108C2×#235—61no—
109C2×C3×M16—31no—
110C22×C3×C8—31no—
111C3×#117—31no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4—31no—
114C2×C3×C4⋊C4C2×D311yescharacteristic
115C2×C3×C4⋊C4C2×D311yescharacteristic
116#930—62no—
117#930—62no—
118#930—64no—
119C4×#67—62no—
120#926C2×D311yescharacteristic
121C2×C4⋊Dic3C2×D311yescharacteristic
122C2×#208C2×D311yescharacteristic
123C22×C3⋊C8C2×D311yescharacteristic
124#897—32no—
125#897—32no—
126#897—62no—
127#897—61no—
128#897—61no—
129#897—62no—
130#895—32no—
131#895—32no—
132#895—62no—
133#895—61no—
134#895—61no—
135#895—62no—
136#893—61no—
137#892—32no—
138#892—32no—
139D3×C4⋊C4—32no—
140D3×C4⋊C4—32no—
141#887—61no—
142#67⋊C4—62no—
143C4×D12—61no—
144C4×Dic6—61no—
145C3×#107—32no—
146C3×#106—32no—
147C3×#106—32no—
148C3×#106—32no—
149#821—31no—
150#821—32no—
151#821—31no—
152(C2×D3)⋊C8—32no—
153(C2×D3)⋊C8—32no—
154(C2×D3)⋊C8—32no—
155(C2×D3)⋊C8—32no—
156#816—31no—
157#816—32no—
158#816—31no—
159#810—32no—
160#810—31no—
161#810—31no—
162#809—32no—
163#809—31no—
164#809—31no—
165#808C2×D312yes—
166#808C2×D312yes—
167#808C2×D312yes—
168#807C2×D312yes—
169C23×C32C2×D411yescharacteristic
170C22×C3⋊D3—21no—
171C22×C3⋊D3C2×D412yes—
172C2×C32×C4—22no—
173C2×C32×C4—81no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4#4112yes—
176C2×C32×C4#4112yes—
177C2×C32×C4C2411yescharacteristic
178#638—24no—
179#638—42no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3C2×D412yes—
182C4⋊(C3⋊D3)—22no—
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)—82no—
186C4×C3⋊D3—22no—
187C4×C3⋊D3—22no—
188C4×C3⋊D3—42no—
189#634—22no—
190#634—21no—
191#634—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—81no—
196#624—82no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—21no—
199C3×C3⋊C8—22no—
200#537—91no—
201#520—91no—
202#485—92no—
203#483—92no—
204#482—92no—
205#481—92no—
206#428—31no—
207C2×#106—31no—
208#417—91no—
209#416—91no—
210#408—91no—
211C22×C3×C4—22no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—61no—
216C22×C3×C4C22×D311yescharacteristic
217C22×C3×C4C22×D311yescharacteristic
218C2×#67—32no—
219C2×#67—32no—
220C2×#67—62no—
221C2×#67—62no—
222C2×#67—61no—
223C2×#67—61no—
224C22×Dic3—21no—
225#235—62no—
226#235—62no—
227#235—62no—
228#235—62no—
229#235—62no—
230#235—62no—
231#235—62no—
232#235—62no—
233#235—122no—
234#235—122no—
235#235—122no—
236#235—122no—
237C2×D12—31no—
238C2×D12—31no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×D12—61no—
243C2×D12—62no—
244C2×D12—62no—
245C2×D12—64no—
246C2×C4×D3—32no—
247C2×C4×D3—32no—
248C2×C4×D3—62no—
249C2×C4×D3—62no—
250C2×C4×D3—61no—
251C2×C4×D3—61no—
252C2×C4×D3—62no—
253C2×C4×D3—62no—
254C2×Dic6—31no—
255C2×Dic6—31no—
256C2×Dic6—31no—
257C2×Dic6—31no—
258C2×Dic6—61no—
259C2×Dic6—61no—
260C3×M16—62no—
261C2×C3×C8—31no—
262C2×C3×C8—31no—
263C2×C3×C8—32no—
264C2×C3×C8—62no—
265C3×C4⋊C4—22no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—22no—
269C3×C4⋊C4—31no—
270C3×C4⋊C4—32no—
271C3×C4⋊C4—31no—
272C3×C4⋊C4—61no—
273C3×C4⋊C4C22×D312yes—
274C3×C4⋊C4C22×D312yes—
275C3×C22⋊C4—61no—
276C3×C42—61no—
277#217—21no—
278(C2×D3)⋊C4—62no—
279(C2×D3)⋊C4—62no—
280(C2×D3)⋊C4—62no—
281(C2×D3)⋊C4—62no—
282(C2×D3)⋊C4—62no—
283(C2×D3)⋊C4—62no—
284(C2×D3)⋊C4—64no—
285(C2×D3)⋊C4—64no—
286C4⋊Dic3—21no—
287C4⋊Dic3—62no—
288C4⋊Dic3—62no—
289C4⋊Dic3—64no—
290C4⋊Dic3C22×D311yescharacteristic
291C4⋊Dic3C22×D311yescharacteristic
292C4⋊Dic3C22×D312yes—
293Dic3⋊C4—62no—
294Dic3⋊C4—62no—
295C4×Dic3—21no—
296C4×Dic3—62no—
297C4×Dic3—62no—
298#208—22no—
299C2×C3⋊C8—22no—
300C2×C3⋊C8C22×D312yes—
301C2×C3⋊C8C22×D311yescharacteristic
302C2×C3⋊C8C22×D311yescharacteristic
303C22×C32—21no—
304C22×C32—21no—
305C22×C32#12312yes—
306C22×C32C22×D411yescharacteristic
307C2×C3⋊D3—22no—
308C2×C3⋊D3—22no—
309C2×C3⋊D3—22no—
310C2×C3⋊D3—41no—
311C2×C3⋊D3—41no—
312C2×C3⋊D3—41no—
313C32×C4—42no—
314C32×C4#12312yes—
315C32×C4C2×#4111yescharacteristic
316C32×C4C2×#4111yescharacteristic
317C3⋊Dic3—22no—
318C3⋊Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—42no—
321C3×Dic3—41no—
322C2×#41—91no—
323C2×#41—91no—
324C2×M16—31no—
325C22×C8—31no—
326C22⋊Q8—92no—
327C22⋊Q8—181no—
328#121—92no—
329#121—181no—
330#121—182no—
331C4×Q8—181no—
332C4×D4—181no—
333C4×D4—182no—
334#117—31no—
335#117—92no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—92no—
338C2×C4⋊C4D3211yescharacteristic
339#107—32no—
340#106—32no—
341#106—32no—
342#106—32no—
343#103—91no—
344#103—91no—
345#103—92no—
346#102—91no—
347#102—91no—
348#102—92no—
349C22⋊C8—92no—
350C22⋊C8—92no—
351C23×C3—21no—
352C23×C3D3×D411yescharacteristic
353C23×C3D3×D411yescharacteristic
354C22×D3—32no—
355C22×D3—32no—
356C22×D3—62no—
357C22×D3—62no—
358C22×D3—61no—
359C22×D3—61no—
360C2×C3×C4—24no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—61no—
368C2×C3×C4—62no—
369C2×C3×C4—61no—
370C2×C3×C4—81no—
371C2×C3×C4—81no—
372C2×C3×C4—82no—
373C2×C3×C4—241no—
374C2×C3×C4#23512yes—
375C2×C3×C4#23512yes—
376C2×C3×C4D3×D411yescharacteristic
377C2×C3×C4D3×D411yescharacteristic
378C2×C3×C4#23912yes—
379C2×C3×C4#23912yes—
380C2×C3×C4C23×D311yescharacteristic
381C2×C3×C4C23×D311yescharacteristic
382#67—64no—
383#67—64no—
384#67—124no—
385#67—124no—
386#67—122no—
387#67—122no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—62no—
394C2×Dic3—62no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—81no—
398D12—62no—
399D12—62no—
400D12—61no—
401D12—61no—
402D12—61no—
403D12—61no—
404D12—121no—
405D12—121no—
406D12—122no—
407D12—242no—
408D12—242no—
409D12—244no—
410C4×D3—62no—
411C4×D3—62no—
412C4×D3—62no—
413C4×D3—62no—
414C4×D3—122no—
415C4×D3—122no—
416C4×D3—124no—
417C4×D3—122no—
418C4×D3—122no—
419C4×D3—242no—
420C4×D3—242no—
421Dic6—62no—
422Dic6—62no—
423Dic6—61no—
424Dic6—61no—
425Dic6—61no—
426Dic6—61no—
427Dic6—122no—
428Dic6—121no—
429Dic6—121no—
430C3×C8—61no—
431C3×C8—61no—
432C3×C8—62no—
433C3⋊C8—21no—
434C3⋊C8—21no—
435C3⋊C8—22no—
436C2×C32—22no—
437C2×C32C2×#12311yescharacteristic
438C2×C32#57711yescharacteristic
439C2×C32#57711yescharacteristic
440C3⋊D3—42no—
441C3⋊D3—81no—
442#41—182no—
443#41—182no—
444#41—182no—
445#41—182no—
446C2×Q8—91no—
447C2×Q8—91no—
448C2×D4—92no—
449C2×D4—91no—
450C2×D4—91no—
451C2×D4—182no—
452C2×D4—181no—
453C22×C4—21no—
454C22×C4—61no—
455C22×C4—92no—
456C22×C4—182no—
457C22×C4—181no—
458C22×C4C2×D3211yescharacteristic
459M16—62no—
460C2×C8—31no—
461C2×C8—31no—
462C2×C8—32no—
463C2×C8—62no—
464C4⋊C4—21no—
465C4⋊C4—32no—
466C4⋊C4—31no—
467C4⋊C4—31no—
468C4⋊C4—61no—
469C4⋊C4—182no—
470C4⋊C4—182no—
471C4⋊C4C2×D3212yes—
472C22⋊C4—61no—
473C22⋊C4—182no—
474C22⋊C4—182no—
475C22⋊C4—182no—
476C42—61no—
477C42—182no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—22no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3#89412yes—
487C22×C3#89412yes—
488C22×C3C2×D3×D411yescharacteristic
489C22×C3C2×D3×D411yescharacteristic
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—62no—
496C2×D3—122no—
497C2×D3—122no—
498C2×D3—122no—
499C2×D3—122no—
500C2×D3—122no—
501C2×D3—122no—
502C2×D3—121no—
503C2×D3—121no—
504C2×D3—121no—
505C2×D3—121no—
506C2×D3—121no—
507C2×D3—121no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—22no—
511C3×C4—44no—
512C3×C4—42no—
513C3×C4—42no—
514C3×C4—121no—
515C3×C4—121no—
516C3×C4—122no—
517C3×C4#89412yes—
518C3×C4#89412yes—
519C3×C4C2×#23511yescharacteristic
520C3×C4C2×#23511yescharacteristic
521C3×C4C2×#23911yescharacteristic
522C3×C4C2×#23911yescharacteristic
523Dic3—42no—
524Dic3—41no—
525Dic3—41no—
526Dic3—62no—
527Dic3—62no—
528Dic3—122no—
529Dic3—122no—
530Dic3—121no—
531Dic3—121no—
532C32#309911yescharacteristic, complemented, Sylow
533C23—92no—
534C23—181no—
535C23#381111yescharacteristic
536Q8—182no—
537Q8—181no—
538Q8—181no—
539D4—184no—
540D4—181no—
541D4—181no—
542D4—182no—
543D4—362no—
544D4—722no—
545C2×C4—22no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—62no—
549C2×C4—81no—
550C2×C4—92no—
551C2×C4—182no—
552C2×C4—182no—
553C2×C4—181no—
554C2×C4—241no—
555C2×C4—362no—
556C2×C4—722no—
557C2×C4#379912yes—
558C2×C4#379912yes—
559C2×C4#381111yescharacteristic
560C2×C4C22×D3211yescharacteristic
561C8—61no—
562C8—61no—
563C8—62no—
564C2×C3—22no—
565C2×C3—22no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—42no—
570C2×C3C2×#89411yescharacteristic
571C2×C3C2×#89411yescharacteristic
572C2×C3#576211yescharacteristic
573C2×C3#576211yescharacteristic
574C2×C3#577511yescharacteristic
575C2×C3#577811yescharacteristic
576D3—122no—
577D3—122no—
578D3—242no—
579D3—242no—
580D3—241no—
581D3—241no—
582C22—21no—
583C22—21no—
584C22—182no—
585C22—182no—
586C22—182no—
587C22—361no—
588C22—361no—
589C22—361no—
590C22#6389712yes—
591C22C2×#381111yescharacteristic
592C4—42no—
593C4—121no—
594C4—122no—
595C4—121no—
596C4—182no—
597C4—361no—
598C4#6389712yes—
599C4C2×#379911yescharacteristic
600C4C2×#379911yescharacteristic
601C3—21no—
602C3#7911211yescharacteristic, complemented
603C3#7922211yescharacteristic, complemented
604C2—22no—
605C2—362no—
606C2—721no—
607C2C2×#6389711yescharacteristic
608C2#1059511511yescharacteristic
609C2#1059532411yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32C22×D4
prime radical#10595981C2
hypercenterC2×C4⋊C4D32
hypocenterC32#3099
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #10594990C2semidirect
3. C3×#4964C2semidirect
4. C2×C3×#808C2semidirect
5. #10591106C2semidirect
6. #10591105C2semidirect
7. #10591073C2semidirect
8. #10590451C2semidirect
9. #10590450C2semidirect
10. #10590445C2semidirect
11. #10590443C2semidirect
56. #4964D3semidirect
57. C2×#808D3semidirect
532. C32#3099semidirect
602. C3#79112semidirect
603. C3#79222semidirect

Maximal subgroups

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

subgroupindex
#105959812
#105949902
C3×#49642
C2×C3×#8082
#105911062
#105911052
#105910732
#105904512
#105904502
#105904452
#105904432
#792223
#791123