Finite Group Databasebeta

← C2×#35997 #7629588

Subgroups — C2×#35997 #7629588

Order 512.

8086 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C22×#519C211yesmaximal, characteristic, complemented
3#35997C2132yesmaximal, direct product factor
4C2×#2584C212yesmaximal, complemented
5C2×#2599C212yesmaximal, complemented
6C2×#2470C214yesmaximal
7C2×#2478C214yesmaximal, complemented
8C2×#2453C212yesmaximal, complemented
9C2×#2429C212yesmaximal, complemented
10C2×#2434C211yesmaximal, characteristic, complemented
11C2×#2409C211yesmaximal, characteristic
12C2×#2625C211yesmaximal, characteristic, complemented
13C2×#2634C212yesmaximal, complemented
14C2×#2396C212yesmaximal, complemented
15C2×#2395C211yesmaximal, characteristic
16C2×#2365C212yesmaximal, complemented
17C2×#2360C212yesmaximal, complemented
18C2×#2347C211yesmaximal, characteristic, complemented
19C24×Q8C2211yescharacteristic
20C2×#552—216no—
21C2×#552C22116yes—
22C22×C4⋊Q8C2212yes—
23C22×#126C2212yes—
24C22×#125C2212yes—
25C22×#125C2211yescharacteristic
26C22×#123C2212yes—
27C22×C22⋊Q8C2211yescharacteristic
28C22×C22⋊Q8C2212yes—
29C22×C22⋊Q8C2211yescharacteristic
30C2×#519—216no—
31C2×#519C22116yescomplemented
32C22×C4×Q8C2212yes—
33C22×C4×Q8C2212yes—
34C22×C4×Q8C2212yes—
35C22×C4×Q8C2211yescharacteristic
36C22×C4×Q8C2212yes—
37C22×C4×D4C2211yescharacteristic
38C22×#117C2212yes—
39C22×#117C2211yescharacteristic
40C22×#117C2214yes—
41C22×#117C2211yescharacteristic
42C23×C4⋊C4C2211yescharacteristic
43#2634C22132yescomplemented
44#2625C22116yescomplemented
45#2599C22132yescomplemented
46#2584C22132yescomplemented
47#2478C22164yescomplemented
48#2470C22164yes—
49#2453C22132yescomplemented
50#2434C22116yescomplemented
51#2429C22132yescomplemented
52#2409C22116yes—
53#2396C22132yescomplemented
54#2395C22116yes—
55C2×#400C2211yescharacteristic
56C2×#400C2212yes—
57C2×#400C2214yes—
58C2×#399C2211yescharacteristic
59C2×#398C2214yes—
60C2×#398C2212yes—
61C2×#398C2211yescharacteristic
62C2×#398C2214yes—
63C2×#397C2211yescharacteristic
64C2×#397C2214yes—
65C2×#397C2212yes—
66C2×#396C2214yes—
67C2×#395C2212yes—
68C2×#395C2212yes—
69C2×#395C2212yes—
70C2×#394C2212yes—
71C2×#393C2212yes—
72#2365C22132yescomplemented
73#2360C22132yescomplemented
74#2347C22116yescomplemented
75C2×#391C2212yes—
76C2×#391C2212yes—
77C2×#391C2212yes—
78C2×#389C2212yes—
79C2×#389C2212yes—
80C2×#389C2214yes—
81C2×#389C2214yes—
82C2×#389C2214yes—
83C2×#389C2214yes—
84C2×#388C2214yes—
85C2×#388C2214yes—
86C2×#387C2214yes—
87C2×#387C2214yes—
88C2×#387C2214yes—
89C2×#387C2214yes—
90C2×#386C2212yes—
91C2×#385C2214yes—
92C2×#385C2212yes—
93C2×#385C2214yes—
94C2×#383C2212yes—
95C2×#380C2211yescharacteristic
96C2×C4×C4⋊C4C2212yes—
97C2×C4×C4⋊C4C2212yes—
98C2×C4×C4⋊C4C2212yes—
99C2×C4×C4⋊C4C2214yes—
100C2×C4×C22⋊C4C2211yescharacteristic
101C2×C4×C22⋊C4C2212yes—
102C2×C4×C22⋊C4C2214yes—
103C23×Q8—216no—
104C23×Q8—28no—
105C23×Q8C2312yes—
106C23×Q8C2318yes—
107C23×Q8C2311yescharacteristic
108C23×Q8C2312yes—
109C23×D4C2311yescharacteristic
110C24×C4C2312yes—
111C24×C4C2311yescharacteristic
112C24×C4C2312yes—
113C24×C4C2311yescharacteristic
114#552—2128no—
115C2×C4⋊Q8—216no—
116C2×C4⋊Q8C23116yes—
117C2×#126—216no—
118C2×#126C23116yes—
119C2×#125—28no—
120C2×#125—216no—
121C2×#125C23116yes—
122C2×#125C2318yes—
123C2×#123—216no—
124C2×#123C23116yes—
125C2×C22⋊Q8—216no—
126C2×C22⋊Q8—28no—
127C2×C22⋊Q8—28no—
128C2×C22⋊Q8C23116yes—
129C2×C22⋊Q8C2318yes—
130C2×C22⋊Q8C2318yes—
131#519—2128no—
132C2×C4×Q8—216no—
133C2×C4×Q8—216no—
134C2×C4×Q8—28no—
135C2×C4×Q8—216no—
136C2×C4×Q8—216no—
137C2×C4×Q8C23116yes—
138C2×C4×Q8C23116yes—
139C2×C4×Q8C2318yes—
140C2×C4×Q8C23116yes—
141C2×C4×Q8C23116yes—
142C2×C4×D4—28no—
143C2×C4×D4C2318yes—
144C2×#117—28no—
145C2×#117—232no—
146C2×#117—216no—
147C2×#117D4116yes—
148C2×#117C23116yes—
149C2×#117C2318yes—
150C2×#117C23132yes—
151C2×#117C2318yes—
152C22×C4⋊C4—28no—
153C22×C4⋊C4C2318yes—
154C22×C4⋊C4C2312yes—
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2314yes—
157C22×C4⋊C4C2314yes—
158C22×C4⋊C4C2314yes—
159C22×C4⋊C4C2314yes—
160C22×C4⋊C4C2314yes—
161C22×C4⋊C4C2311yescharacteristic
162C22×C4⋊C4C2312yes—
163C22×C4⋊C4C2312yes—
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2311yescharacteristic
166C22×C4⋊C4C2312yes—
167C22×C4⋊C4C2311yescharacteristic
168C22×C4⋊C4C2312yes—
169C22×C4⋊C4C2314yes—
170C22×C4⋊C4C2312yes—
171C22×C4⋊C4C2312yes—
172C22×C4⋊C4C2312yes—
173C22×C4⋊C4C2314yes—
174C22×C4⋊C4C2311yescharacteristic
175C22×C4⋊C4C2311yescharacteristic
176C22×C22⋊C4C2311yescharacteristic
177C22×C22⋊C4C2314yes—
178C22×C22⋊C4C2312yes—
179C22×C22⋊C4C2312yes—
180C22×C22⋊C4C2312yes—
181C22×C22⋊C4C2312yes—
182C22×C22⋊C4C2312yes—
183C22×C22⋊C4C2311yescharacteristic
184C22×C22⋊C4C2312yes—
185C22×C22⋊C4C2312yes—
186C22×C22⋊C4C2312yes—
187C22×C42C2314yes—
188C22×C42C2312yes—
189C22×C42C2314yes—
190C22×C42C2312yes—
191C22×C42C2311yescharacteristic
192C22×C42C2312yes—
193C22×C42C2314yes—
194C22×C42C2312yes—
195#400C23116yes—
196#400C2318yes—
197#400C23132yes—
198#399C2318yes—
199#398C2318yes—
200#398C23116yes—
201#398C23132yes—
202#398C23132yes—
203#397C2318yes—
204#397C23132yes—
205#397C23116yes—
206#396C23132yes—
207#395C23116yes—
208#395C23116yes—
209#395C23116yes—
210#394C23116yes—
211#393C23116yes—
212#391C23116yes—
213#391C23116yes—
214#391C23116yes—
215#389C23132yes—
216#389C23132yes—
217#389C23132yes—
218#389C23132yes—
219#389C23116yes—
220#389C23116yes—
221#388C23132yes—
222#388C23132yes—
223#387C23132yes—
224#387C23132yes—
225#387C23132yes—
226#387C23132yes—
227#386C23116yes—
228#385C23132yes—
229#385C23116yes—
230#385C23132yes—
231#383C23116yes—
232#380C2318yes—
233C4×C4⋊C4C23116yes—
234C4×C4⋊C4C23116yes—
235C4×C4⋊C4C23116yes—
236C4×C4⋊C4C23132yes—
237C4×C22⋊C4C2318yes—
238C4×C22⋊C4C23116yes—
239C4×C22⋊C4C23132yes—
240C2×#95C2312yes—
241C2×#95C2314yes—
242C2×#95C2312yes—
243C2×#95C2311yescharacteristic
244C2×#95C2311yescharacteristic
245C2×#95C2314yes—
246C2×#95C2314yes—
247C2×#95C2314yes—
248C2×#95C2314yes—
249C2×#95C2312yes—
250C2×#95C2312yes—
251C2×#95C2314yes—
252C2×#95C2312yes—
253C2×#95C2314yes—
254C2×#95C2314yes—
255C2×#95C2314yes—
256C25C2411yescharacteristic
257C25C2411yescharacteristic
258C22×Q8—2128no—
259C22×Q8—264no—
260C22×Q8—28no—
261C22×Q8—28no—
262C22×Q8—232no—
263C22×Q8—24no—
264C22×Q8—216no—
265C22×Q8—28no—
266C22×Q8—28no—
267C22×Q8#41116yes—
268C22×Q8C2418yes—
269C22×Q8C2414yes—
270C22×Q8C2418yes—
271C22×D4—24no—
272C22×D4—24no—
273C22×D4—42no—
274C22×D4C2414yes—
275C23×C4—28no—
276C23×C4—216no—
277C23×C4—28no—
278C23×C4—28no—
279C23×C4—28no—
280C23×C4—24no—
281C23×C4—28no—
282C23×C4—44no—
283C23×C4C2×D418yes—
284C23×C4C2414yes—
285C23×C4C2418yes—
286C23×C4C2412yes—
287C23×C4C2412yes—
288C23×C4C2414yes—
289C23×C4C2411yescharacteristic
290C23×C4C2412yes—
291C23×C4C2412yes—
292C23×C4C2411yescharacteristic
293C23×C4C2412yes—
294C23×C4C2411yescharacteristic
295C23×C4C2418yes—
296C23×C4C2414yes—
297C23×C4C2412yes—
298C23×C4C2411yescharacteristic
299C23×C4C2412yes—
300C23×C4C2414yes—
301C23×C4C2412yes—
302C23×C4C2411yescharacteristic
303C4⋊Q8—264no—
304#126—264no—
305#125—232no—
306#125—264no—
307#123—264no—
308C22⋊Q8—264no—
309C22⋊Q8—232no—
310C22⋊Q8—232no—
311C4×Q8—264no—
312C4×Q8—264no—
313C4×Q8—232no—
314C4×Q8—264no—
315C4×Q8—264no—
316C4×D4—232no—
317#117—232no—
318#117—2128no—
319#117—264no—
320#117C2×D4164yes—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—216no—
324C2×C4⋊C4—216no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—216no—
335C2×C4⋊C4—216no—
336C2×C4⋊C4—216no—
337C2×C4⋊C4—216no—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—232no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4—48no—
342C2×C4⋊C4C2×D418yes—
343C2×C4⋊C4C2×D418yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2418yes—
346C2×C4⋊C4C24116yes—
347C2×C4⋊C4C24116yes—
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2418yes—
351C2×C4⋊C4C2418yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2418yes—
354C2×C4⋊C4C2418yes—
355C2×C4⋊C4C2418yes—
356C2×C4⋊C4C24116yes—
357C2×C4⋊C4C2418yes—
358C2×C4⋊C4C24116yes—
359C2×C4⋊C4C24116yes—
360C2×C4⋊C4C2418yes—
361C2×C4⋊C4C24116yes—
362C2×C4⋊C4C2418yes—
363C2×C4⋊C4C24116yes—
364C2×C4⋊C4C2414yes—
365C2×C4⋊C4C2414yes—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—216no—
373C2×C22⋊C4—28no—
374C2×C22⋊C4—28no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4C2×D4116yes—
377C2×C22⋊C4C2418yes—
378C2×C22⋊C4C2414yes—
379C2×C22⋊C4C2418yes—
380C2×C22⋊C4C2414yes—
381C2×C22⋊C4C2418yes—
382C2×C22⋊C4C24116yes—
383C2×C22⋊C4C2418yes—
384C2×C22⋊C4C2418yes—
385C2×C22⋊C4C2418yes—
386C2×C22⋊C4C2418yes—
387C2×C22⋊C4C2418yes—
388C2×C42—216no—
389C2×C42—216no—
390C2×C42—28no—
391C2×C42—28no—
392C2×C42—24no—
393C2×C42—28no—
394C2×C42—48no—
395C2×C42C2×D4116yes—
396C2×C42C24116yes—
397C2×C42C2418yes—
398C2×C42C24116yes—
399C2×C42C2418yes—
400C2×C42C2414yes—
401C2×C42C2418yes—
402C2×C42C24116yes—
403C2×C42C2418yes—
404#95C2414yes—
405#95C2414yes—
406#95C24116yes—
407#95C2418yes—
408#95C24116yes—
409#95C2418yes—
410#95C2418yes—
411#95C24116yes—
412#95C24116yes—
413#95C2418yes—
414#95C2418yes—
415#95C24116yes—
416#95C24116yes—
417#95C24116yes—
418#95C24116yes—
419#95C24116yes—
420C24—22no—
421C24—24no—
422C24—24no—
423C24—24no—
424C24—24no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24C22×D414yes—
429C242−1+414yes—
430C24C2512yes—
431C24C2512yes—
432C24C2511yescharacteristic
433C2×Q8—2256no—
434C2×Q8—264no—
435C2×Q8—264no—
436C2×Q8—216no—
437C2×Q8—216no—
438C2×Q8—28no—
439C2×Q8—232no—
440C2×Q8—216no—
441C2×Q8—216no—
442C2×Q8—416no—
443C2×Q8C2×#41132yes—
444C2×D4—28no—
445C2×D4—28no—
446C2×D4—44no—
447C2×D4—416no—
448C22×C4—264no—
449C22×C4—28no—
450C22×C4—216no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—232no—
459C22×C4—232no—
460C22×C4—216no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—24no—
471C22×C4—28no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—28no—
479C22×C4—216no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—28no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—216no—
487C22×C4—28no—
488C22×C4—216no—
489C22×C4—48no—
490C22×C4—432no—
491C22×C4—416no—
492C22×C4C22×D418yes—
493C22×C4C22×D418yes—
494C22×C4C22×D414yes—
495C22×C4C22×D4116yes—
496C22×C4C22×D414yes—
497C22×C4C2×#41116yes—
498C22×C4C2×#4118yes—
499C22×C42−1+418yes—
500C22×C4C2514yes—
501C22×C4C2518yes—
502C22×C4C2512yes—
503C22×C4C2514yes—
504C22×C4C2512yes—
505C22×C4C2514yes—
506C22×C4C2514yes—
507C22×C4C2514yes—
508C22×C4C2512yes—
509C22×C4C2512yes—
510C22×C4C2514yes—
511C22×C4C2514yes—
512C22×C4C2518yes—
513C22×C4C2514yes—
514C22×C4C2512yes—
515C4⋊C4—28no—
516C4⋊C4—216no—
517C4⋊C4—232no—
518C4⋊C4—232no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—28no—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—216no—
528C4⋊C4—232no—
529C4⋊C4—232no—
530C4⋊C4—232no—
531C4⋊C4—232no—
532C4⋊C4—216no—
533C4⋊C4—48no—
534C4⋊C4—416no—
535C4⋊C4C22×D4116yes—
536C4⋊C4C22×D4116yes—
537C22⋊C4—216no—
538C22⋊C4—28no—
539C22⋊C4—216no—
540C22⋊C4—216no—
541C22⋊C4—28no—
542C22⋊C4—216no—
543C22⋊C4—232no—
544C22⋊C4—216no—
545C22⋊C4—216no—
546C22⋊C4—48no—
547C22⋊C4C22×D4132yes—
548C42—232no—
549C42—232no—
550C42—216no—
551C42—216no—
552C42—28no—
553C42—216no—
554C42—416no—
555C42C22×D4132yes—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—28no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—216no—
569C23—44no—
570C23—44no—
571C23—48no—
572C23—48no—
573C23—48no—
574C23C23×D414yes—
575C23C23×D412yes—
576C23C22×#4114yes—
577C23C2×2−1+414yes—
578C23C2×2−1+412yes—
579C23C2×2−1+414yes—
580C23#58512yes—
581C23C2611yescharacteristic
582Q8—264no—
583Q8—264no—
584Q8—416no—
585D4—416no—
586C2×C4—264no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—232no—
597C2×C4—216no—
598C2×C4—216no—
599C2×C4—216no—
600C2×C4—232no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—432no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—416no—
634C2×C4—48no—
635C2×C4—416no—
636C2×C4#525116yes—
637C2×C4C23×D418yes—
638C2×C4C23×D418yes—
639C2×C4C23×D414yes—
640C2×C4C23×D414yes—
641C2×C4C22×#41116yes—
642C2×C4C22×#4118yes—
643C2×C4C2×2−1+418yes—
644C22—28no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—416no—
654C22—416no—
655C22—42no—
656C22C2×#52514yes—
657C22#348914yes—
658C22#349914yes—
659C22#351618yes—
660C22#352318yes—
661C22C24×D411yescharacteristic
662C22C23×#4112yes—
663C22C22×2−1+412yes—
664C22C22×2−1+411yescharacteristic
665C22C2×#58511yescharacteristic
666C4—28no—
667C4—28no—
668C4—28no—
669C4—24no—
670C4—216no—
671C4—48no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—44no—
680C4C2×#52518yes—
681C2—44no—
682C2—44no—
683C2#3599718yesdirect product factor
684C2C22×#52511yescharacteristic
685C2C2×#351612yes—
686C2C2×#352312yes—
687C2C2×#348911yescharacteristic
688C2C2×#349911yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#519C2semidirect
3. #35997C2direct (N × H)
4. C2×#2584C2semidirect
5. C2×#2599C2semidirect
7. C2×#2478C2semidirect
8. C2×#2453C2semidirect
9. C2×#2429C2semidirect
10. C2×#2434C2semidirect
12. C2×#2625C2semidirect
13. C2×#2634C2semidirect
14. C2×#2396C2semidirect
16. C2×#2365C2semidirect
17. C2×#2360C2semidirect
18. C2×#2347C2semidirect
31. C2×#519C22semidirect
43. #2634C22semidirect
44. #2625C22semidirect
45. #2599C22semidirect
46. #2584C22semidirect
47. #2478C22semidirect
49. #2453C22semidirect
50. #2434C22semidirect
51. #2429C22semidirect
53. #2396C22semidirect
72. #2365C22semidirect
73. #2360C22semidirect
74. #2347C22semidirect
683. C2#35997direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5522
C22×#5192
#359972
C2×#25842
C2×#25992
C2×#24702
C2×#24782
C2×#24532
C2×#24292
C2×#24342
C2×#24092
C2×#26252
C2×#26342
C2×#23962
C2×#23952
C2×#23652
C2×#23602
C2×#23472