Finite Group Databasebeta

← #31934091

Subgroups — #31934091

Order 18176.

2006 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985921C214yesmaximal, complemented
2C2×#17985901C211yesmaximal, characteristic
3C2×#17985901C211yesmaximal, characteristic
4C2×#17985901C211yesmaximal, characteristic
5C2×#17985897C211yesmaximal, characteristic, complemented
6C2×#17985869C211yesmaximal, characteristic, complemented
7#22586586C214yesmaximal, complemented
8#22586570C211yesmaximal, characteristic, complemented
9C2×C4×C4⋊C4×C71C211yesmaximal, characteristic, complemented
10C23×D284C2211yescharacteristic
11C22×C4⋊Dic71C2211yescharacteristic
12C22×C4⋊Dic71C2211yescharacteristic
13C22×C4⋊Dic71C2214yes—
14C22×C4⋊Dic71C2211yescharacteristic
15C22×Dic71⋊C4C2212yes—
16C22×(C2×D71)⋊C4C2212yes—
17C22×(C2×D71)⋊C4C2214yes—
18C22×(C2×D71)⋊C4C2214yes—
19C22×(C2×D71)⋊C4C2212yes—
20#17985921—216no—
21#17985901—24no—
22#17985901—24no—
23#17985901—24no—
24#17985897—24no—
25#17985869—24no—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2214yes—
28C22×C4⋊C4×C71C2212yes—
29C22×C42×C71C2211yescharacteristic
30C22×C42×C71C2211yescharacteristic
31C22×C42×C71C2211yescharacteristic
32C4×C4⋊C4×C71—24no—
33C2×#95×C71C2212yes—
34C24×D71C2312yes—
35C22×D284—22no—
36C22×D284—22no—
37C22×D284—22no—
38C22×D284—22no—
39C22×D284—22no—
40C22×D284—22no—
41C22×D284Q814yes—
42C23×Dic71C2312yes—
43C23×Dic71C2312yes—
44C23×Dic71C2312yes—
45C2×C4⋊Dic71—28no—
46C2×C4⋊Dic71—22no—
47C2×C4⋊Dic71—22no—
48C2×C4⋊Dic71—22no—
49C2×C4⋊Dic71—22no—
50C2×C4⋊Dic71—22no—
51C2×C4⋊Dic71—28no—
52C2×C4⋊Dic71—22no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71—22no—
55C2×C4⋊Dic71—44no—
56C2×C4⋊Dic71D414yes—
57C2×Dic71⋊C4—24no—
58C2×Dic71⋊C4—28no—
59C2×(C2×D71)⋊C4—28no—
60C2×(C2×D71)⋊C4—28no—
61C2×(C2×D71)⋊C4—24no—
62C2×(C2×D71)⋊C4—24no—
63C2×(C2×D71)⋊C4—28no—
64C2×(C2×D71)⋊C4—28no—
65C2×(C2×D71)⋊C4—28no—
66C2×(C2×D71)⋊C4—28no—
67C2×(C2×D71)⋊C4—44no—
68C2×(C2×D71)⋊C4—44no—
69C71⋊#95—24no—
70C71⋊#95—28no—
71C23×C4×C71C2312yes—
72C23×C4×C71C2311yescharacteristic
73C23×C4×C71C2311yescharacteristic
74C23×C4×C71C2312yes—
75C23×C4×C71C2311yescharacteristic
76C2×C4⋊C4×C71—24no—
77C2×C4⋊C4×C71—24no—
78C2×C4⋊C4×C71D418yescomplemented
79C2×C42×C71—22no—
80C2×C42×C71—22no—
81C2×C42×C71—22no—
82C2×C42×C71—22no—
83C2×C42×C71—22no—
84C2×C42×C71—22no—
85C2×C42×C71—22no—
86C2×C42×C71D414yes—
87C2×C42×C71D414yes—
88#95×C71—24no—
89C23×D71—22no—
90C23×D71—22no—
91C23×D71—22no—
92C23×D71—24no—
93C23×D71—22no—
94C23×D71—22no—
95C23×D71—22no—
96C23×D71—24no—
97C23×D71—24no—
98C23×D71—24no—
99C23×D71C2×Q814yes—
100C2×D284—28no—
101C2×D284—28no—
102C2×D284—28no—
103C2×D284—44no—
104C2×D284—44no—
105C2×D284—44no—
106C2×D284—44no—
107C22×Dic71—22no—
108C22×Dic71—24no—
109C22×Dic71—22no—
110C22×Dic71—24no—
111C22×Dic71—24no—
112C22×Dic71—24no—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—24no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—24no—
121C22×Dic71C2×D414yes—
122C4⋊Dic71—28no—
123C4⋊Dic71—416no—
124C4⋊Dic71—44no—
125C4⋊Dic71—44no—
126Dic71⋊C4—48no—
127(C2×D71)⋊C4—416no—
128(C2×D71)⋊C4—48no—
129(C2×D71)⋊C4—416no—
130(C2×D71)⋊C4—48no—
131C24×C71C2411yescharacteristic
132C22×C4×C71—21no—
133C22×C4×C71—22no—
134C22×C4×C71—22no—
135C22×C4×C71—21no—
136C22×C4×C71—22no—
137C22×C4×C71—22no—
138C22×C4×C71—22no—
139C22×C4×C71—21no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—21no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71C2×D412yes—
150C22×C4×C71C2×D414yes—
151C22×C4×C71C2×D412yes—
152C22×C4×C71C2×D412yes—
153C22×C4×C71C2×D414yes—
154C22×C4×C71C2×D414yes—
155C22×C4×C71C2×D412yes—
156C22×C4×C71C2×D414yes—
157C22×C4×C71C2×D412yes—
158C22×C4×C71C2×Q812yes—
159C22×C4×C71#4112yes—
160C22×C4×C71#4114yes—
161C22×C4×C71#4112yes—
162C22×C4×C71#4112yes—
163C4⋊C4×C71—216no—
164C42×C71—28no—
165C42×C71—28no—
166C42×C71—44no—
167C23×C71C22×D411yescharacteristic
168C23×C71C22×D411yescharacteristic
169C23×C71C22×D411yescharacteristic
170C23×C71C22×D412yes—
171C23×C71C22×Q811yescharacteristic
172C23×C71C2×#4111yescharacteristic
173C23×C71C2×#4111yescharacteristic
174C23×C71C2×#4112yes—
175C23×C71C2×#4111yescharacteristic
176C23×C71H(5,𝔽2)12yes—
177C23×C712−1+412yes—
178C22×D71—24no—
179C22×D71—24no—
180C22×D71—24no—
181C22×D71—28no—
182C22×D71—28no—
183C22×D71—44no—
184C22×D71—42no—
185C22×D71—42no—
186C22×D71—42no—
187C22×D71—42no—
188C22×D71—42no—
189C22×D71—42no—
190C22×D71—44no—
191C22×D71—42no—
192C22×D71—42no—
193C22×D71—44no—
194C22×D71—44no—
195C22×D71—44no—
196C22×D71—44no—
197C22×D71—44no—
198C22×D71—44no—
199C22×D71—44no—
200C22×D71—44no—
201C2×C4×C71—24no—
202C2×C4×C71—22no—
203C2×C4×C71—24no—
204C2×C4×C71—22no—
205C2×C4×C71—24no—
206C2×C4×C71—24no—
207C2×C4×C71—24no—
208C2×C4×C71—24no—
209C2×C4×C71—24no—
210C2×C4×C71—24no—
211C2×C4×C71—22no—
212C2×C4×C71—24no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71—42no—
220C2×C4×C71—42no—
221C2×C4×C71—42no—
222C2×C4×C71—42no—
223C2×C4×C71#12114yes—
224C2×C4×C71#12118yes—
225C2×C4×C71#12114yes—
226C2×C4×C71#12114yes—
227C2×C4×C71C22⋊Q814yes—
228C2×C4×C71C22⋊Q814yes—
229C2×C4×C71C22⋊Q814yes—
230C2×C4×C71#12614yes—
231C2×C4×C71#12714yes—
232C2×Dic71—24no—
233C2×Dic71—28no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—44no—
237C2×Dic71—44no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—42no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—44no—
244C2×Dic71—42no—
245C2×Dic71—44no—
246C2×Dic71—44no—
247D284—416no—
248C22×C71C2×#12111yescharacteristic
249C22×C71C2×#12111yescharacteristic
250C22×C71C2×#12112yes—
251C22×C71C2×#12111yescharacteristic
252C22×C71C2×C22⋊Q811yescharacteristic
253C22×C71C2×C22⋊Q811yescharacteristic
254C22×C71C2×C22⋊Q811yescharacteristic
255C22×C71C2×#12611yescharacteristic
256C22×C71C2×#12711yescharacteristic
257C22×C71#53211yescharacteristic
258C22×C71#53212yes—
259C22×C71#53511yescharacteristic
260C22×C71#53611yescharacteristic
261C22×C71#53712yes—
262C22×C71#54212yes—
263C22×C71D4212yes—
264C22×C71#54712yes—
265C22×C71#54812yes—
266C22×C71D4×Q812yes—
267C22×C71#55012yes—
268C22×C71#55312yes—
269C22×C71#55412yes—
270C22×C71#55612yes—
271C2×D71—44no—
272C2×D71—44no—
273C2×D71—48no—
274C2×D71—48no—
275C2×D71—44no—
276C2×D71—82no—
277C2×D71—82no—
278C2×D71—84no—
279C2×D71—84no—
280C2×D71—82no—
281C2×D71—82no—
282C4×C71—24no—
283C4×C71—28no—
284C4×C71—42no—
285C4×C71—44no—
286C4×C71#39618yescomplemented
287Dic71—44no—
288Dic71—82no—
289Dic71—82no—
290#15134—711nomaximal, Sylow
291C2×C71C2×#39611yescharacteristic
292C2×C71#241612yes—
293C2×C71#245212yes—
294C2×C71#251212yes—
295C2×C71#253812yes—
296C2×C71#254111yescharacteristic
297C2×C71#254211yescharacteristic
298C2×C71#254511yescharacteristic
299C2×C71#256611yescharacteristic
300C2×C71#262011yescharacteristic
301C2×C71#285611yescharacteristic
302D71—84no—
303C2×#396—714no—
304C2×#390—711no—
305C2×#390—711no—
306C2×#390—711no—
307C2×#389—711no—
308C2×#384—711no—
309C2×C4×C4⋊C4D7111yescharacteristic, complemented
310#1439—714no—
311C4⋊#95—711no—
312C71#1513411yescharacteristic, complemented, Sylow
313C23×D4—711no—
314C22×C4⋊C4—714no—
315C22×C4⋊C4—711no—
316C22×C4⋊C4—712no—
317C22×C4⋊C4—711no—
318C22×C4⋊C4—711no—
319C22×C4⋊C4C2×D7112yes—
320C22×C22⋊C4—712no—
321C22×C22⋊C4—714no—
322C22×C22⋊C4—712no—
323C22×C22⋊C4—714no—
324C22×C42C2×D7111yescharacteristic
325C22×C42C2×D7111yescharacteristic
326C22×C42C2×D7111yescharacteristic
327#396—14216no—
328#390—1424no—
329#390—1424no—
330#390—1424no—
331#389—1424no—
332#384—1424no—
333C4×C4⋊C4—24no—
334C2×#95—714no—
335C2×#95—712no—
336C2×#95C2×D7112yes—
337C25—712no—
338C22×D4—714no—
339C22×D4—1422no—
340C22×D4—1422no—
341C22×D4—1422no—
342C22×D4—1422no—
343C22×D4—1422no—
344C22×D4—1422no—
345C23×C4—712no—
346C23×C4—712no—
347C23×C4—712no—
348C23×C4C22×D7112yes—
349C23×C4C22×D7111yescharacteristic
350C23×C4C22×D7112yes—
351C23×C4C22×D7111yescharacteristic
352C23×C4C22×D7111yescharacteristic
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—2844no—
356C2×C4⋊C4—714no—
357C2×C4⋊C4—1428no—
358C2×C4⋊C4—1422no—
359C2×C4⋊C4—1422no—
360C2×C4⋊C4—1424no—
361C2×C4⋊C4—1422no—
362C2×C4⋊C4—1422no—
363C2×C4⋊C4—1428no—
364C2×C4⋊C4—1428no—
365C2×C4⋊C4—1422no—
366C2×C4⋊C4—1422no—
367C2×C4⋊C4—1422no—
368C2×C4⋊C4—1422no—
369C2×C4⋊C4D28418yescomplemented
370C2×C22⋊C4—2844no—
371C2×C22⋊C4—2844no—
372C2×C22⋊C4—1428no—
373C2×C22⋊C4—1428no—
374C2×C22⋊C4—1424no—
375C2×C22⋊C4—1424no—
376C2×C22⋊C4—1428no—
377C2×C22⋊C4—1428no—
378C2×C22⋊C4—1428no—
379C2×C22⋊C4—1428no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42D28414yes—
388C2×C42C22⋊D7114yes—
389#95—24no—
390#95—1424no—
391#95—1428no—
392C24—714no—
393C24—1422no—
394C24—1422no—
395C24—1424no—
396C24—1422no—
397C24—1422no—
398C24—1424no—
399C24—1422no—
400C24—1422no—
401C24—1424no—
402C24—1424no—
403C24C23×D7111yescharacteristic
404C2×D4—2844no—
405C2×D4—2844no—
406C2×D4—2844no—
407C2×D4—2844no—
408C2×D4—1428no—
409C2×D4—1428no—
410C2×D4—1428no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—714no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1424no—
432C22×C4—1424no—
433C22×C4—1422no—
434C22×C4—1422no—
435C22×C4—1424no—
436C22×C4—1422no—
437C22×C4—1422no—
438C22×C4—1424no—
439C22×C4—1422no—
440C22×C4—1424no—
441C22×C4—1422no—
442C22×C4—1424no—
443C22×C4D4×D7112yes—
444C22×C4C2×C22⋊D7114yes—
445C22×C4C2×C22⋊D7112yes—
446C22×C4C2×D28412yes—
447C22×C4C2×D28414yes—
448C22×C4C2×D28414yes—
449C22×C4C2×D28412yes—
450C22×C4C2×D28414yes—
451C22×C4C2×D28412yes—
452C22×C4Q8×D7112yes—
453C22×C4#1176586412yes—
454C22×C4#1176586514yes—
455C22×C4#1176586512yes—
456C22×C4#1176586612yes—
457C4⋊C4—216no—
458C4⋊C4—28416no—
459C4⋊C4—2844no—
460C4⋊C4—2844no—
461C4⋊C4—2848no—
462C4⋊C4—1428no—
463C22⋊C4—2848no—
464C22⋊C4—28416no—
465C22⋊C4—28416no—
466C22⋊C4—2848no—
467C42—28no—
468C42—28no—
469C42—44no—
470C23—2842no—
471C23—2842no—
472C23—2844no—
473C23—2844no—
474C23—2844no—
475C23—2844no—
476C23—2844no—
477C23—2844no—
478C23—2844no—
479C23—2844no—
480C23—2842no—
481C23—2844no—
482C23—2842no—
483C23—2842no—
484C23—2842no—
485C23—2842no—
486C23—2842no—
487C23—2844no—
488C23—1428no—
489C23—1428no—
490C23—1424no—
491C23—1424no—
492C23—1424no—
493C23C2×D4×D7111yescharacteristic
494C23C22×C22⋊D7111yescharacteristic
495C23C22×D28411yescharacteristic
496C23C22×D28412yes—
497C23C2×Q8×D7111yescharacteristic
498C23C2×#1176586411yescharacteristic
499C23C2×#1176586511yescharacteristic
500C23C2×#1176586512yes—
501C23C2×#1176586611yescharacteristic
502C23#1455617412yes—
503C23#1455617512yes—
504D4—28416no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—2842no—
528C2×C4—2844no—
529C2×C4—2842no—
530C2×C4—2842no—
531C2×C4—2844no—
532C2×C4—2844no—
533C2×C4—2844no—
534C2×C4—2842no—
535C2×C4—2842no—
536C2×C4—2842no—
537C2×C4—2842no—
538C2×C4—2842no—
539C2×C4—2844no—
540C2×C4—1428no—
541C2×C4—1424no—
542C2×C4#1455610014yes—
543C2×C4#1455610218yes—
544C2×C4#1455610214yes—
545C2×C4#1455610314yes—
546C2×C4#1455610414yes—
547C2×C4#1455610614yes—
548C2×C4#1455610814yes—
549C2×C4#1455612314yes—
550C2×C4#1455612514yes—
551C22—2844no—
552C22—2844no—
553C22—2844no—
554C22—2848no—
555C22—2848no—
556C22—5682no—
557C22—5684no—
558C22—5682no—
559C22—5682no—
560C22—5682no—
561C22—5684no—
562C22C2×#1455610011yescharacteristic
563C22C2×#1455610212yes—
564C22C2×#1455610211yescharacteristic
565C22C2×#1455610311yescharacteristic
566C22C2×#1455610411yescharacteristic
567C22C2×#1455610611yescharacteristic
568C22C2×#1455610811yescharacteristic
569C22C2×#1455612311yescharacteristic
570C22C2×#1455612511yescharacteristic
571C22#1798654911yescharacteristic
572C22#1798654912yes—
573C22#1798656811yescharacteristic
574C22#1798657511yescharacteristic
575C22#1798657712yes—
576C22#1798662012yes—
577C22D4×D28412yes—
578C22#1798665312yes—
579C22#1798665812yes—
580C22Q8×D28412yes—
581C22#1798667112yes—
582C22#1798669512yes—
583C22#1798670912yes—
584C22#1798672212yes—
585C4—28no—
586C4—24no—
587C4—44no—
588C4—42no—
589C4—2844no—
590C4—5682no—
591C4—5682no—
592C4#1798592118yescomplemented
593C2—5684no—
594C2C2×#1798592111yescharacteristic
595C2#2259178012yes—
596C2#2259219012yes—
597C2#2259292612yes—
598C2#2259325812yes—
599C2#2259328211yescharacteristic
600C2#2259328911yescharacteristic
601C2#2259332011yescharacteristic
602C2#2259354011yescharacteristic
603C2#2259409711yescharacteristic
604C2#2259668511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutator, socle, cosocleC24×C71C24
FittingC2×C4×C4⋊C4×C71C2
prime radicalC23×D284C22
hypercenterC2×C4×C4⋊C4D71
hypocenterC71#15134
LCSG ▷ C24×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#17985921C2semidirect
5. C2×#17985897C2semidirect
6. C2×#17985869C2semidirect
7. #22586586C2semidirect
8. #22586570C2semidirect
9. C2×C4×C4⋊C4×C71C2semidirect
78. C2×C4⋊C4×C71D4semidirect
286. C4×C71#396semidirect
309. C2×C4×C4⋊C4D71semidirect
312. C71#15134semidirect
369. C2×C4⋊C4D284semidirect
592. C4#17985921semidirect

Maximal subgroups

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

subgroupindex
C2×#179859212
C2×#179859012
C2×#179858972
C2×#179858692
#225865862
#225865702
C2×C4×C4⋊C4×C712
#1513471