Finite Group Databasebeta

← Q8×#6513 #14234259

Subgroups — Q8×#6513 #14234259

Order 1792.

1786 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×#41C211yesmaximal, characteristic, complemented
2Q8×#1157C211yesmaximal, characteristic, complemented
3Q8×#1143C211yesmaximal, characteristic, complemented
4Q8×#1144C211yesmaximal, characteristic, complemented
5#11741138C213yesmaximal, complemented
6#11741127C213yesmaximal, complemented
7#11741074C213yesmaximal, complemented
8#11741073C213yesmaximal, complemented
9Q8×#1142C211yesmaximal, characteristic, complemented
10Q8×#1141C211yesmaximal, characteristic, complemented
11#11740794C213yesmaximal, complemented
12#11740784C213yesmaximal, complemented
13#11739952C213yesmaximal, complemented
14C4×#6513C213yesmaximal
15C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
16C7×Q82C2211yescharacteristic
17C7×#557C2213yes—
18C7×#554—23no—
19C7×#554C2213yes—
20C7×D4×Q8—21no—
21C7×D4×Q8C2211yescharacteristic
22C7×#533—23no—
23C7×#533C2213yes—
24C4×C7×#41C2213yes—
25C2×C4×C7×Q8—21no—
26C2×C4×C7×Q8C2211yescharacteristic
27C2×#6513C2211yescharacteristic
28Q8×C22⋊D7—21no—
29#90270C2213yes—
30#90269—23no—
31Q8×D28C2211yescharacteristic
32C4×Q8×D7—21no—
33#90258C2213yes—
34Q8×Dic14C2211yescharacteristic
35#90202—23no—
36C4×#1157C2213yes—
37#90161C2213yes—
38#90005C2213yes—
39#89866C2213yes—
40#89863C2213yes—
41#89859C2213yes—
42#89856C2213yes—
43#89851C2213yes—
44#89851C2213yes—
45#89850C2213yes—
46#89850C2213yes—
47#89838C2216yes—
48#89836C2216yes—
49#89836C2216yes—
50#89835C2216yes—
51C4×#1144C2213yes—
52C4×#1144C2213yes—
53C4×#1143C2213yes—
54C4×#1143C2213yes—
55Q8×C7⋊C8C2211yescharacteristic
56Q8×C7⋊C8C2211yescharacteristic
57Q8×C7⋊C8C2213yes—
58Q8×C7⋊C8C2213yes—
59#89794C2216yes—
60#89793C2213yes—
61#89792C2213yes—
62C4×#1142C2213yes—
63C4×#1142C2213yes—
64C4×#1141C2213yes—
65C4×#1141C2213yes—
66#89781C2216yes—
67#89780C2213yes—
68#89779C2213yes—
69#89774C2213yes—
70#89770C2213yes—
71#89770C2216yes—
72C2×C4⋊(C7⋊C8)C2213yes—
73C2×C4×C7⋊C8C2213yes—
74Q8×#135—71nomaximal, Sylow
75C2×C7×#41C2311yescharacteristic
76C22×C7×Q8—21no—
77C22×C7×Q8D411yescharacteristic
78C2×Q8×D7—21no—
79C2×#1157C2311yescharacteristic
80C7×C4⋊Q8—23no—
81C7×C4⋊Q8—23no—
82C7×C4⋊Q8C2313yes—
83C7×C4⋊Q8C2313yes—
84C7×#125—26no—
85C7×#125C2316yes—
86C7×C22⋊Q8—23no—
87C7×C22⋊Q8—23no—
88C7×C22⋊Q8—23no—
89C7×C22⋊Q8—23no—
90C7×C22⋊Q8—26no—
91C4×C7×Q8—21no—
92C4×C7×Q8—23no—
93C4×C7×Q8—23no—
94C4×C7×Q8—23no—
95C4×C7×Q8D411yescharacteristic
96C4×C7×Q8D413yes—
97C4×C7×Q8C2313yes—
98C4×C7×Q8C2313yes—
99C4×C7×Q8C2311yescharacteristic
100C4×C7×Q8C2313yes—
101C4×C7×D4—23no—
102C4×C7×D4—23no—
103C4×C7×D4C2313yes—
104C4×C7×D4C2313yes—
105C7×#117—23no—
106C7×#117C2313yes—
107C2×C7×C4⋊C4—23no—
108C2×C7×C4⋊C4D413yes—
109C2×C42×C7—23no—
110C2×C42×C7C2313yes—
111#6513Q818yesdirect product factor
112#6509—23no—
113Q8×Dic7—21no—
114#6507—23no—
115C2×#1144C2311yescharacteristic
116C2×#1143C2311yescharacteristic
117C2×#1142C2311yescharacteristic
118C2×#1141C2311yescharacteristic
119C4×C22⋊D7—23no—
120C4×C22⋊D7—23no—
121#6489C2313yes—
122#6487—23no—
123#6487—23no—
124C22×C7⋊C8C2313yes—
125C22×C7⋊C8C2311yescharacteristic
126#6460—23no—
127#6459—26no—
128D7×C4⋊C4—23no—
129#6452—26no—
130#6451—23no—
131#6450—23no—
132C4×D28C2313yes—
133C4×D28C2313yes—
134#6435—23no—
135C42×D7—23no—
136#6433C2316yes—
137#6432C2313yes—
138#6432C2313yes—
139C4×Dic14—23no—
140C4×Dic14—23no—
141C4×Dic14C2313yes—
142C4×Dic14C2313yes—
143#6409C2313yes—
144#6409C2313yes—
145#6406C2313yes—
146#6406C2313yes—
147#6384C2313yes—
148#6384C2313yes—
149#6383C2313yes—
150#6383C2313yes—
151#6382C2316yes—
152#6382C2316yes—
153#6382C2313yes—
154#6382C2313yes—
155#6381C2316yes—
156#6381C2316yes—
157#6381C2313yes—
158#6381C2313yes—
159C4⋊(C7⋊C8)C2313yes—
160C4⋊(C7⋊C8)C2313yes—
161C4⋊(C7⋊C8)C2316yes—
162C4×C7⋊C8C2313yes—
163C4×C7⋊C8C2313yes—
164C4×C7⋊C8C2313yes—
165C4×C7⋊C8C2313yes—
166Q8×#41—71no—
167Q8×#41D711yescharacteristic, complemented
168#3383—73no—
169Q8×Q16—71no—
170#3381—73no—
171#3381—73no—
172#3380—73no—
173Q8×SD16—71no—
174Q8×SD16—71no—
175Q8×D8—71no—
176#3160—73no—
177#3076—73no—
178#3076—73no—
179C2×C8×Q8—71no—
180C4×#135—73no—
181C7×#41—22no—
182C7×#41C2×Q814yescomplemented
183C2×C7×Q8—24no—
184C2×C7×Q8—23no—
185C2×C7×Q8—23no—
186C2×C7×Q8—44no—
187C2×C7×Q8C2×D413yes—
188C2×C7×Q8C2×D411yescharacteristic
189C2×C7×Q8C2411yescharacteristic
190C2×C7×D4—21no—
191C2×C7×D4C2411yescharacteristic
192C22×C4×C7—21no—
193C22×C4×C7—23no—
194C22×C4×C7—23no—
195C22×C4×C7C2×D413yes—
196C22×C4×C7C2×D413yes—
197C22×C4×C7C2411yescharacteristic
198C2×C22⋊D7—21no—
199Q8×D7—44no—
200#1157—22no—
201#1157C2×Q814yescomplemented
202C2×D28C2411yescharacteristic
203C2×C4×D7—21no—
204C2×C4×D7—23no—
205C2×C4×D7—23no—
206C2×Dic14—23no—
207C2×Dic14—23no—
208C2×Dic14C2411yescharacteristic
209C7×C4⋊C4—26no—
210C7×C4⋊C4—23no—
211C7×C4⋊C4—23no—
212C7×C4⋊C4—23no—
213C7×C4⋊C4—23no—
214C7×C4⋊C4—26no—
215C7×C4⋊C4C2×D413yes—
216C7×C4⋊C4C2×D413yes—
217C7×C4⋊C4C2×D416yes—
218C7×C4⋊C4C2413yes—
219C7×C4⋊C4C2413yes—
220C7×C22⋊C4—23no—
221C7×C22⋊C4—23no—
222C7×C22⋊C4—23no—
223C42×C7—23no—
224C42×C7—23no—
225C42×C7—23no—
226C42×C7C2×D413yes—
227C42×C7C2×D413yes—
228C42×C7C2413yes—
229C42×C7C2413yes—
230C22⋊Dic7—23no—
231#1144C2×Q814yescomplemented
232#1143C2×Q814yescomplemented
233#1142C2×Q814yescomplemented
234#1141C2×Q814yescomplemented
235(C2×D7)⋊C4—23no—
236(C2×D7)⋊C4—23no—
237C4⋊Dic7—23no—
238C4⋊Dic7—23no—
239C4⋊Dic7C2413yes—
240C4⋊Dic7C2413yes—
241Dic7⋊C4—26no—
242Dic7⋊C4—26no—
243Dic7⋊C4—23no—
244Dic7⋊C4—23no—
245C4×Dic7—23no—
246C4×Dic7—23no—
247C4×Dic7—23no—
248C2×C7⋊C8—26no—
249C2×C7⋊C8C2×Q814yescomplemented
250C2×C7⋊C8C2413yes—
251C2×C7⋊C8C2413yes—
252C2×C7⋊C8C2411yescharacteristic
253C2×C7⋊C8C2411yescharacteristic
254C2×#135—71no—
255Q82—71no—
256Q82C2×D711yescharacteristic
257#557—73no—
258#557C2×D713yes—
259#554—23no—
260#554—73no—
261#554—143no—
262#554C2×D713yes—
263D4×Q8—21no—
264D4×Q8—71no—
265D4×Q8—141no—
266D4×Q8C2×D711yescharacteristic
267#533—23no—
268#533—73no—
269#533—143no—
270#533C2×D713yes—
271C4×#41—73no—
272C4×#41C2×D713yes—
273C2×C4×Q8—21no—
274C2×C4×Q8—141no—
275C2×C4×Q8C2×D711yescharacteristic
276#500—73no—
277#500—73no—
278#499—76no—
279#499—76no—
280#498—73no—
281#498—73no—
282#479—76no—
283#479—76no—
284#478—76no—
285#478—76no—
286#477—73no—
287#477—73no—
288#476—73no—
289#476—73no—
290#475—73no—
291#475—73no—
292#474—73no—
293#474—73no—
294C8×Q8—71no—
295C8×Q8—71no—
296C8×Q8—73no—
297C8×Q8—73no—
298C4×Q16—73no—
299C4×Q16—73no—
300C4×SD16—73no—
301C4×SD16—73no—
302C4×SD16—73no—
303C4×SD16—73no—
304C4×D8—73no—
305C4×D8—73no—
306#427—76no—
307#427—73no—
308C2×C4⋊C8—73no—
309#416—73no—
310#416—73no—
311C2×C4×C8—73no—
312C23×C7—21no—
313C23×C7C22×D411yescharacteristic
314C22×D7—21no—
315C7×Q8—26no—
316C7×Q8—21no—
317C7×Q8—46no—
318C7×Q8—83no—
319C7×Q8#13514yescomplemented
320C7×Q8C22×Q812yes—
321C7×D4—24no—
322C7×D4—21no—
323C7×D4C22×Q812yes—
324C2×C4×C7—21no—
325C2×C4×C7—24no—
326C2×C4×C7—26no—
327C2×C4×C7—23no—
328C2×C4×C7—23no—
329C2×C4×C7—43no—
330C2×C4×C7—46no—
331C2×C4×C7—83no—
332C2×C4×C7C22×D413yes—
333C2×C4×C7C22×D413yes—
334C2×C4×C7C22×D413yes—
335C2×C4×C7C22×D413yes—
336C2×C4×C7C22×D411yescharacteristic
337C2×C4×C7C22×Q812yes—
338C2×C4×C72−1+412yes—
339C2×C4×C7C2511yescharacteristic
340C22⋊D7—24no—
341C2×Dic7—21no—
342C2×Dic7—23no—
343C2×Dic7—23no—
344D28—21no—
345D28C22×Q812yes—
346C4×D7—24no—
347C4×D7—46no—
348C4×D7—83no—
349Dic14—21no—
350Dic14—46no—
351Dic14—83no—
352Dic14C22×Q812yes—
353C7⋊C8—23no—
354C7⋊C8—23no—
355C7⋊C8C22×Q812yes—
356C7⋊C8C22×Q812yes—
357C2×#41—71no—
358C2×#41C22×D711yescharacteristic
359C22×Q8—21no—
360C22×Q8—141no—
361C22×Q8C22⋊D711yescharacteristic
362#135—78no—
363C2×Q16—71no—
364C2×SD16—71no—
365C2×SD16—71no—
366C2×D8—71no—
367C22×C8—71no—
368C22×C8—73no—
369C4⋊Q8—23no—
370C4⋊Q8—23no—
371C4⋊Q8—73no—
372C4⋊Q8—73no—
373C4⋊Q8—143no—
374C4⋊Q8—143no—
375C4⋊Q8C22×D713yes—
376C4⋊Q8C22×D713yes—
377#125—26no—
378#125—76no—
379#125—146no—
380#125C22×D716yes—
381C22⋊Q8—23no—
382C22⋊Q8—23no—
383C22⋊Q8—23no—
384C22⋊Q8—23no—
385C22⋊Q8—26no—
386C22⋊Q8—143no—
387C22⋊Q8—143no—
388C22⋊Q8—143no—
389C22⋊Q8—143no—
390C22⋊Q8—146no—
391C4×Q8—21no—
392C4×Q8—23no—
393C4×Q8—23no—
394C4×Q8—23no—
395C4×Q8—73no—
396C4×Q8—73no—
397C4×Q8—141no—
398C4×Q8—143no—
399C4×Q8—143no—
400C4×Q8—143no—
401C4×Q8C22⋊D711yescharacteristic
402C4×Q8C22⋊D713yes—
403C4×Q8C22×D713yes—
404C4×Q8C22×D713yes—
405C4×Q8C22×D711yescharacteristic
406C4×Q8C22×D713yes—
407C4×D4—23no—
408C4×D4—23no—
409C4×D4—73no—
410C4×D4—73no—
411C4×D4—143no—
412C4×D4—143no—
413C4×D4C22×D713yes—
414C4×D4C22×D713yes—
415#117—23no—
416#117—73no—
417#117—143no—
418#117C22×D713yes—
419C2×C4⋊C4—23no—
420C2×C4⋊C4—143no—
421C2×C4⋊C4C22⋊D713yes—
422C2×C42—23no—
423C2×C42—143no—
424C2×C42C22×D713yes—
425#107—73no—
426#107—73no—
427#107—76no—
428#107—76no—
429#106—73no—
430#106—73no—
431#106—76no—
432#106—76no—
433C4⋊C8—73no—
434C4⋊C8—73no—
435C4⋊C8—76no—
436#103—73no—
437#103—73no—
438#103—73no—
439#103—73no—
440#102—73no—
441#102—73no—
442#102—73no—
443#102—73no—
444C4×C8—73no—
445C4×C8—73no—
446C4×C8—73no—
447C4×C8—73no—
448C22×C7—21no—
449C22×C7—21no—
450C22×C7—22no—
451C22×C7—41no—
452C22×C7—41no—
453C22×C7D4×Q812yes—
454C22×C7C23×D411yescharacteristic
455C2×D7—22no—
456C2×D7—41no—
457C2×D7—41no—
458C4×C7—23no—
459C4×C7—23no—
460C4×C7—22no—
461C4×C7—23no—
462C4×C7—43no—
463C4×C7—43no—
464C4×C7D4×Q812yes—
465C4×C7C2×#13516yes—
466C4×C7C23×Q811yescharacteristic
467C4×C7C2×2−1+411yescharacteristic
468Dic7—22no—
469Dic7—43no—
470Dic7—43no—
471#41—22no—
472#41—74no—
473#41—142no—
474#41Q8×D714yescomplemented
475C2×Q8—24no—
476C2×Q8—23no—
477C2×Q8—23no—
478C2×Q8—44no—
479C2×Q8—71no—
480C2×Q8—143no—
481C2×Q8—143no—
482C2×Q8—284no—
483C2×Q8C2×C22⋊D713yes—
484C2×Q8C2×C22⋊D711yescharacteristic
485C2×Q8C23×D711yescharacteristic
486C2×D4—21no—
487C2×D4—71no—
488C2×D4—141no—
489C2×D4C23×D711yescharacteristic
490C22×C4—21no—
491C22×C4—23no—
492C22×C4—23no—
493C22×C4—141no—
494C22×C4—143no—
495C22×C4—143no—
496C22×C4C2×C22⋊D713yes—
497C22×C4C2×C22⋊D713yes—
498C22×C4C23×D711yescharacteristic
499Q16—74no—
500SD16—74no—
501SD16—74no—
502D8—74no—
503C2×C8—74no—
504C2×C8—71no—
505C2×C8—71no—
506C2×C8—73no—
507C2×C8—73no—
508C2×C8—146no—
509C4⋊C4—26no—
510C4⋊C4—23no—
511C4⋊C4—23no—
512C4⋊C4—23no—
513C4⋊C4—23no—
514C4⋊C4—26no—
515C4⋊C4—73no—
516C4⋊C4—73no—
517C4⋊C4—146no—
518C4⋊C4—143no—
519C4⋊C4—143no—
520C4⋊C4—146no—
521C4⋊C4—143no—
522C4⋊C4—143no—
523C4⋊C4C2×C22⋊D713yes—
524C4⋊C4C2×C22⋊D716yes—
525C4⋊C4C2×C22⋊D713yes—
526C4⋊C4C23×D713yes—
527C4⋊C4C23×D713yes—
528C22⋊C4—23no—
529C22⋊C4—23no—
530C22⋊C4—23no—
531C22⋊C4—143no—
532C22⋊C4—143no—
533C22⋊C4—143no—
534C42—23no—
535C42—23no—
536C42—23no—
537C42—143no—
538C42—143no—
539C42—143no—
540C42C2×C22⋊D713yes—
541C42C2×C22⋊D713yes—
542C42C23×D713yes—
543C42C23×D713yes—
544C2×C7—22no—
545C2×C7—42no—
546C2×C7C2×D4×Q811yescharacteristic
547C2×C7C22×#13511yescharacteristic
548C2×C7#358411yescharacteristic
549D7—42no—
550C23—21no—
551C23—141no—
552C23C22×C22⋊D711yescharacteristic
553Q8—26no—
554Q8—21no—
555Q8—46no—
556Q8—72no—
557Q8—83no—
558Q8—141no—
559Q8—286no—
560Q8—563no—
561Q8#651314yesdirect product factor
562Q8C2×Q8×D712yes—
563D4—24no—
564D4—21no—
565D4—72no—
566D4—144no—
567D4—141no—
568D4C2×Q8×D712yes—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—26no—
572C2×C4—23no—
573C2×C4—23no—
574C2×C4—43no—
575C2×C4—46no—
576C2×C4—83no—
577C2×C4—141no—
578C2×C4—144no—
579C2×C4—143no—
580C2×C4—143no—
581C2×C4—286no—
582C2×C4—563no—
583C2×C4C2×Q8×D712yes—
584C2×C4#655112yes—
585C2×C4C22×C22⋊D713yes—
586C2×C4C22×C22⋊D713yes—
587C2×C4C22×C22⋊D713yes—
588C2×C4C22×C22⋊D713yes—
589C2×C4C22×C22⋊D711yescharacteristic
590C2×C4C24×D711yescharacteristic
591C8—72no—
592C8—72no—
593C8—143no—
594C8—143no—
595C7Q8×#13511yescharacteristic, complemented, Sylow
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—142no—
602C22—281no—
603C22—281no—
604C22Q8×C22⋊D712yes—
605C22C23×C22⋊D711yescharacteristic
606C4—22no—
607C4—23no—
608C4—23no—
609C4—23no—
610C4—43no—
611C4—43no—
612C4—142no—
613C4—283no—
614C4—283no—
615C4Q8×C22⋊D712yes—
616C4C2×#651316yes—
617C4C22×Q8×D711yescharacteristic
618C4C2×#655111yescharacteristic
619C2—22no—
620C2—42no—
621C2—282no—
622C2C2×Q8×C22⋊D711yescharacteristic
623C2C22×#651311yescharacteristic
624C2#1174436511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC7×Q8×#41C2
socleC22×C7C23×D4
prime radicalC2×#6513C22
hypercenterQ8×#41D7
hypocenterC7Q8×#135
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. C7×Q8×#41C2semidirect
2. Q8×#1157C2semidirect
3. Q8×#1143C2semidirect
4. Q8×#1144C2semidirect
5. #11741138C2semidirect
6. #11741127C2semidirect
7. #11741074C2semidirect
8. #11741073C2semidirect
9. Q8×#1142C2semidirect
10. Q8×#1141C2semidirect
11. #11740794C2semidirect
12. #11740784C2semidirect
13. #11739952C2semidirect
15. C2×Q8×C7⋊C8C2semidirect
111. #6513Q8direct (N × H)
167. Q8×#41D7semidirect
182. C7×#41C2×Q8semidirect
201. #1157C2×Q8semidirect
231. #1144C2×Q8semidirect
232. #1143C2×Q8semidirect
233. #1142C2×Q8semidirect
234. #1141C2×Q8semidirect
249. C2×C7⋊C8C2×Q8semidirect
319. C7×Q8#135semidirect
474. #41Q8×D7semidirect
561. Q8#6513direct (N × H)
595. C7Q8×#135semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×#412
Q8×#11572
Q8×#11432
Q8×#11442
#117411382
#117411272
#117410742
#117410732
Q8×#11422
Q8×#11412
#117407942
#117407842
#117399522
C4×#65132
C2×Q8×C7⋊C82
Q8×#1357