Finite Group Databasebeta

← D4×#6409 #13369228

Subgroups — D4×#6409 #13369228

Order 1792.

1530 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic
2C22×#6409C212yesmaximal, complemented
3D4×C4⋊Dic7C211yesmaximal, characteristic
4C2×D4×C7⋊C8C211yesmaximal, characteristic
5#11729886C212yesmaximal, complemented
6#11729539C211yesmaximal, characteristic, complemented
7#11729532C211yesmaximal, characteristic, complemented
8#11729450C212yesmaximal, complemented
9#11729446C212yesmaximal, complemented
10C4×#6409C211yesmaximal, characteristic, complemented
11#11729380C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2212yes—
13C7×D4×Q8—24no—
14C7×D4×Q8C418yescomplemented
15C2×C7×C4⋊Q8—21no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8—22no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×D4×Dic7—21no—
23C22×C4⋊Dic7C2212yes—
24C23×C7⋊C8C2212yes—
25C2×#6409—28no—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6409C2211yescharacteristic
30C2×#6409C2212yes—
31C2×#6409C2212yes—
32C2×#6409C2211yescharacteristic
33#89984C2212yes—
34C2×C22⋊(C7⋊C8)C2212yes—
35D4×C7⋊C8—24no—
36#89757—21no—
37#89722—22no—
38#89707C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2211yescharacteristic
43#89324C2212yes—
44#89324C2211yescharacteristic
45D4×#103—71nomaximal, Sylow
46C22×C7×Q8—22no—
47C22×C7×Q8—24no—
48C22×C7×Q8—24no—
49C22×C7×Q8—24no—
50C22×C7×Q8—42no—
51C22×C7×Q8—42no—
52C22×C7×Q8C2×C418yescomplemented
53C22×C7×Q8C2312yes—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4D411yescharacteristic
56C23×C4×C7—22no—
57C23×C4×C7C2312yes—
58C23×Dic7—22no—
59C7×C4⋊Q8—28no—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8C2×C414yescomplemented
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—216no—
64C7×C22⋊Q8C2×C418yescomplemented
65C4×C7×Q8—22no—
66C4×C7×Q8C2×C414yescomplemented
67C4×C7×D4—28no—
68C4×C7×D4C2×C414yes—
69C4×C7×D4D414yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4D412yes—
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—22no—
81D4×Dic7—44no—
82C2×C4⋊Dic7—24no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—42no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2312yes—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4×Dic7—21no—
91C22×C7⋊C8—24no—
92C22×C7⋊C8—24no—
93C22×C7⋊C8—42no—
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2312yes—
96C22×C7⋊C8C2311yescharacteristic
97#6409—24no—
98#6409—22no—
99#6409—24no—
100#6409—24no—
101#6409—42no—
102#6409—42no—
103#6409D414yesdirect product factor
104C7⋊#95—21no—
105C7⋊#95—22no—
106C22⋊(C7⋊C8)—24no—
107C4⋊(C7⋊C8)—22no—
108C4×C7⋊C8—22no—
109C2×D4×Q8D711yescharacteristic
110C2×C8×D4—71no—
111C22×#103—72no—
112D4×C4⋊C4—71no—
113#1981—71no—
114#1956—71no—
115#1892—72no—
116#1872—72no—
117#1803—71no—
118#1787—72no—
119C4×#103—71no—
120C24×C7C2×D412yes—
121C2×C7×Q8—28no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—216no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8—22no—
131C2×C7×Q8—44no—
132C2×C7×Q8—44no—
133C2×C7×Q8—48no—
134C2×C7×Q8—48no—
135C2×C7×Q8—48no—
136C2×C7×Q8C22×C414yes—
137C2×C7×Q8C22×C412yes—
138C2×C7×Q8C2×D412yes—
139C2×C7×Q8#4112yes—
140C2×C7×D4C22⋊C414yes—
141C2×C7×D4SD1614yes—
142C2×C7×D4Q1614yes—
143C22×C4×C7—21no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—28no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—44no—
152C22×C4×C7—44no—
153C22×C4×C7C22×C414yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D414yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2411yescharacteristic
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—22no—
162C22×Dic7—44no—
163C22×Dic7—44no—
164C22×Dic7—82no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—28no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4C22⋊C414yes—
171C7×C4⋊C4C22×C412yes—
172C7×C22⋊C4—28no—
173C7×C22⋊C4C22⋊C418yes—
174C42×C7—24no—
175C42×C7C22×C412yes—
176C42×C7C2×D412yes—
177C22⋊Dic7—44no—
178C4⋊Dic7—21no—
179C4⋊Dic7—22no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—22no—
183C4⋊Dic7—44no—
184C4⋊Dic7—42no—
185C4⋊Dic7—81no—
186C4⋊Dic7C2×D412yes—
187C4×Dic7—42no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—21no—
190C2×C7⋊C8—21no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—21no—
194C2×C7⋊C8—21no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—48no—
197C2×C7⋊C8C2×D412yes—
198C23×Q8C2×D712yes—
199C23×C8—72no—
200D4×Q8—24no—
201D4×Q8Dic718yescomplemented
202C2×C4⋊Q8—21no—
203C2×C4⋊Q8C2×D711yescharacteristic
204C2×C22⋊Q8—22no—
205C2×C22⋊Q8C2×D712yes—
206C2×C4×Q8C2×D711yescharacteristic
207C2×C4×D4—21no—
208C2×C4×D4—141no—
209C2×C4×D4C2×D711yescharacteristic
210C22×C4⋊C4—72no—
211C8×D4—144no—
212C2×C4⋊C8—71no—
213C2×#103—71no—
214C2×#103—72no—
215C2×#103—72no—
216C2×#103—72no—
217C2×#103—71no—
218C2×#103—72no—
219C2×#103—72no—
220C2×#103—148no—
221C2×C22⋊C8—72no—
222C2×C4×C8—71no—
223#389—141no—
224#385—142no—
225#384—71no—
226#380—72no—
227C4×C4⋊C4—71no—
228#340—72no—
229#340—71no—
230#340—71no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7C2×C22⋊C414yes—
236C23×C7C2×SD1614yes—
237C23×C7C2×Q1614yes—
238C23×C7C22×D411yescharacteristic
239C7×Q8—22no—
240C7×Q8—22no—
241C7×Q8—22no—
242C7×Q8—24no—
243C7×Q8—24no—
244C7×Q8—416no—
245C7×Q8—42no—
246C7×Q8—42no—
247C7×Q8C4×D414yescomplemented
248C7×D4#103116yescomplemented
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—416no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7C2×C22⋊C412yes—
268C2×C4×C7C2×C22⋊C414yes—
269C2×C4×C7C2×C22⋊C412yes—
270C2×C4×C7C2×SD1612yes—
271C2×C4×C7C2×Q1612yes—
272C2×C4×C7C23×C411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7H(5,𝔽2)11yescharacteristic
277C2×C4×C7H(5,𝔽2)11yescharacteristic
278C2×Dic7—22no—
279C2×Dic7—41no—
280C2×Dic7—41no—
281C2×Dic7—42no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—41no—
287C2×Dic7—88no—
288C7⋊C8—22no—
289C7⋊C8—42no—
290C7⋊C8—41no—
291C22×Q8—22no—
292C22×Q8—24no—
293C22×Q8—24no—
294C22×Q8—24no—
295C22×Q8—42no—
296C22×Q8—42no—
297C22×Q8C2×Dic718yescomplemented
298C22×Q8C22×D712yes—
299C22×Q8C22×D711yescharacteristic
300C22×D4C22⋊D711yescharacteristic
301C23×C4—22no—
302C23×C4—142no—
303C23×C4C22×D712yes—
304C22×C8—71no—
305C22×C8—72no—
306C22×C8—71no—
307C22×C8—144no—
308C22×C8—144no—
309C22×C8—282no—
310C4⋊Q8—28no—
311C4⋊Q8—22no—
312C4⋊Q8C2×Dic714yescomplemented
313C22⋊Q8—24no—
314C22⋊Q8—216no—
315C22⋊Q8C2×Dic718yescomplemented
316C4×Q8—22no—
317C4×Q8C2×Dic714yescomplemented
318C4×D4—28no—
319C4×D4—284no—
320C4×D4C2×Dic714yes—
321C4×D4C22⋊D714yes—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—72no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—144no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—282no—
334C2×C4⋊C4C22⋊D711yescharacteristic
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—142no—
338C2×C22⋊C4C22⋊D712yes—
339C2×C42—21no—
340C2×C42—141no—
341C2×C42C22×D711yescharacteristic
342C4⋊C8—142no—
343#103—74no—
344#103—144no—
345#103—144no—
346#103—144no—
347#103—142no—
348#103—282no—
349#103—282no—
350C22⋊C8—144no—
351C4×C8—142no—
352#95—141no—
353#95—142no—
354C22×C7—24no—
355C22×C7—24no—
356C22×C7—24no—
357C22×C7—24no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7C2×#10318yescomplemented
361C22×C7C22×C22⋊C411yescharacteristic
362C22×C7D4211yescharacteristic
363C22×C7#54611yescharacteristic
364C22×C7C22×SD1611yescharacteristic
365C22×C7C22×Q1611yescharacteristic
366C22×C7#57711yescharacteristic
367C22×C7#57811yescharacteristic
368C4×C7—24no—
369C4×C7—24no—
370C4×C7—22no—
371C4×C7—44no—
372C4×C7—42no—
373C4×C7C2×#10314yescomplemented
374C4×C7C2×C4×D411yescharacteristic
375C4×C7#51811yescharacteristic
376C4×C7D4211yescharacteristic
377C4×C7#54611yescharacteristic
378Dic7—42no—
379Dic7—82no—
380Dic7—81no—
381C24C2×C22⋊D712yes—
382C2×Q8—28no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—216no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8D4×D712yes—
398C2×Q8#115912yes—
399C2×Q8C22×Dic714yes—
400C2×Q8C22×Dic712yes—
401C2×D4#114314yes—
402C2×D4#114414yes—
403C2×D4C22⋊Dic714yes—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—142no—
417C22×C4—284no—
418C22×C4—284no—
419C22×C4—562no—
420C22×C4C22×Dic714yes—
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C2×C22⋊D712yes—
423C22×C4C2×C22⋊D714yes—
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C23×D711yescharacteristic
426C2×C8—72no—
427C2×C8—141no—
428C2×C8—141no—
429C2×C8—142no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—142no—
433C2×C8—141no—
434C2×C8—142no—
435C2×C8—288no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—21no—
441C4⋊C4—72no—
442C4⋊C4—141no—
443C4⋊C4—142no—
444C4⋊C4—141no—
445C4⋊C4—141no—
446C4⋊C4—142no—
447C4⋊C4—284no—
448C4⋊C4—282no—
449C4⋊C4—561no—
450C4⋊C4C22⋊Dic714yes—
451C4⋊C4C22×Dic712yes—
452C22⋊C4—28no—
453C22⋊C4—284no—
454C22⋊C4C22⋊Dic718yes—
455C42—24no—
456C42—282no—
457C42C22×Dic712yes—
458C42C2×C22⋊D712yes—
459C2×C7—28no—
460C2×C7D4×C22⋊C411yescharacteristic
461C2×C7C22×#10311yescharacteristic
462C2×C7#289811yescharacteristic
463C2×C7D4×SD1611yescharacteristic
464C2×C7D4×Q1611yescharacteristic
465C2×C7#329811yescharacteristic
466C2×C7#329911yescharacteristic
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23C2×#114314yes—
472C23C2×#114414yes—
473C23C2×C22⋊Dic714yes—
474C23C22×C22⋊D711yescharacteristic
475Q8—22no—
476Q8—22no—
477Q8—22no—
478Q8—24no—
479Q8—24no—
480Q8—416no—
481Q8—42no—
482Q8—42no—
483Q8D4×Dic714yescomplemented
484D4#6409116yesdirect product factor
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—416no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—142no—
504C2×C4—281no—
505C2×C4—281no—
506C2×C4—282no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4—282no—
510C2×C4—282no—
511C2×C4—281no—
512C2×C4—568no—
513C2×C4C2×#114312yes—
514C2×C4C2×#114412yes—
515C2×C4C2×C22⋊Dic712yes—
516C2×C4C2×C22⋊Dic714yes—
517C2×C4C2×C22⋊Dic712yes—
518C2×C4C2×D4×D711yescharacteristic
519C2×C4C2×#115911yescharacteristic
520C2×C4#654811yescharacteristic
521C2×C4#654811yescharacteristic
522C2×C4C23×Dic711yescharacteristic
523C2×C4C22×C22⋊D711yescharacteristic
524C8—142no—
525C8—282no—
526C8—281no—
527C7D4×#10311yescharacteristic, complemented, Sylow
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22C2×#640918yescomplemented
535C22D4×C22⋊D711yescharacteristic
536C22#9049711yescharacteristic
537C22C22×#114311yescharacteristic
538C22C22×#114411yescharacteristic
539C22#9052811yescharacteristic
540C22#9052911yescharacteristic
541C22C22×C22⋊Dic711yescharacteristic
542C4—24no—
543C4—24no—
544C4—22no—
545C4—44no—
546C4—42no—
547C4—282no—
548C4—561no—
549C4—562no—
550C4C2×#640914yescomplemented
551C4C2×D4×Dic711yescharacteristic
552C4#9048911yescharacteristic
553C4D4×C22⋊D711yescharacteristic
554C4#9049711yescharacteristic
555C2—28no—
556C2#1174099511yescharacteristic
557C2#1174099611yescharacteristic
558C2D4×#114311yescharacteristic
559C2D4×#114411yescharacteristic
560C2D4×C22⋊Dic711yescharacteristic
561C2C22×#640911yescharacteristic
562C2#1174155611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×D4×Q8C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
cosocleC22×C4×C7C24
hypercenterC2×D4×Q8D7
hypocenterC7D4×#103
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C22×#6409C2semidirect
5. #11729886C2semidirect
6. #11729539C2semidirect
7. #11729532C2semidirect
8. #11729450C2semidirect
9. #11729446C2semidirect
10. C4×#6409C2semidirect
11. #11729380C2semidirect
14. C7×D4×Q8C4semidirect
52. C22×C7×Q8C2×C4semidirect
61. C7×C4⋊Q8C2×C4semidirect
64. C7×C22⋊Q8C2×C4semidirect
66. C4×C7×Q8C2×C4semidirect
103. #6409D4direct (N × H)
201. D4×Q8Dic7semidirect
247. C7×Q8C4×D4semidirect
248. C7×D4#103semidirect
297. C22×Q8C2×Dic7semidirect
312. C4⋊Q8C2×Dic7semidirect
315. C22⋊Q8C2×Dic7semidirect
317. C4×Q8C2×Dic7semidirect
360. C22×C7C2×#103semidirect
373. C4×C7C2×#103semidirect
483. Q8D4×Dic7semidirect
484. D4#6409direct (N × H)
527. C7D4×#103semidirect
534. C22C2×#6409semidirect
550. C4C2×#6409semidirect

Maximal subgroups

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

subgroupindex
C2×C7×D4×Q82
C22×#64092
D4×C4⋊Dic72
C2×D4×C7⋊C82
#117298862
#117295392
#117295322
#117294502
#117294462
C4×#64092
#117293802
D4×#1037