Finite Group Databasebeta

← #10834057

Subgroups — #10834057

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4840C211yesmaximal, characteristic, complemented
2C2×#4806C211yesmaximal, characteristic
3#76610C211yesmaximal, characteristic, complemented
4#76249C211yesmaximal, characteristic
5C3×#2066C211yesmaximal, characteristic, complemented
6#70580C211yesmaximal, characteristic, complemented
7#70574C211yesmaximal, characteristic, complemented
8#70482C211yesmaximal, characteristic, complemented
9#70266C211yesmaximal, characteristic, complemented
10#70260C211yesmaximal, characteristic, complemented
11#70254C211yesmaximal, characteristic, complemented
12#69680C211yesmaximal, characteristic, complemented
13#69635C211yesmaximal, characteristic, complemented
14#68881C211yesmaximal, characteristic
15#68854C211yesmaximal, characteristic, complemented
16#24518—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#895C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#815C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4993—21no—
38#4991—21no—
39#4988C2211yescharacteristic
40#4987C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48#4929C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4840—24no—
51#4806—24no—
52#4675—21no—
53#4672—21no—
54#4664—21no—
55#4664—21no—
56#4646—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#448—31no—
63C2×#433—31no—
64#2624—31no—
65#2310—31no—
66#2068—31no—
67#2066D311yescharacteristic, complemented
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1935—31no—
73#1933—31no—
74#1875—31no—
75#1854—31no—
76#1787—31no—
77C22×C3×Q8C2311yescharacteristic
78C2×D3×Q8—22no—
79C2×D3×Q8—22no—
80C2×D3×Q8—22no—
81C2×D3×Q8—22no—
82C2×D3×Q8—41no—
83C2×D3×Q8—41no—
84C22×D12—21no—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×C4×D3C2311yescharacteristic
88C22×Dic6—21no—
89C22×Dic6C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×#215—22no—
98C2×#215—22no—
99C2×#215—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4D411yescharacteristic
108C2×(C2×D3)⋊C4D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4D411yescharacteristic
113C2×Dic3⋊C4D411yescharacteristic
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C2×#204—22no—
119C2×#204—22no—
120C2×#204—41no—
121#895—22no—
122#895—22no—
123#892—41no—
124#892—41no—
125#872—22no—
126#872—22no—
127C3×#106—41no—
128C3×C4⋊C8—22no—
129C3×#103—22no—
130C3×#103—41no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134#835—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#821—41no—
144(C2×D3)⋊C8—22no—
145(C2×D3)⋊C8D414yes—
146#816—22no—
147#815—22no—
148C8×Dic3—22no—
149#810—22no—
150#810—41no—
151#809—41no—
152#807—41no—
153C4⋊(C3⋊C8)—22no—
154C23×Q8—31no—
155C22×SD16—31no—
156C22×SD16—31no—
157C2×#121—31no—
158C2×#117—31no—
159C2×#117—31no—
160#448—64no—
161#433—64no—
162C2×#107—31no—
163C2×#106C2×D311yescharacteristic
164C2×C4⋊C8—31no—
165C2×C4⋊C8C2×D311yescharacteristic
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103C2×D311yescharacteristic
171C2×#103C2×D311yescharacteristic
172C2×#102—31no—
173C2×#102—31no—
174C2×C22⋊C8—31no—
175C2×C22⋊C8—31no—
176C2×#97—31no—
177C2×C4×C8—31no—
178#399—61no—
179#397—61no—
180#394—61no—
181#394—61no—
182#391—31no—
183#391—31no—
184#391—31no—
185#391—61no—
186#391—61no—
187#391C2×D311yescharacteristic
188#390—31no—
189#390—31no—
190#389—31no—
191#389C2×D311yescharacteristic
192#381—61no—
193#381—61no—
194#380—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199C23×D3—21no—
200C23×D3C2×D411yescharacteristic
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8#4112yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220D3×Q8—44no—
221D3×Q8—44no—
222D3×Q8—44no—
223D3×Q8—44no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—42no—
231C2×D12—81no—
232C2×D12—81no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—24no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3C2×D412yes—
241C2×C4×D3#4112yes—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—24no—
247C2×Dic6—21no—
248C2×Dic6—42no—
249C2×Dic6—42no—
250C2×Dic6#4112yes—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8C2×D412yes—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—81no—
264C3×C42—21no—
265C3×C42—21no—
266#215—44no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—22no—
269(C2×D3)⋊C4—42no—
270(C2×D3)⋊C4—42no—
271(C2×D3)⋊C4—81no—
272(C2×D3)⋊C4—81no—
273(C2×D3)⋊C4—81no—
274(C2×D3)⋊C4—81no—
275(C2×D3)⋊C4—81no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280Dic3⋊C4—22no—
281Dic3⋊C4—22no—
282Dic3⋊C4—81no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—42no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8C2×D412yes—
292#204—44no—
293C22×Q8—31no—
294C22×Q8—61no—
295C22×Q8—62no—
296C22×Q8—62no—
297C22×Q8—62no—
298C22×Q8—62no—
299C22×Q8—121no—
300C22×Q8—121no—
301C22×Q8C22×D311yescharacteristic
302C22×D4—31no—
303C22×D4—61no—
304C23×C4—31no—
305C23×C4—61no—
306C2×SD16—62no—
307C2×SD16—62no—
308C2×SD16—62no—
309C2×SD16—62no—
310C2×SD16—121no—
311C2×SD16—121no—
312C22×C8—31no—
313C22×C8—31no—
314C22×C8C22×D311yescharacteristic
315C22×C8C22×D311yescharacteristic
316#121—62no—
317#121—62no—
318#117—62no—
319#117—62no—
320#117—121no—
321#117—121no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C22⋊C4—31no—
332C2×C22⋊C4—31no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C42—31no—
341C2×C42—61no—
342C2×C42C22×D311yescharacteristic
343#107—121no—
344#106—41no—
345C4⋊C8—22no—
346C4⋊C8—62no—
347#103—22no—
348#103—41no—
349#103—62no—
350#103—62no—
351#103—62no—
352#103—121no—
353#102—121no—
354#102—121no—
355C22⋊C8—34no—
356C22⋊C8—62no—
357#97—62no—
358C4×C8—62no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366#95—61no—
367#95—61no—
368#95—61no—
369#95—61no—
370C23×C3C22×D411yescharacteristic
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3#12112yes—
385C3×Q8—22no—
386C3×Q8—22no—
387C3×Q8—41no—
388C3×Q8—41no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4H(5,𝔽2)11yescharacteristic
412C2×C3×C4H(5,𝔽2)11yescharacteristic
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—22no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3#12112yes—
431D12—41no—
432D12—41no—
433D12—41no—
434D12—41no—
435D12—84no—
436C4×D3—22no—
437C4×D3—22no—
438C4×D3—22no—
439C4×D3—22no—
440C4×D3—48no—
441Dic6—22no—
442Dic6—22no—
443Dic6—48no—
444Dic6—41no—
445Dic6—41no—
446C3×C8—22no—
447C3×C8—41no—
448C3⋊C8—22no—
449C3⋊C8—41no—
450C24—31no—
451C24—61no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—32no—
458C2×Q8—64no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—122no—
465C2×Q8—122no—
466C2×Q8—124no—
467C2×Q8—124no—
468C2×Q8—124no—
469C2×Q8—124no—
470C2×Q8#23912yes—
471C2×D4—61no—
472C2×D4—61no—
473C2×D4—61no—
474C2×D4—61no—
475C2×D4—61no—
476C2×D4—61no—
477C2×D4—122no—
478C2×D4—241no—
479C2×D4—241no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—31no—
485C22×C4—32no—
486C22×C4—32no—
487C22×C4—31no—
488C22×C4—31no—
489C22×C4—64no—
490C22×C4—61no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—122no—
498C22×C4—122no—
499C22×C4D3×D411yescharacteristic
500C22×C4D3×D411yescharacteristic
501C22×C4C23×D311yescharacteristic
502SD16—124no—
503SD16—124no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—32no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8D3×D412yes—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—62no—
523C4⋊C4—62no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—81no—
529C4⋊C4—241no—
530C22⋊C4—62no—
531C22⋊C4—62no—
532C22⋊C4—122no—
533C22⋊C4—122no—
534C22⋊C4—241no—
535C22⋊C4—241no—
536C22⋊C4—241no—
537C22⋊C4—241no—
538C22⋊C4—241no—
539C42—21no—
540C42—21no—
541C42—61no—
542C42—61no—
543C42—122no—
544C22×C3C2×#12111yescharacteristic
545C22×C3#52511yescharacteristic
546C22×C3#53511yescharacteristic
547C22×C3#57711yescharacteristic
548C22×C3#57711yescharacteristic
549C22×C3#57711yescharacteristic
550C22×C3#57811yescharacteristic
551C2×D3—22no—
552C2×D3—22no—
553C2×D3—22no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—81no—
559C2×D3—81no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C3×C4—42no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—81no—
569C3×C4—81no—
570C3×C4—81no—
571C3×C4#54611yescharacteristic
572C3×C4#54611yescharacteristic
573C3×C4#55111yescharacteristic
574C3×C4#55111yescharacteristic
575Dic3—22no—
576Dic3—42no—
577Dic3—41no—
578Dic3—41no—
579Dic3—81no—
580Dic3—81no—
581C23—32no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23C2×D3×D411yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—41no—
599Q8—41no—
600Q8—62no—
601Q8—62no—
602Q8—128no—
603Q8—121no—
604Q8—121no—
605D4—121no—
606D4—121no—
607D4—121no—
608D4—121no—
609D4—244no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—32no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—62no—
642C2×C4—128no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4C2×D3×D411yescharacteristic
652C2×C4#100411yescharacteristic
653C2×C4C2×#23911yescharacteristic
654C2×C4D3×#4111yescharacteristic
655C2×C4D3×#4111yescharacteristic
656C2×C4#100911yescharacteristic
657C8—22no—
658C8—41no—
659C8—62no—
660C8—121no—
661C2×C3#241111yescharacteristic
662C2×C3#305711yescharacteristic
663C2×C3#306311yescharacteristic
664C2×C3#312111yescharacteristic
665C2×C3#312611yescharacteristic
666C2×C3#322111yescharacteristic
667C2×C3#322511yescharacteristic
668D3—42no—
669D3—161no—
670C22—62no—
671C22—62no—
672C22—62no—
673C22—121no—
674C22—121no—
675C22—121no—
676C22—121no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22D3×#12111yescharacteristic
685C22#560711yescharacteristic
686C22#562711yescharacteristic
687C22#576211yescharacteristic
688C22#577011yescharacteristic
689C22#577811yescharacteristic
690C22#577911yescharacteristic
691C4—41no—
692C4—41no—
693C4—42no—
694C4—62no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—121no—
699C4—121no—
700C4—122no—
701C4—241no—
702C4—241no—
703C4#561011yescharacteristic
704C4#563011yescharacteristic
705C4#563711yescharacteristic
706C4#570511yescharacteristic
707C3#2451811yescharacteristic, complemented, Sylow
708C2—122no—
709C2—481no—
710C2#7741411yescharacteristic
711C2#8043611yescharacteristic
712C2#8061111yescharacteristic
713C2#8065911yescharacteristic
714C2#8090711yescharacteristic
715C2#8100011yescharacteristic
716C2#8121011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2066C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2066D3
hypocenterC3#24518
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2066

Complements

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

NH ≅ G/Ntype
1. C2×#4840C2semidirect
3. #76610C2semidirect
5. C3×#2066C2semidirect
6. #70580C2semidirect
7. #70574C2semidirect
8. #70482C2semidirect
9. #70266C2semidirect
10. #70260C2semidirect
11. #70254C2semidirect
12. #69680C2semidirect
13. #69635C2semidirect
15. #68854C2semidirect
67. #2066D3semidirect
707. C3#24518semidirect

Maximal subgroups

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

subgroupindex
C2×#48402
C2×#48062
#766102
#762492
C3×#20662
#705802
#705742
#704822
#702662
#702602
#702542
#696802
#696352
#688812
#688542
#245183