Finite Group Databasebeta

← #13285771

Subgroups — #13285771

Order 1792.

1982 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic
2C2×#90001C211yesmaximal, characteristic, complemented
3C2×#89982C211yesmaximal, characteristic
4C2×C4×C22⋊Dic7C211yesmaximal, characteristic
5C2×#89744C212yesmaximal, complemented
6#11729888C211yesmaximal, characteristic, complemented
7#11729322C211yesmaximal, characteristic, complemented
8#11729320C211yesmaximal, characteristic, complemented
9#11729318C212yesmaximal, complemented
10#11729301C212yesmaximal
11#11729301C212yesmaximal
12C23×C7×Q8C2211yescharacteristic
13C24×C4×C7C2211yescharacteristic
14C2×C7×C22⋊Q8—24no—
15C2×C7×C22⋊Q8—28no—
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C22⋊Dic7C2211yescharacteristic
20C22×C22⋊Dic7C2212yes—
21C22×C22⋊Dic7C2211yescharacteristic
22C22×Dic7⋊C4C2211yescharacteristic
23C22×Dic7⋊C4C2211yescharacteristic
24C22×Dic7⋊C4C2212yes—
25C22×Dic7⋊C4C2212yes—
26C22×C4×Dic7C2212yes—
27#90001—24no—
28#89982—24no—
29C4×C22⋊Dic7—24no—
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38C2×C7⋊#95C2212yes—
39C2×C7⋊#95C2211yescharacteristic
40#89744—28no—
41#14763—71nomaximal, Sylow
42C25×C7C2311yescharacteristic
43C22×C7×Q8—24no—
44C22×C7×Q8—22no—
45C22×C7×Q8—22no—
46C22×C7×Q8—22no—
47C22×C7×Q8—42no—
48C23×C4×C7—22no—
49C23×C4×C7—24no—
50C23×C4×C7—22no—
51C23×C4×C7—42no—
52C23×C4×C7D414yes—
53C23×C4×C7C2312yes—
54C23×C4×C7C2312yes—
55C23×C4×C7C2311yescharacteristic
56C23×C4×C7C2311yescharacteristic
57C23×Dic7C2311yescharacteristic
58C23×Dic7C2311yescharacteristic
59C23×Dic7C2312yes—
60C23×Dic7C2311yescharacteristic
61C23×Dic7C2311yescharacteristic
62C23×Dic7C2312yes—
63C7×C22⋊Q8—232no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—24no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4D418yes—
69C2×C7×C22⋊C4—24no—
70C2×C7×C22⋊C4—24no—
71C2×C7×C22⋊C4—24no—
72C2×C22⋊Dic7—22no—
73C2×C22⋊Dic7—24no—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7—22no—
76C2×C22⋊Dic7—24no—
77C2×C22⋊Dic7—22no—
78C2×C22⋊Dic7—42no—
79C2×C22⋊Dic7Q818yescomplemented
80C2×Dic7⋊C4—24no—
81C2×Dic7⋊C4—24no—
82C2×Dic7⋊C4—24no—
83C2×Dic7⋊C4—22no—
84C2×Dic7⋊C4—22no—
85C2×Dic7⋊C4—24no—
86C2×Dic7⋊C4—24no—
87C2×Dic7⋊C4—22no—
88C2×Dic7⋊C4—22no—
89C2×Dic7⋊C4—24no—
90C2×Dic7⋊C4—42no—
91C2×C4×Dic7—24no—
92C2×C4×Dic7—24no—
93C2×C4×Dic7—42no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C7⋊#95—22no—
97C7⋊#95—24no—
98C7⋊#95—24no—
99C7⋊#95—24no—
100C7⋊#95—24no—
101C7⋊#95—24no—
102C7⋊#95—42no—
103C7⋊#95D414yes—
104C22×C22⋊Q8D711yescharacteristic
105C2×#395—71no—
106C2×#389—72no—
107C2×#385—71no—
108C2×C4×C22⋊C4—71no—
109#1446—71no—
110#1444—71no—
111#1441—71no—
112#1439—72no—
113#1439—72no—
114#1436—72no—
115C24×C7—21no—
116C24×C7—22no—
117C24×C7—22no—
118C24×C7—21no—
119C24×C7—21no—
120C24×C7—21no—
121C24×C7—22no—
122C24×C7—21no—
123C24×C7—21no—
124C24×C7C2×D412yes—
125C24×C7C2×Q814yes—
126C24×C7C2411yescharacteristic
127C2×C7×Q8—28no—
128C2×C7×Q8—48no—
129C2×C7×Q8—48no—
130C2×C7×Q8—44no—
131C2×C7×Q8—44no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—28no—
150C22×C4×C7—28no—
151C22×C4×C7—44no—
152C22×C4×C7—48no—
153C22×C4×C7—48no—
154C22×C4×C7C2×D414yes—
155C22×C4×C7C2×D414yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D414yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7#4112yes—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—21no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—21no—
174C22×Dic7—22no—
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—22no—
178C22×Dic7—21no—
179C22×Dic7—22no—
180C22×Dic7—21no—
181C22×Dic7—21no—
182C22×Dic7—22no—
183C22×Dic7—21no—
184C22×Dic7—21no—
185C22×Dic7—22no—
186C22×Dic7—21no—
187C22×Dic7—22no—
188C22×Dic7—22no—
189C22×Dic7—21no—
190C22×Dic7—21no—
191C22×Dic7C2×D414yes—
192C22×Dic7C2×Q814yes—
193C22×Dic7C2×Q814yes—
194C22×Dic7#4112yes—
195C22×Dic7#4112yes—
196C22×Dic7#4114yes—
197C7×C4⋊C4—28no—
198C7×C4⋊C4—216no—
199C7×C22⋊C4—216no—
200C22⋊Dic7—28no—
201C22⋊Dic7—216no—
202C22⋊Dic7—44no—
203Dic7⋊C4—216no—
204Dic7⋊C4—48no—
205Dic7⋊C4—44no—
206Dic7⋊C4—44no—
207C4×Dic7—48no—
208C23×Q8C2×D711yescharacteristic
209C24×C4C2×D711yescharacteristic
210C2×C22⋊Q8—24no—
211C2×C22⋊Q8—28no—
212C22×C4⋊C4—71no—
213C22×C4⋊C4—72no—
214C22×C4⋊C4—71no—
215C22×C4⋊C4—72no—
216C22×C4⋊C4C2×D711yescharacteristic
217C22×C4⋊C4C2×D712yes—
218C22×C22⋊C4—72no—
219C22×C22⋊C4—71no—
220C22×C22⋊C4—71no—
221C22×C22⋊C4C2×D712yes—
222C22×C42—72no—
223#395—144no—
224#389—148no—
225#385—144no—
226C4×C22⋊C4—144no—
227C2×#95—71no—
228C2×#95—72no—
229C2×#95—71no—
230C2×#95—72no—
231C2×#95—72no—
232C2×#95—72no—
233C2×#95—72no—
234C2×#95—71no—
235C2×#95—71no—
236C2×#95—72no—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—24no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—24no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—24no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7C4⋊Q814yes—
260C23×C7C22×D411yescharacteristic
261C23×C7C22×D411yescharacteristic
262C23×C7C22×D412yes—
263C23×C7C22×Q812yes—
264C23×C7C2×#4111yescharacteristic
265C23×C7C2×#4112yes—
266C23×C7C2×#4111yescharacteristic
267C23×C7C2×#4111yescharacteristic
268C23×C7H(5,𝔽2)12yes—
269C23×C72−1+411yescharacteristic
270C23×C72−1+411yescharacteristic
271C7×Q8—416no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—24no—
277C2×C4×C7—24no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—24no—
284C2×C4×C7—24no—
285C2×C4×C7—24no—
286C2×C4×C7—416no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7#12014yes—
296C2×C4×C7#12114yes—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—24no—
300C2×Dic7—24no—
301C2×Dic7—24no—
302C2×Dic7—24no—
303C2×Dic7—24no—
304C2×Dic7—24no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—24no—
309C2×Dic7—22no—
310C2×Dic7—24no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—41no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—41no—
320C2×Dic7—42no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—42no—
329C2×Dic7—42no—
330C2×Dic7—42no—
331C2×Dic7—42no—
332C2×Dic7C22⋊Q818yescomplemented
333C25C22×D711yescharacteristic
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—24no—
337C22×Q8—22no—
338C22×Q8—42no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—24no—
342C23×C4—42no—
343C23×C4—72no—
344C23×C4—71no—
345C23×C4—71no—
346C23×C4—72no—
347C23×C4—71no—
348C23×C4—71no—
349C23×C4C22⋊D714yes—
350C23×C4C22×D712yes—
351C23×C4C22×D712yes—
352C23×C4C22×D711yescharacteristic
353C23×C4C22×D711yescharacteristic
354C22⋊Q8—232no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—144no—
360C2×C4⋊C4—144no—
361C2×C4⋊C4—144no—
362C2×C4⋊C4—144no—
363C2×C4⋊C4—142no—
364C2×C4⋊C4—142no—
365C2×C4⋊C4—142no—
366C2×C4⋊C4—142no—
367C2×C4⋊C4—144no—
368C2×C4⋊C4—144no—
369C2×C4⋊C4—282no—
370C2×C4⋊C4C22⋊D718yes—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—78no—
375C2×C22⋊C4—144no—
376C2×C22⋊C4—142no—
377C2×C22⋊C4—142no—
378C2×C22⋊C4—144no—
379C2×C22⋊C4—142no—
380C2×C22⋊C4—142no—
381C2×C22⋊C4—282no—
382C2×C42—144no—
383C2×C42—144no—
384C2×C42—282no—
385#95—74no—
386#95—144no—
387#95—144no—
388#95—144no—
389#95—142no—
390#95—142no—
391#95—142no—
392#95—144no—
393#95—144no—
394#95—282no—
395C22×C7—24no—
396C22×C7—24no—
397C22×C7—24no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—44no—
403C22×C7—44no—
404C22×C7—44no—
405C22×C7—44no—
406C22×C7C2×#12011yescharacteristic
407C22×C7C2×#12111yescharacteristic
408C22×C7C2×C22⋊Q812yes—
409C22×C7#52511yescharacteristic
410C22×C7#52512yes—
411C22×C7C2×C4⋊Q811yescharacteristic
412C22×C7#53211yescharacteristic
413C22×C7#53212yes—
414C22×C7#53312yes—
415C22×C7#53411yescharacteristic
416C22×C7#53611yescharacteristic
417C22×C7#53912yes—
418C22×C7#54212yes—
419C22×C7D4212yes—
420C22×C7#54612yes—
421C22×C7#54711yescharacteristic
422C22×C7#54811yescharacteristic
423C22×C7D4×Q812yes—
424C22×C7D4×Q812yes—
425C22×C7#55211yescharacteristic
426C22×C7#55412yes—
427C22×C7#55412yes—
428C22×C7#55611yescharacteristic
429C4×C7—24no—
430C4×C7—24no—
431C4×C7—44no—
432C4×C7—44no—
433Dic7—28no—
434Dic7—42no—
435Dic7—42no—
436Dic7—44no—
437Dic7—81no—
438Dic7—81no—
439C24—21no—
440C24—21no—
441C24—22no—
442C24—21no—
443C24—21no—
444C24—21no—
445C24—21no—
446C24—22no—
447C24—22no—
448C24Q8×D714yes—
449C24C2×C22⋊D712yes—
450C24C23×D711yescharacteristic
451C2×Q8—28no—
452C2×Q8—48no—
453C2×Q8—48no—
454C2×Q8—44no—
455C2×Q8—44no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—74no—
479C22×C4—74no—
480C22×C4—74no—
481C22×C4—74no—
482C22×C4—72no—
483C22×C4—72no—
484C22×C4—142no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—142no—
488C22×C4—141no—
489C22×C4—142no—
490C22×C4—141no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—142no—
496C22×C4—141no—
497C22×C4—141no—
498C22×C4—142no—
499C22×C4—142no—
500C22×C4—142no—
501C22×C4—141no—
502C22×C4—141no—
503C22×C4—142no—
504C22×C4—141no—
505C22×C4—141no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4—141no—
509C22×C4—141no—
510C22×C4—142no—
511C22×C4—141no—
512C22×C4—142no—
513C22×C4—141no—
514C22×C4D4×D712yes—
515C22×C4#115912yes—
516C22×C4C2×C22⋊D714yes—
517C22×C4C2×C22⋊D714yes—
518C22×C4C2×C22⋊D712yes—
519C22×C4C2×C22⋊D712yes—
520C22×C4C2×C22⋊D714yes—
521C4⋊C4—28no—
522C4⋊C4—216no—
523C4⋊C4—1416no—
524C4⋊C4—288no—
525C4⋊C4—284no—
526C4⋊C4—284no—
527C22⋊C4—216no—
528C22⋊C4—1416no—
529C22⋊C4—148no—
530C22⋊C4—284no—
531C42—288no—
532C2×C7—44no—
533C2×C7#238811yescharacteristic
534C2×C7#241011yescharacteristic
535C2×C7#241511yescharacteristic
536C2×C7#243411yescharacteristic
537C2×C7#244912yes—
538C2×C7#245212yes—
539C2×C7#246512yes—
540C2×C7#255612yes—
541C2×C7#258311yescharacteristic
542C2×C7#266311yescharacteristic
543C2×C7#285711yescharacteristic
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23#650714yes—
567C23C2×D4×D711yescharacteristic
568C23C2×#115911yescharacteristic
569C23#654812yes—
570C23C2×Q8×D712yes—
571C23#655111yescharacteristic
572C23#655111yescharacteristic
573C23D7×#4111yescharacteristic
574C23D7×#4112yes—
575C23D7×#4111yescharacteristic
576C23C22×C22⋊D712yes—
577C23C22×C22⋊D711yescharacteristic
578Q8—416no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—78no—
603C2×C4—144no—
604C2×C4—144no—
605C2×C4—144no—
606C2×C4—144no—
607C2×C4—144no—
608C2×C4—144no—
609C2×C4—144no—
610C2×C4—142no—
611C2×C4—142no—
612C2×C4—144no—
613C2×C4—144no—
614C2×C4—142no—
615C2×C4—144no—
616C2×C4—142no—
617C2×C4—282no—
618C2×C4—281no—
619C2×C4—281no—
620C2×C4—282no—
621C2×C4—282no—
622C2×C4—281no—
623C2×C4—281no—
624C2×C4—281no—
625C2×C4—282no—
626C2×C4—282no—
627C2×C4—281no—
628C2×C4—282no—
629C2×C4—281no—
630C2×C4—281no—
631C2×C4—281no—
632C2×C4—282no—
633C2×C4—281no—
634C2×C4—281no—
635C2×C4—281no—
636C2×C4—282no—
637C2×C4—282no—
638C2×C4#650214yes—
639C2×C4#651614yes—
640C7#1476311yescharacteristic, complemented, Sylow
641C22—24no—
642C22—24no—
643C22—24no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22#9031312yes—
653C22D7×C22⋊Q812yes—
654C22#9032211yescharacteristic
655C22#9032411yescharacteristic
656C22#9032712yes—
657C22#9032712yes—
658C22#9032912yes—
659C22#9033011yescharacteristic
660C22#9033112yes—
661C22#9033211yescharacteristic
662C22D4×C22⋊D712yes—
663C22C2×#650211yescharacteristic
664C22#9049511yescharacteristic
665C22#9049712yes—
666C22C2×#650711yescharacteristic
667C22Q8×C22⋊D712yes—
668C22Q8×C22⋊D712yes—
669C22#9050711yescharacteristic
670C22#9051912yes—
671C22#9051911yescharacteristic
672C22#9052312yes—
673C22#9052311yescharacteristic
674C22C2×#651611yescharacteristic
675C4—24no—
676C4—24no—
677C4—44no—
678C4—44no—
679C4—148no—
680C4—284no—
681C4—282no—
682C4—282no—
683C4—561no—
684C4—561no—
685C2—44no—
686C2#1174021112yes—
687C2#1174021312yes—
688C2#1174022712yes—
689C2#1174023011yescharacteristic
690C2#1174025611yescharacteristic
691C2#1174025911yescharacteristic
692C2#1174026212yes—
693C2#1174027011yescharacteristic
694C2#1174149311yescharacteristic
695C2#1174151011yescharacteristic
696C2#1174154011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutator, socle, cosocleC24×C7C24
FittingC22×C7×C22⋊Q8C2
prime radicalC25×C7C23
hypercenterC22×C22⋊Q8D7
hypocenterC7#14763
LCSG ▷ C24×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#90001C2semidirect
5. C2×#89744C2semidirect
6. #11729888C2semidirect
7. #11729322C2semidirect
8. #11729320C2semidirect
9. #11729318C2semidirect
79. C2×C22⋊Dic7Q8semidirect
332. C2×Dic7C22⋊Q8semidirect
640. C7#14763semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C2×#900012
C2×#899822
C2×C4×C22⋊Dic72
C2×#897442
#117298882
#117293222
#117293202
#117293182
#117293012
#147637