Finite Group Databasebeta

← #11792274

Subgroups — #11792274

Order 1152.

869 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64151C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic, complemented
3#10595043C211yesmaximal, characteristic
4C3×#5229C211yesmaximal, characteristic, complemented
5C2×#63570C211yesmaximal, characteristic, complemented
6C2×#63569C211yesmaximal, characteristic, complemented
7#10589167C211yesmaximal, characteristic, complemented
8#72110—31nomaximal
9#69694—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×#3828C2211yescharacteristic
12C2×C3×#235C2211yescharacteristic
13C2×#3756—21no—
14#64151—22no—
15#64151—22no—
16C2×#3755—21no—
17C2×#3748C2211yescharacteristic
18C2×C3×#216—21no—
19C3×#942—22no—
20C3×#942—22no—
21C2×C3×#215—21no—
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25#63978—22no—
26#63916—21no—
27#63909—21no—
28#63871—21no—
29#63857—21no—
30C3×C3⋊#95—22no—
31#63570—22no—
32#63569—22no—
33C2×C3×#137—31no—
34#5833—31no—
35C2×#942—31no—
36C2×#942—31no—
37C2×#942—61no—
38C2×#942D311yescharacteristic, complemented
39#5510—31no—
40C3×#391—31no—
41C2×#837—31no—
42#5229D311yescharacteristic, complemented
43C2×#825—31no—
44C2×#805—31no—
45C2×#805—31no—
46#4554—31no—
47#4530—31no—
48C2×C32×Q8—21no—
49C2×C32×Q8—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8D412yescomplemented
54C22×C32×C4—21no—
55C22×C32×C4C2311yescharacteristic
56C2×#638—21no—
57#3828—21no—
58#3828—21no—
59#3828—21no—
60#3828—21no—
61C2×C4⋊(C3⋊D3)D411yescharacteristic
62C2×C4×C3⋊D3—21no—
63C2×#634D411yescharacteristic
64C2×C3×#67—21no—
65C22×C3×Dic3—21no—
66C22×C3×Dic3D412yes—
67C3×#235—21no—
68C3×#235—21no—
69C3×#235—21no—
70C3×#235—21no—
71C2×C3×D12D411yescharacteristic
72C2×C3×C4×D3—21no—
73C2×C3×Dic6D411yescharacteristic
74C2×#626—22no—
75#3756—24no—
76#3755—24no—
77#3748—21no—
78#3748D412yescomplemented
79C2×C3⋊(C3⋊C8)—21no—
80C3×#216—24no—
81C3×#215—24no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—21no—
84C3×C4×Dic3—21no—
85C3×C4×Dic3—21no—
86C3×#208—21no—
87C3×#208D412yescomplemented
88C2×C3×C3⋊C8—21no—
89Dic3⋊Dic3—22no—
90#3722—22no—
91(C2×D3)⋊Dic3—22no—
92#2007—91noSylow
93C2×C3×#41—31no—
94C22×C3×Q8—21no—
95C22×C3×Q8C2×D311yescharacteristic
96C22×C3×Q8C2×D311yescharacteristic
97C2×#235—31no—
98C2×#235—31no—
99C2×#235—61no—
100C2×#235C2×D311yescharacteristic
101C3×#137—62no—
102C3×#137—62no—
103C2×C3×Q16—61no—
104C2×C3×SD16—61no—
105C2×C3×M16—31no—
106C2×C3×C4⋊C4—31no—
107C2×C3×C42—31no—
108#946—61no—
109C2×#216—21no—
110C2×#216—61no—
111C2×#216—61no—
112C2×#216—62no—
113#942—22no—
114#942—22no—
115#942—62no—
116#942—62no—
117#942—64no—
118#942—64no—
119#942—62no—
120#942—62no—
121C2×#215—21no—
122C2×#215—61no—
123C2×#215—61no—
124C2×#215—62no—
125#939—62no—
126#938—61no—
127#930—61no—
128#930—62no—
129C2×C4⋊Dic3C2×D311yescharacteristic
130#924—61no—
131C2×C4×Dic3C2×D311yescharacteristic
132C2×#208—31no—
133C2×#208—31no—
134C2×#208—61no—
135C2×#208C2×D311yescharacteristic
136#897—61no—
137#897—61no—
138#895—61no—
139#895—61no—
140C3×#95—62no—
141#837—62no—
142C3⋊#95—22no—
143#825—62no—
144#805—62no—
145#805—62no—
146C23×C32C2×D411yescharacteristic
147C22×C3⋊D3—21no—
148C22×C3×D3—21no—
149C32×Q8—22no—
150C32×Q8—22no—
151C32×Q8—41no—
152C32×Q8—41no—
153C2×C32×C4—22no—
154C2×C32×C4—41no—
155C2×C32×C4—41no—
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4C2×D411yescharacteristic
158C2×C32×C4C2×D411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4#4111yescharacteristic
162#638—41no—
163#638—41no—
164C2×C3⋊Dic3—21no—
165C4⋊(C3⋊D3)—21no—
166C4⋊(C3⋊D3)—21no—
167C4×C3⋊D3—41no—
168C4×C3⋊D3—41no—
169#634—21no—
170#634—21no—
171C3×#67—41no—
172C3×#67—41no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—41no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3—42no—
179C3×D12—21no—
180C3×D12—21no—
181C3×C4×D3—41no—
182C3×C4×D3—41no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185#626—82no—
186C3⋊(C3⋊C8)—22no—
187C3×C3⋊C8—22no—
188C2×#137—31no—
189C2×#137—91no—
190#537—91no—
191C2×#104—91no—
192C2×#104—91no—
193#391—31no—
194#343—91no—
195C3×#41—61no—
196C3×#41—61no—
197C3×#41—61no—
198C3×#41—61no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—22no—
202C2×C3×Q8—22no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—22no—
212C2×C3×Q8—31no—
213C2×C3×Q8—41no—
214C2×C3×Q8—41no—
215C2×C3×Q8—41no—
216C2×C3×Q8D1212yescomplemented
217C2×C3×Q8#6712yescomplemented
218C2×C3×D4—31no—
219C2×C3×D4—61no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—32no—
224C22×C3×C4—41no—
225C22×C3×C4—61no—
226C22×C3×C4—61no—
227C22×C3×C4C22×D311yescharacteristic
228C22×C3×C4C22×D311yescharacteristic
229C2×#67—21no—
230C2×#67—62no—
231C2×#67—61no—
232C2×#67—61no—
233C2×#67—121no—
234C22×Dic3—21no—
235C22×Dic3#6712yes—
236#235—21no—
237#235—21no—
238#235—21no—
239#235—21no—
240#235—62no—
241#235—62no—
242#235—61no—
243#235—61no—
244#235—61no—
245#235—61no—
246#235—61no—
247#235—61no—
248#235—61no—
249#235—61no—
250#235—121no—
251#235—121no—
252C2×D12—31no—
253C2×D12—31no—
254C2×D12—61no—
255C2×D12—62no—
256C2×D12#6711yescharacteristic
257C2×C4×D3—21no—
258C2×C4×D3—61no—
259C2×C4×D3—61no—
260C2×C4×D3—121no—
261C2×Dic6—31no—
262C2×Dic6—31no—
263C2×Dic6—61no—
264C2×Dic6#6711yescharacteristic
265C3×Q16—64no—
266C3×SD16—64no—
267C3×M16—32no—
268C3×M16—61no—
269C2×C3×C8—61no—
270C3×C4⋊C4—61no—
271C3×C4⋊C4—61no—
272C3×C42—61no—
273C3×C42—61no—
274#217—62no—
275#216—24no—
276#216—68no—
277#216—64no—
278#216—64no—
279#215—24no—
280#215—68no—
281#215—64no—
282#215—64no—
283(C2×D3)⋊C4—62no—
284(C2×D3)⋊C4—62no—
285(C2×D3)⋊C4—62no—
286C4⋊Dic3—21no—
287C4⋊Dic3—21no—
288C4⋊Dic3—62no—
289Dic3⋊C4—62no—
290C4×Dic3—21no—
291C4×Dic3—21no—
292#208—21no—
293#208—32no—
294#208—32no—
295#208—62no—
296#208—62no—
297#208—61no—
298#208—61no—
299#208#6712yescomplemented
300C2×C3⋊C8—21no—
301C2×C3⋊C8—62no—
302C2×C3⋊C8—61no—
303C2×C3⋊C8—61no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32#12011yescharacteristic
307C22×C32#12111yescharacteristic
308C22×C32#12711yescharacteristic
309C2×C3⋊D3—41no—
310C2×C3⋊D3—41no—
311C2×C3⋊D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C32×C4—42no—
316C32×C4#12011yescharacteristic
317C32×C4#12011yescharacteristic
318C32×C4#12111yescharacteristic
319C32×C4#12111yescharacteristic
320C3⋊Dic3—41no—
321C3×Dic3—42no—
322C3×Dic3—41no—
323C3×Dic3—42no—
324C2×#41—31no—
325C2×#41—91no—
326C22×Q8D3211yescharacteristic
327#137—62no—
328#137—62no—
329#137—182no—
330#137—182no—
331C2×Q16—61no—
332C2×Q16—181no—
333C2×SD16—61no—
334C2×SD16—181no—
335C2×M16—31no—
336C2×M16—91no—
337C22⋊Q8—181no—
338C22⋊Q8—181no—
339#121—181no—
340#121—181no—
341#121—182no—
342C2×C4⋊C4—31no—
343C2×C42—31no—
344#104—182no—
345#104—182no—
346#95—62no—
347C23×C3—21no—
348C23×C3—61no—
349C23×C3D3×D411yescharacteristic
350C23×C3C2×#6711yescharacteristic
351C22×D3—21no—
352C22×D3—61no—
353C22×D3—61no—
354C22×D3—121no—
355C3×Q8—24no—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—42no—
361C3×Q8—42no—
362C3×Q8—42no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—61no—
368C3×Q8—61no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—121no—
372C3×D4—121no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—62no—
389C2×C3×C4—61no—
390C2×C3×C4—62no—
391C2×C3×C4—121no—
392C2×C3×C4—121no—
393C2×C3×C4—121no—
394C2×C3×C4—121no—
395C2×C3×C4—122no—
396C2×C3×C4C2×D1211yescharacteristic
397C2×C3×C4#23511yescharacteristic
398C2×C3×C4D3×D411yescharacteristic
399C2×C3×C4D3×D411yescharacteristic
400C2×C3×C4D3×D411yescharacteristic
401C2×C3×C4D3×D411yescharacteristic
402C2×C3×C4D3×D411yescharacteristic
403C2×C3×C4D3×D411yescharacteristic
404C2×C3×C4#23711yescharacteristic
405C2×C3×C4C2×#6711yescharacteristic
406C2×C3×C4C2×#6711yescharacteristic
407C2×C3×C4C2×#6711yescharacteristic
408#67—41no—
409#67—41no—
410#67—122no—
411#67—122no—
412#67—121no—
413#67—121no—
414#67—121no—
415#67—121no—
416#67—242no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—42no—
423C2×Dic3—61no—
424C2×Dic3—61no—
425C2×Dic3—121no—
426D12—21no—
427D12—21no—
428D12—62no—
429D12—61no—
430D12—61no—
431D12—61no—
432D12—61no—
433D12—121no—
434D12—242no—
435C4×D3—41no—
436C4×D3—41no—
437C4×D3—122no—
438C4×D3—122no—
439C4×D3—121no—
440C4×D3—121no—
441C4×D3—121no—
442C4×D3—121no—
443Dic6—21no—
444Dic6—21no—
445Dic6—62no—
446Dic6—61no—
447Dic6—61no—
448Dic6—61no—
449Dic6—61no—
450Dic6—121no—
451C3×C8—62no—
452C3⋊C8—22no—
453C3⋊C8—64no—
454C3⋊C8—62no—
455C3⋊C8—62no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32#39211yescharacteristic
459C2×C32#45511yescharacteristic
460C2×C32#45611yescharacteristic
461C3⋊D3—81no—
462C3×D3—81no—
463#41—61no—
464#41—61no—
465#41—61no—
466#41—61no—
467#41—181no—
468#41—181no—
469#41—181no—
470#41—181no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—31no—
477C2×Q8—91no—
478C2×Q8#62612yescomplemented
479C2×D4—31no—
480C2×D4—61no—
481C2×D4—91no—
482C2×D4—182no—
483C2×D4—181no—
484C22×C4—21no—
485C22×C4—32no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—181no—
489C22×C4C2×D3211yescharacteristic
490Q16—64no—
491Q16—184no—
492SD16—64no—
493SD16—184no—
494M16—32no—
495M16—61no—
496M16—92no—
497M16—181no—
498C2×C8—61no—
499C2×C8—181no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—182no—
503C22⋊C4—182no—
504C22⋊C4—182no—
505C42—61no—
506C42—61no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—21no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—121no—
517C22×C3—121no—
518C22×C3—121no—
519C22×C3#93011yescharacteristic
520C22×C3#93711yescharacteristic
521C22×C3#93711yescharacteristic
522C22×C3#93911yescharacteristic
523C22×C3#94011yescharacteristic
524C22×C3#94011yescharacteristic
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—122no—
529C2×D3—121no—
530C2×D3—121no—
531C2×D3—121no—
532C2×D3—121no—
533C2×D3—121no—
534C2×D3—121no—
535C2×D3—241no—
536C2×D3—241no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—44no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—122no—
545C3×C4—122no—
546C3×C4—121no—
547C3×C4#88211yescharacteristic
548C3×C4#88211yescharacteristic
549C3×C4#88411yescharacteristic
550C3×C4#88411yescharacteristic
551C3×C4#93711yescharacteristic
552C3×C4#93811yescharacteristic
553C3×C4#93911yescharacteristic
554C3×C4#95311yescharacteristic
555Dic3—42no—
556Dic3—41no—
557Dic3—42no—
558Dic3—122no—
559Dic3—121no—
560Dic3—121no—
561C32#200711yescharacteristic, complemented, Sylow
562C23—61no—
563C23—181no—
564C23D3×#6711yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—61no—
570Q8—61no—
571Q8—181no—
572Q8—181no—
573D4—61no—
574D4—61no—
575D4—121no—
576D4—121no—
577D4—181no—
578D4—181no—
579D4—361no—
580D4—361no—
581D4—722no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—61no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—122no—
593C2×C4—181no—
594C2×C4—361no—
595C2×C4—361no—
596C2×C4#380411yescharacteristic
597C2×C4C2×#62611yescharacteristic
598C2×C4D3×#6711yescharacteristic
599C2×C4D3×#6711yescharacteristic
600C2×C4#381111yescharacteristic
601C2×C4#381111yescharacteristic
602C8—62no—
603C8—182no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—41no—
612C2×C3—41no—
613C2×C3—241no—
614C2×C3#496011yescharacteristic
615C2×C3#506111yescharacteristic
616C2×C3#509511yescharacteristic
617C2×C3#520211yescharacteristic
618C2×C3#520511yescharacteristic
619C2×C3#522311yescharacteristic
620D3—81no—
621D3—242no—
622D3—241no—
623D3—241no—
624C22—21no—
625C22—21no—
626C22—121no—
627C22—121no—
628C22—121no—
629C22—361no—
630C22—361no—
631C22—361no—
632C22#6397811yescharacteristic
633C22#6397911yescharacteristic
634C22#6398311yescharacteristic
635C4—42no—
636C4—122no—
637C4—122no—
638C4—121no—
639C4—361no—
640C4#6395811yescharacteristic
641C4#6397111yescharacteristic
642C4#6397711yescharacteristic
643C4#6398111yescharacteristic
644C3—21no—
645C3#6969411yescharacteristic, complemented
646C3#7211011yescharacteristic, complemented
647C2—21no—
648C2—21no—
649C2—241no—
650C2—721no—
651C2#1059165911yescharacteristic
652C2#1059166111yescharacteristic
653C2#1059172411yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#120
prime radical#10595043C2
hypercenterC22×Q8D32
hypocenterC32#2007
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#64151C2semidirect
2. C2×C3×#942C2semidirect
4. C3×#5229C2semidirect
5. C2×#63570C2semidirect
6. C2×#63569C2semidirect
7. #10589167C2semidirect
38. C2×#942D3semidirect
42. #5229D3semidirect
53. C2×C32×Q8D4semidirect
78. #3748D4semidirect
87. C3×#208D4semidirect
216. C2×C3×Q8D12semidirect
217. C2×C3×Q8#67semidirect
299. #208#67semidirect
478. C2×Q8#626semidirect
561. C32#2007semidirect
645. C3#69694semidirect
646. C3#72110semidirect

Maximal subgroups

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

subgroupindex
C2×#641512
C2×C3×#9422
#105950432
C3×#52292
C2×#635702
C2×#635692
#105891672
#721103
#696943