Finite Group Databasebeta

← #31919724

Subgroups — #31919724

Order 18176.

1774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985896C212yesmaximal
2C2×#17985894C212yesmaximal
3C2×#17985868C212yesmaximal
4C2×#17985868C211yesmaximal, characteristic
5C2×#17985850C211yesmaximal, characteristic
6C2×#17985850C211yesmaximal, characteristic
7#22586605C211yesmaximal, characteristic
8#22586604C211yesmaximal, characteristic
9#22586603C211yesmaximal, characteristic
10Dic71⋊#95C212yesmaximal
11C71×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic71C2212yes—
13C22×C4⋊Dic71C2211yescharacteristic
14C22×Dic71⋊C4C2212yes—
15C22×Dic71⋊C4C2212yes—
16C22×Dic71⋊C4C2212yes—
17C22×Dic71⋊C4C2212yes—
18C22×Dic71⋊C4C2211yescharacteristic
19C22×Dic71⋊C4C2211yescharacteristic
20C22×Dic71⋊C4C2212yes—
21C22×C4×Dic71C2211yescharacteristic
22C22×C4×Dic71C2211yescharacteristic
23C22×C4×Dic71C2212yes—
24C22×C4×Dic71C2211yescharacteristic
25#17985896—28no—
26#17985894—28no—
27#17985868—28no—
28#17985868—24no—
29#17985850—24no—
30#17985850C418yescomplemented
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2212yes—
33C2×C71⋊#95C2212yes—
34C2×C71⋊#95C2211yescharacteristic
35C2×C71⋊#95C2211yescharacteristic
36C22×C4⋊C4×C71C2212yes—
37C22×C4⋊C4×C71C2211yescharacteristic
38C2×#95×C71C2212yes—
39C2×#95×C71C2211yescharacteristic
40C2×#95×C71C2211yescharacteristic
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2311yescharacteristic
44C23×Dic71C2312yes—
45C23×Dic71C2311yescharacteristic
46C23×Dic71C2311yescharacteristic
47C2×C4⋊Dic71—24no—
48C2×C4⋊Dic71—22no—
49C2×C4⋊Dic71—24no—
50C2×C4⋊Dic71—42no—
51C2×C4⋊Dic71—42no—
52C2×Dic71⋊C4—22no—
53C2×Dic71⋊C4—22no—
54C2×Dic71⋊C4—22no—
55C2×Dic71⋊C4—22no—
56C2×Dic71⋊C4—24no—
57C2×Dic71⋊C4—24no—
58C2×Dic71⋊C4—24no—
59C2×Dic71⋊C4—24no—
60C2×Dic71⋊C4—24no—
61C2×Dic71⋊C4—24no—
62C2×Dic71⋊C4—24no—
63C2×Dic71⋊C4—24no—
64C2×Dic71⋊C4—24no—
65C2×Dic71⋊C4—24no—
66C2×Dic71⋊C4—42no—
67C2×Dic71⋊C4—42no—
68C2×Dic71⋊C4C2×C418yes—
69C2×Dic71⋊C4D414yes—
70C2×Dic71⋊C4D418yes—
71C2×Dic71⋊C4Q814yes—
72C2×Dic71⋊C4Q818yes—
73C2×C4×Dic71—24no—
74C2×C4×Dic71—24no—
75C2×C4×Dic71—22no—
76C2×C4×Dic71—22no—
77C2×C4×Dic71—24no—
78C2×C4×Dic71—24no—
79C2×C4×Dic71C2×C418yes—
80C2×C4×Dic71C2×C414yes—
81C2×C4×Dic71D414yes—
82C2×C4×Dic71Q814yes—
83C71⋊#95—24no—
84C71⋊#95—24no—
85C71⋊#95—22no—
86C71⋊#95—24no—
87C71⋊#95C2×C414yes—
88C23×C4×C71C2312yes—
89C23×C4×C71C2311yescharacteristic
90C23×C4×C71C2311yescharacteristic
91C23×C4×C71C2311yescharacteristic
92C23×C4×C71C2312yes—
93C2×C4⋊C4×C71—24no—
94C2×C4⋊C4×C71—22no—
95C2×C4⋊C4×C71—24no—
96C2×C4⋊C4×C71—42no—
97C2×C4⋊C4×C71—42no—
98#95×C71—22no—
99#95×C71—24no—
100#95×C71C2×C414yes—
101C22×Dic71—22no—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—21no—
107C22×Dic71—21no—
108C22×Dic71—22no—
109C22×Dic71—21no—
110C22×Dic71—21no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71C22×C414yes—
121C22×Dic71C22×C412yes—
122C22×Dic71C22×C412yes—
123C22×Dic71C2×D412yes—
124C22×Dic71C2×D414yes—
125C22×Dic71C2×D412yes—
126C22×Dic71C2×D412yes—
127C22×Dic71C2×D414yes—
128C22×Dic71C2×D412yes—
129C22×Dic71C2×Q812yes—
130C22×Dic71C2×Q814yes—
131C22×Dic71C2×Q812yes—
132C22×Dic71C2×Q812yes—
133C22×Dic71C2×Q814yes—
134C22×Dic71C2×Q812yes—
135C22×Dic71#4112yes—
136C22×Dic71#4112yes—
137C4⋊Dic71—48no—
138C4⋊Dic71—44no—
139Dic71⋊C4—28no—
140Dic71⋊C4—28no—
141Dic71⋊C4—48no—
142Dic71⋊C4—48no—
143Dic71⋊C4—48no—
144Dic71⋊C4—48no—
145Dic71⋊C4C4⋊C4132yescomplemented
146C4×Dic71—216no—
147C4×Dic71—44no—
148C4×Dic71—44no—
149C4×Dic71C4⋊C4116yescomplemented
150C24×C71C2411yescharacteristic
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—22no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71C22×C412yes—
173C22×C4×C71C22×C414yes—
174C22×C4×C71C2×D412yes—
175C22×C4×C71C2×D412yes—
176C22×C4×C71C2×D414yes—
177C22×C4×C71C2×Q812yes—
178C22×C4×C71C2×Q812yes—
179C22×C4×C71C2×Q814yes—
180C4⋊C4×C71—48no—
181C4⋊C4×C71—44no—
182C23×C71C23×C411yescharacteristic
183C23×C71C22×D411yescharacteristic
184C23×C71C22×D412yes—
185C23×C71C22×D411yescharacteristic
186C23×C71C22×Q811yescharacteristic
187C23×C71C22×Q812yes—
188C23×C71C22×Q811yescharacteristic
189C23×C71C2×#4111yescharacteristic
190C23×C71C2×#4111yescharacteristic
191C23×C71H(5,𝔽2)12yes—
192C23×C712−1+412yes—
193C2×C4×C71—24no—
194C2×C4×C71—24no—
195C2×C4×C71—24no—
196C2×C4×C71—22no—
197C2×C4×C71—22no—
198C2×C4×C71—24no—
199C2×C4×C71—24no—
200C2×C4×C71—24no—
201C2×C4×C71—24no—
202C2×C4×C71—24no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—24no—
206C2×C4×C71—24no—
207C2×C4×C71—41no—
208C2×C4×C71—41no—
209C2×C4×C71—41no—
210C2×C4×C71—41no—
211C2×C4×C71—42no—
212C2×C4×C71—42no—
213C2×C4×C71—42no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71—42no—
220C2×C4×C71C2×C4⋊C414yes—
221C2×C4×C71C2×C4⋊C418yes—
222C2×Dic71—24no—
223C2×Dic71—24no—
224C2×Dic71—24no—
225C2×Dic71—24no—
226C2×Dic71—24no—
227C2×Dic71—24no—
228C2×Dic71—22no—
229C2×Dic71—22no—
230C2×Dic71—24no—
231C2×Dic71—24no—
232C2×Dic71—41no—
233C2×Dic71—41no—
234C2×Dic71—41no—
235C2×Dic71—41no—
236C2×Dic71—42no—
237C2×Dic71—42no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—42no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71C2×C4⋊C418yes—
246C2×Dic71C2×C4⋊C414yes—
247C2×Dic71C2×C4⋊C418yes—
248C2×Dic71C2×C4⋊C414yes—
249C2×Dic71C4×D414yes—
250C2×Dic71C4×Q814yes—
251C2×Dic71#12114yes—
252C2×Dic71C22⋊Q814yes—
253C2×Dic71#12514yes—
254C2×Dic71#12714yes—
255C2×Dic71C4⋊Q814yes—
256C2×Dic71C4⋊Q818yes—
257C2×Dic71C4⋊Q814yes—
258C22×C71C22×C4⋊C411yescharacteristic
259C22×C71C22×C4⋊C412yes—
260C22×C71C2×C4×D411yescharacteristic
261C22×C71C2×C4×Q811yescharacteristic
262C22×C71#52012yes—
263C22×C71C2×#12111yescharacteristic
264C22×C71C2×C22⋊Q811yescharacteristic
265C22×C71C2×#12511yescharacteristic
266C22×C71C2×#12711yescharacteristic
267C22×C71C2×C4⋊Q811yescharacteristic
268C22×C71C2×C4⋊Q812yes—
269C22×C71C2×C4⋊Q811yescharacteristic
270C22×C71#53511yescharacteristic
271C22×C71#53611yescharacteristic
272C22×C71#54412yes—
273C22×C71D4212yes—
274C22×C71#54712yes—
275C22×C71D4×Q812yes—
276C22×C71D4×Q812yes—
277C22×C71#55012yes—
278C22×C71#55412yes—
279C22×C71#55712yes—
280C22×C71Q8212yes—
281C4×C71—24no—
282C4×C71—28no—
283C4×C71—42no—
284C4×C71—42no—
285C4×C71—82no—
286Dic71—28no—
287Dic71—42no—
288Dic71—42no—
289Dic71—82no—
290Dic71#38418yescomplemented
291Dic71#38918yescomplemented
292#13910—711nomaximal, Sylow
293C2×C71C2×#38411yescharacteristic
294C2×C71C2×#38911yescharacteristic
295C2×C71#231711yescharacteristic
296C2×C71D4×C4⋊C412yes—
297C2×C71Q8×C4⋊C412yes—
298C2×C71#254111yescharacteristic
299C2×C71#254211yescharacteristic
300C2×C71#254611yescharacteristic
301C2×C71#255112yes—
302C2×C71#255212yes—
303C2×C71#256711yescharacteristic
304C2×#389—712no—
305C2×#389—712no—
306C2×#384—711no—
307C2×#384—712no—
308C2×C4×C4⋊C4—711no—
309C2×C4×C4⋊C4—711no—
310#1444—711no—
311#1444—711no—
312#1444—711no—
313#1444D7111yescharacteristic
314C4⋊#95—712no—
315C71#1391011yescharacteristic, complemented, Sylow
316C22×C4⋊C4—711no—
317C22×C4⋊C4—712no—
318C22×C4⋊C4—712no—
319C22×C4⋊C4—712no—
320C22×C4⋊C4—712no—
321C22×C4⋊C4—711no—
322C22×C4⋊C4—711no—
323C22×C4⋊C4—712no—
324C22×C4⋊C4—712no—
325C22×C4⋊C4C2×D7112yes—
326C22×C4⋊C4C2×D7111yescharacteristic
327C22×C42—711no—
328C22×C42—712no—
329C22×C42—711no—
330C22×C42—711no—
331#389—1428no—
332#389—1428no—
333#384—1424no—
334#384—1428no—
335C4×C4⋊C4—718no—
336C4×C4⋊C4—1424no—
337C2×#95—711no—
338C2×#95—712no—
339C2×#95—712no—
340C2×#95—712no—
341C2×#95—711no—
342C2×#95C2×D7112yes—
343C2×#95C2×D7111yescharacteristic
344C2×#95C2×D7111yescharacteristic
345C23×C4—711no—
346C23×C4—711no—
347C23×C4—712no—
348C23×C4—711no—
349C23×C4—711no—
350C23×C4—712no—
351C23×C4C22×D7112yes—
352C23×C4C22×D7111yescharacteristic
353C23×C4C22×D7111yescharacteristic
354C23×C4C22×D7112yes—
355C23×C4C22×D7111yescharacteristic
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—2842no—
362C2×C4⋊C4—2842no—
363C2×C4⋊C4—2842no—
364C2×C4⋊C4—2842no—
365C2×C4⋊C4—718no—
366C2×C4⋊C4—718no—
367C2×C4⋊C4—718no—
368C2×C4⋊C4—714no—
369C2×C4⋊C4—714no—
370C2×C4⋊C4—1424no—
371C2×C4⋊C4—1424no—
372C2×C4⋊C4—1424no—
373C2×C4⋊C4—1424no—
374C2×C4⋊C4—1424no—
375C2×C4⋊C4—1424no—
376C2×C4⋊C4—1424no—
377C2×C4⋊C4—1424no—
378C2×C4⋊C4—1424no—
379C2×C4⋊C4—1424no—
380C2×C4⋊C4—1424no—
381C2×C4⋊C4—1422no—
382C2×C4⋊C4—1424no—
383C2×C4⋊C4—1422no—
384C2×C4⋊C4—1422no—
385C2×C4⋊C4—1422no—
386C2×C4⋊C4—1422no—
387C2×C42—714no—
388C2×C42—714no—
389C2×C42—714no—
390C2×C42—718no—
391C2×C42—1422no—
392C2×C42—1422no—
393C2×C42—1424no—
394C2×C42—1424no—
395C2×C42—1424no—
396C2×C42—1424no—
397#95—22no—
398#95—24no—
399#95—714no—
400#95—1424no—
401#95—1424no—
402#95—1422no—
403#95—1424no—
404#95C4×D7114yes—
405C24C23×D7111yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—712no—
428C22×C4—712no—
429C22×C4—712no—
430C22×C4—712no—
431C22×C4—714no—
432C22×C4—712no—
433C22×C4—712no—
434C22×C4—712no—
435C22×C4—714no—
436C22×C4—712no—
437C22×C4—712no—
438C22×C4—712no—
439C22×C4—714no—
440C22×C4—714no—
441C22×C4—714no—
442C22×C4—712no—
443C22×C4—712no—
444C22×C4—1422no—
445C22×C4—1422no—
446C22×C4—1422no—
447C22×C4—1422no—
448C22×C4—1422no—
449C22×C4—1422no—
450C22×C4—1422no—
451C22×C4—1422no—
452C22×C4—1421no—
453C22×C4—1421no—
454C22×C4—1422no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1422no—
458C22×C4—1422no—
459C22×C4—1422no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4C2×C4×D7112yes—
464C22×C4C2×C4×D7114yes—
465C22×C4D4×D7112yes—
466C22×C4D4×D7112yes—
467C22×C4D4×D7114yes—
468C22×C4Q8×D7112yes—
469C22×C4Q8×D7112yes—
470C22×C4Q8×D7114yes—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—2844no—
474C4⋊C4—2848no—
475C4⋊C4—2848no—
476C4⋊C4—2848no—
477C4⋊C4—2848no—
478C4⋊C4—2848no—
479C4⋊C4—7132no—
480C4⋊C4—1428no—
481C4⋊C4—1428no—
482C42—2844no—
483C42—2844no—
484C42—7116no—
485C42—14216no—
486C23C22×C4×D7111yescharacteristic
487C23C2×D4×D7111yescharacteristic
488C23C2×D4×D7111yescharacteristic
489C23C2×D4×D7112yes—
490C23C2×Q8×D7111yescharacteristic
491C23C2×Q8×D7111yescharacteristic
492C23C2×Q8×D7112yes—
493C23#41×D7111yescharacteristic
494C23#41×D7111yescharacteristic
495C23#1455617312yes—
496C23#1455617512yes—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2842no—
527C2×C4—2842no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2842no—
531C2×C4—2842no—
532C2×C4—2842no—
533C2×C4—2842no—
534C2×C4—2842no—
535C2×C4—2842no—
536C2×C4—2842no—
537C2×C4—714no—
538C2×C4—714no—
539C2×C4—718no—
540C2×C4—714no—
541C2×C4—714no—
542C2×C4—714no—
543C2×C4—714no—
544C2×C4—714no—
545C2×C4—714no—
546C2×C4—714no—
547C2×C4—718no—
548C2×C4—714no—
549C2×C4—718no—
550C2×C4—1424no—
551C2×C4—1424no—
552C2×C4—1422no—
553C2×C4—1422no—
554C2×C4—1424no—
555C2×C4—1424no—
556C2×C4—1424no—
557C2×C4—1424no—
558C2×C4—1424no—
559C2×C4—1424no—
560C2×C4C4⋊C4×D7114yes—
561C2×C4C4⋊C4×D7118yes—
562C22C2×C4⋊C4×D7112yes—
563C22C2×C4⋊C4×D7111yescharacteristic
564C22C4×D4×D7111yescharacteristic
565C22C4×Q8×D7111yescharacteristic
566C22#1798647812yes—
567C22#121×D7111yescharacteristic
568C22C22⋊Q8×D7111yescharacteristic
569C22#125×D7111yescharacteristic
570C22#127×D7111yescharacteristic
571C22C4⋊Q8×D7111yescharacteristic
572C22C4⋊Q8×D7112yes—
573C22C4⋊Q8×D7111yescharacteristic
574C22#1798656411yescharacteristic
575C22#1798657111yescharacteristic
576C22#1798662812yes—
577C22#1798663112yes—
578C22#1798664712yes—
579C22#1798666212yes—
580C22#1798666212yes—
581C22#1798666712yes—
582C22#1798670312yes—
583C22#1798672612yes—
584C22#1798672912yes—
585C4—28no—
586C4—24no—
587C4—42no—
588C4—42no—
589C4—82no—
590C4—2842no—
591C4—2842no—
592C4—5682no—
593C4—718no—
594C4—718no—
595C4—1428no—
596C2#384×D7111yescharacteristic
597C2#389×D7111yescharacteristic
598C2#2259095911yescharacteristic
599C2#2259123812yes—
600C2#2259125412yes—
601C2#2259328111yescharacteristic
602C2#2259328811yescharacteristic
603C2#2259332411yescharacteristic
604C2#2259336512yes—
605C2#2259337012yes—
606C2#2259354411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×#1444C2
socle, prime radical, cosocleC24×C71C24
hypercenter#1444D71
hypocenterC71#13910
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
30. #17985850C4semidirect
145. Dic71⋊C4C4⋊C4semidirect
149. C4×Dic71C4⋊C4semidirect
290. Dic71#384semidirect
291. Dic71#389semidirect
315. C71#13910semidirect

Maximal subgroups

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

subgroupindex
C2×#179858962
C2×#179858942
C2×#179858682
C2×#179858502
#225866052
#225866042
#225866032
Dic71⋊#952
C71×#14442
#1391071