Finite Group Databasebeta

← #13416822

Subgroups — #13416822

Order 1792.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#89792C211yesmaximal, characteristic, complemented
3C2×C4×#1143C211yesmaximal, characteristic
4#11731120C211yesmaximal, characteristic
5#11729387C211yesmaximal, characteristic
6#11729387C211yesmaximal, characteristic
7#11729383C211yesmaximal, characteristic, complemented
8#11729383C211yesmaximal, characteristic, complemented
9#11727383C211yesmaximal, characteristic
10#11727380C211yesmaximal, characteristic
11#11727372C211yesmaximal, characteristic, complemented
12#11727359C211yesmaximal, characteristic, complemented
13#11727341C211yesmaximal, characteristic
14#11727324C211yesmaximal, characteristic, complemented
15#11727236C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C22×#1143C2211yescharacteristic
20C2×#6459—21no—
21C2×C4×D28C2211yescharacteristic
22C2×#6432C2211yescharacteristic
23C7×#389—21no—
24C7×#389—21no—
25C7×#389C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#384—21no—
28C7×#384C2211yescharacteristic
29C4×C7×C4⋊C4—21no—
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C4×#1143—24no—
35#89792—24no—
36#89766—21no—
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45#89711C2211yescharacteristic
46#89711C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89445—21no—
52#89440—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26373—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×D28C2311yescharacteristic
64C22×C4×D7—21no—
65C22×Dic14—21no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C4×C7×Q8—41no—
71C4×C7×Q8—41no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4D411yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C42×C7—21no—
85C2×C42×C7—21no—
86C2×C42×C7—21no—
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2311yescharacteristic
90C2×#1143—22no—
91C2×#1143—22no—
92C2×#1143Q814yescomplemented
93C2×(C2×D7)⋊C4—21no—
94C2×(C2×D7)⋊C4—21no—
95C2×(C2×D7)⋊C4—21no—
96C2×(C2×D7)⋊C4—21no—
97C2×(C2×D7)⋊C4—21no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4⋊Dic7C2311yescharacteristic
102C2×Dic7⋊C4—21no—
103C2×Dic7⋊C4—21no—
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108#6459—44no—
109C4×D28—22no—
110C4×D28—41no—
111#6432—22no—
112#6432—41no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95D411yescharacteristic
116C7×#95D411yescharacteristic
117#6409—22no—
118#6409—22no—
119#6409D414yes—
120C7⋊#95—21no—
121C7⋊#95—21no—
122#6383—22no—
123#6383—22no—
124#6383—41no—
125#6382—22no—
126#6382—22no—
127#6382—22no—
128#6382—22no—
129#6382—41no—
130C4⋊(C7⋊C8)—22no—
131C4×C7⋊C8—22no—
132C2×#475—71no—
133C2×C4×SD16—71no—
134#2452—71no—
135Q8×C4⋊C4D711yescharacteristic, complemented
136#2094—71no—
137#2084—71no—
138#2065—71no—
139#2058—71no—
140#1941—71no—
141#1916—71no—
142#1880—71no—
143#1880—71no—
144#1865—71no—
145#1865—71no—
146#1864—71no—
147C23×D7—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8C2×D412yes—
152C2×C7×Q8C2×Q812yes—
153C2×C7×Q8#4112yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C2×D28—21no—
169C2×D28—21no—
170C2×D28—21no—
171C2×D28—21no—
172C2×D28—21no—
173C2×D28C2×Q812yes—
174C2×C4×D7—42no—
175C2×C4×D7—42no—
176C2×C4×D7—81no—
177C2×Dic14—42no—
178C2×Dic14—42no—
179C2×Dic14—81no—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—41no—
187C7×C4⋊C4—42no—
188C7×C4⋊C4—42no—
189C7×C4⋊C4—42no—
190C7×C4⋊C4—81no—
191C7×C4⋊C4—81no—
192C7×C4⋊C4—81no—
193C7×C4⋊C4SD1614yescomplemented
194C7×C4⋊C4#4112yes—
195C42×C7—24no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—42no—
203C42×C7—42no—
204#1143—28no—
205(C2×D7)⋊C4—42no—
206(C2×D7)⋊C4—42no—
207(C2×D7)⋊C4—42no—
208(C2×D7)⋊C4—42no—
209(C2×D7)⋊C4—81no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—42no—
216C4⋊Dic7C2×D412yes—
217Dic7⋊C4—42no—
218Dic7⋊C4—42no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8C2×D412yes—
230C2×C7⋊C8C2×Q812yes—
231C22×SD16—71no—
232C2×C4⋊Q8—71no—
233C2×C22⋊Q8—141no—
234C2×C4×Q8C2×D711yescharacteristic
235C2×C4×Q8C2×D711yescharacteristic
236C2×C4×Q8C2×D711yescharacteristic
237C2×C4×D4—71no—
238#475—144no—
239C4×SD16—144no—
240C2×#106—71no—
241C2×#106—71no—
242C2×#106—71no—
243C2×#106—71no—
244C2×#106—71no—
245C2×C4⋊C8—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#102—71no—
250C2×#102—71no—
251C2×#102—71no—
252C2×C4×C8—71no—
253#396—141no—
254#393—141no—
255#390—71no—
256#390—71no—
257#389—21no—
258#389—21no—
259#389—71no—
260#389—71no—
261#389C2×D711yescharacteristic
262#389C2×D711yescharacteristic
263#385—141no—
264#384—21no—
265#384C2×D711yescharacteristic
266C4×C4⋊C4—21no—
267C4×C4⋊C4C2×D711yescharacteristic
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#336—71no—
273#336—71no—
274#336—71no—
275#336—71no—
276C23×C7C22×D411yescharacteristic
277C22×D7—22no—
278C22×D7—41no—
279C22×D7—41no—
280C22×D7—41no—
281C22×D7—41no—
282C22×D7—41no—
283C22×D7—41no—
284C7×Q8—22no—
285C7×Q8—22no—
286C7×Q8—22no—
287C7×Q8C22⋊Q814yescomplemented
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7C2×SD1612yes—
316C2×C4×C7C2×SD1612yes—
317C2×C4×C7C2×SD1612yes—
318C2×C4×C7C22×D411yescharacteristic
319C2×C4×C7C22×Q811yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7H(5,𝔽2)11yescharacteristic
323C2×C4×C72−1+411yescharacteristic
324C2×Dic7—22no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333D28—22no—
334D28—22no—
335D28—41no—
336D28—41no—
337C4×D7—84no—
338Dic14—84no—
339C7⋊C8—22no—
340C7⋊C8—22no—
341C7⋊C8—41no—
342C7⋊C8—41no—
343C22×Q8—141no—
344C22×Q8C22×D711yescharacteristic
345C22×D4—71no—
346C23×C4—141no—
347C2×SD16—74no—
348C2×SD16—142no—
349C2×SD16—142no—
350C22×C8—71no—
351C22×C8—71no—
352C22×C8—71no—
353C22×C8—71no—
354C4⋊Q8—142no—
355C4⋊Q8—281no—
356C22⋊Q8—284no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—41no—
362C4×Q8—41no—
363C4×D4—142no—
364C4×D4—281no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4C22⋊D711yescharacteristic
379C2×C4⋊C4C22⋊D711yescharacteristic
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C4⋊C4C22×D711yescharacteristic
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C22⋊C4—141no—
384C2×C22⋊C4—141no—
385C2×C22⋊C4—141no—
386C2×C22⋊C4—141no—
387C2×C22⋊C4—141no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42C22×D711yescharacteristic
392C2×C42C22×D711yescharacteristic
393C2×C42C22×D711yescharacteristic
394#106—142no—
395#106—142no—
396#106—142no—
397#106—142no—
398#106—281no—
399C4⋊C8—142no—
400#103—74no—
401#103—142no—
402#103—142no—
403#102—142no—
404#102—142no—
405#102—281no—
406C4×C8—142no—
407#95—21no—
408#95—21no—
409#95—141no—
410#95—141no—
411#95C22⋊D711yescharacteristic
412#95C22⋊D711yescharacteristic
413C22×C7#52511yescharacteristic
414C22×C7D4211yescharacteristic
415C22×C7D4×Q811yescharacteristic
416C22×C7C22×SD1611yescharacteristic
417C22×C7#57511yescharacteristic
418C22×C7#57711yescharacteristic
419C22×C7#57711yescharacteristic
420C2×D7—42no—
421C2×D7—42no—
422C2×D7—42no—
423C2×D7—81no—
424C2×D7—81no—
425C2×D7—81no—
426C2×D7—81no—
427C4×C7—22no—
428C4×C7—24no—
429C4×C7—22no—
430C4×C7—22no—
431C4×C7—41no—
432C4×C7—42no—
433C4×C7—41no—
434C4×C7—41no—
435C4×C7—81no—
436C4×C7—81no—
437C4×C7C2×C22⋊Q811yescharacteristic
438C4×C7#53911yescharacteristic
439C4×C7#54711yescharacteristic
440C4×C7#55411yescharacteristic
441Dic7—42no—
442Dic7—81no—
443Dic7—81no—
444C24—141no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—282no—
449C2×Q8—282no—
450C2×Q8—561no—
451C2×Q8#115712yes—
452C2×Q8D4×D712yes—
453C2×Q8Q8×D712yes—
454C2×D4—72no—
455C2×D4—141no—
456C2×D4—141no—
457C2×D4—141no—
458C2×D4—141no—
459C2×D4—141no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—141no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—282no—
468C22×C4—282no—
469C22×C4—561no—
470C22×C4C2×C22⋊D711yescharacteristic
471C22×C4C2×C22⋊D711yescharacteristic
472C22×C4C2×C22⋊D711yescharacteristic
473C22×C4C2×C22⋊D711yescharacteristic
474C22×C4C2×C22⋊D711yescharacteristic
475C22×C4C2×C22⋊D711yescharacteristic
476C22×C4C23×D711yescharacteristic
477SD16—148no—
478C2×C8—72no—
479C2×C8—72no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—141no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—72no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—141no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—282no—
510C4⋊C4—282no—
511C4⋊C4—282no—
512C4⋊C4#114314yescomplemented
513C4⋊C4#115712yes—
514C22⋊C4—282no—
515C22⋊C4—282no—
516C22⋊C4—282no—
517C22⋊C4—282no—
518C22⋊C4—561no—
519C42—24no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—42no—
527C42—42no—
528C2×C7#244911yescharacteristic
529C2×C7#310311yescharacteristic
530C2×C7#312511yescharacteristic
531C2×C7#327111yescharacteristic
532C2×C7D4×SD1611yescharacteristic
533C2×C7Q8×SD1611yescharacteristic
534C2×C7#338811yescharacteristic
535D7—82no—
536C23—142no—
537C23—281no—
538C23—281no—
539C23—281no—
540C23—281no—
541C23—281no—
542C23—281no—
543C23C22×C22⋊D711yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—22no—
547Q8—564no—
548Q8#645914yescomplemented
549D4—142no—
550D4—142no—
551D4—281no—
552D4—281no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—142no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4—564no—
590C2×C4C2×#114312yes—
591C2×C4C2×#114312yes—
592C2×C4C2×#114312yes—
593C2×C4C2×#115711yescharacteristic
594C2×C4C2×#115711yescharacteristic
595C2×C4C2×D4×D711yescharacteristic
596C2×C4#654811yescharacteristic
597C2×C4C2×Q8×D711yescharacteristic
598C2×C4#655111yescharacteristic
599C8—142no—
600C8—142no—
601C8—281no—
602C8—281no—
603C7#2637311yescharacteristic, complemented, Sylow
604C22—282no—
605C22—282no—
606C22—282no—
607C22—561no—
608C22—561no—
609C22—561no—
610C22—561no—
611C22#9048211yescharacteristic
612C22D4×C22⋊D711yescharacteristic
613C22C22×#114311yescharacteristic
614C22Q8×C22⋊D711yescharacteristic
615C22#9051311yescharacteristic
616C22#9052711yescharacteristic
617C22#9052811yescharacteristic
618C4—22no—
619C4—22no—
620C4—24no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—282no—
629C4—561no—
630C4—561no—
631C4#9019811yescharacteristic
632C4C2×#645911yescharacteristic
633C4#9020211yescharacteristic
634C4#9020611yescharacteristic
635C2—562no—
636C2#1173961811yescharacteristic
637C2#1173988711yescharacteristic
638C2#1174002111yescharacteristic
639C2D4×#114311yescharacteristic
640C2#1174102811yescharacteristic
641C2#1174114011yescharacteristic
642C2Q8×#114311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×Q8×C4⋊C4C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenterQ8×C4⋊C4D7
hypocenterC7#26373
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C7×Q8×C4⋊C4C2semidirect
2. C2×#89792C2semidirect
7. #11729383C2semidirect
8. #11729383C2semidirect
11. #11727372C2semidirect
12. #11727359C2semidirect
14. #11727324C2semidirect
15. #11727236C2semidirect
92. C2×#1143Q8semidirect
135. Q8×C4⋊C4D7semidirect
193. C7×C4⋊C4SD16semidirect
287. C7×Q8C22⋊Q8semidirect
512. C4⋊C4#1143semidirect
548. Q8#6459semidirect
603. C7#26373semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×C4⋊C42
C2×#897922
C2×C4×#11432
#117311202
#117293872
#117293832
#117273832
#117273802
#117273722
#117273592
#117273412
#117273242
#117272362
#263737