Finite Group Databasebeta

← C2×#22586688 #31963396

Subgroups — C2×#22586688 #31963396

Order 18176.

1490 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985963C211yesmaximal, characteristic, complemented
2C22×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
3#22586688C218yesmaximal, direct product factor
4C2×#17985725C212yesmaximal, complemented
5C2×#17985723C212yesmaximal, complemented
6C22×#123×C71C211yesmaximal, characteristic
7C22×#11765849—21no—
8C23×C71⋊C8—21no—
9#17985963—22no—
10#17985963C2214yescomplemented
11C2×C22⋊(C71⋊C8)—22no—
12C2×C22⋊(C71⋊C8)—21no—
13C2×C22⋊(C71⋊C8)—21no—
14C2×C22⋊(C71⋊C8)C2212yes—
15C2×C22⋊(C71⋊C8)C2212yes—
16C2×C22⋊(C71⋊C8)C2214yescomplemented
17#17985725C2218yescomplemented
18#17985723C2218yescomplemented
19C23×D4×C71C411yescharacteristic
20C24×C4×C71C2211yescharacteristic
21C2×#123×C71—21no—
22C2×#123×C71—21no—
23C2×#123×C71—21no—
24C2×#123×C71—21no—
25C2×#123×C71—22no—
26C2×#123×C71—22no—
27C2×#123×C71C414yes—
28C2×#123×C71C2214yes—
29C22×C4⋊C4×C71C411yescharacteristic
30C22×C4⋊C4×C71C2211yescharacteristic
31C22×C22⋊C4×C71C411yescharacteristic
32C22×C22⋊C4×C71C411yescharacteristic
33C22×C22⋊C4×C71C2211yescharacteristic
34C2×#11765849—24no—
35C2×#11765849—44no—
36C22×C71⋊C8—22no—
37C22×C71⋊C8—22no—
38C22×C71⋊C8—24no—
39C22×C71⋊C8—44no—
40C22⋊(C71⋊C8)—22no—
41C22⋊(C71⋊C8)—24no—
42C22⋊(C71⋊C8)—24no—
43C22⋊(C71⋊C8)—22no—
44C22⋊(C71⋊C8)C2314yes—
45C22⋊(C71⋊C8)C2314yes—
46C25×C71C2×C411yescharacteristic
47C25×C71C2×C411yescharacteristic
48C22×D4×C71—21no—
49C22×D4×C71—22no—
50C22×D4×C71—21no—
51C22×D4×C71—44no—
52C22×D4×C71C2×C414yes—
53C23×C4×C71—44no—
54C23×C4×C71C2×C411yescharacteristic
55C23×C4×C71C2×C411yescharacteristic
56C23×C4×C71C2×C411yescharacteristic
57C23×C4×C71C2×C411yescharacteristic
58C23×C4×C71C2×C412yes—
59C23×C4×C71D411yescharacteristic
60C23×C4×C71D411yescharacteristic
61C23×C4×C71D411yescharacteristic
62C23×C4×C71D411yescharacteristic
63C23×C4×C71D412yes—
64C23×C4×C71D412yes—
65C23×C4×C71C2311yescharacteristic
66C23×C4×C71C2312yes—
67#123×C71—24no—
68#123×C71—24no—
69#123×C71—24no—
70#123×C71—24no—
71#123×C71—24no—
72#123×C71—24no—
73#123×C71—24no—
74#123×C71—24no—
75C2×C4⋊C4×C71—21no—
76C2×C4⋊C4×C71—21no—
77C2×C4⋊C4×C71—22no—
78C2×C4⋊C4×C71—21no—
79C2×C4⋊C4×C71—21no—
80C2×C4⋊C4×C71—21no—
81C2×C4⋊C4×C71—21no—
82C2×C4⋊C4×C71C2×C414yes—
83C2×C4⋊C4×C71C2×C412yes—
84C2×C4⋊C4×C71C2312yes—
85C2×C22⋊C4×C71—21no—
86C2×C22⋊C4×C71—22no—
87C2×C22⋊C4×C71—21no—
88C2×C22⋊C4×C71—21no—
89C2×C22⋊C4×C71—21no—
90C2×C22⋊C4×C71—22no—
91C2×C22⋊C4×C71—21no—
92C2×C22⋊C4×C71—21no—
93C2×C22⋊C4×C71—21no—
94C2×C22⋊C4×C71—21no—
95C2×C22⋊C4×C71C2×C414yes—
96C2×C22⋊C4×C71C2×C414yes—
97C2×C22⋊C4×C71C2×C412yes—
98C2×C22⋊C4×C71C2312yes—
99#11765849—48no—
100C2×C71⋊C8—24no—
101C2×C71⋊C8—24no—
102C2×C71⋊C8—44no—
103C2×C71⋊C8—48no—
104C2×C71⋊C8—44no—
105C24×C71—24no—
106C24×C71—24no—
107C24×C71—21no—
108C24×C71—21no—
109C24×C71—24no—
110C24×C71—42no—
111C24×C71—42no—
112C24×C71—42no—
113C24×C71C22⋊C411yescharacteristic
114C24×C71C22⋊C412yes—
115C24×C71C22⋊C411yescharacteristic
116C24×C71C22×C412yes—
117C24×C71C22×C411yescharacteristic
118C24×C71C22×C412yes—
119C2×D4×C71—24no—
120C2×D4×C71—24no—
121C2×D4×C71—48no—
122C2×D4×C71—44no—
123C2×D4×C71—44no—
124C2×D4×C71—48no—
125C22×C4×C71—24no—
126C22×C4×C71—24no—
127C22×C4×C71—21no—
128C22×C4×C71—21no—
129C22×C4×C71—21no—
130C22×C4×C71—21no—
131C22×C4×C71—21no—
132C22×C4×C71—21no—
133C22×C4×C71—24no—
134C22×C4×C71—42no—
135C22×C4×C71—44no—
136C22×C4×C71—44no—
137C22×C4×C71—42no—
138C22×C4×C71—42no—
139C22×C4×C71—44no—
140C22×C4×C71—42no—
141C22×C4×C71—44no—
142C22×C4×C71—42no—
143C22×C4×C71—42no—
144C22×C4×C71C22⋊C411yescharacteristic
145C22×C4×C71C22⋊C411yescharacteristic
146C22×C4×C71C22⋊C412yes—
147C22×C4×C71C22⋊C412yes—
148C22×C4×C71C22⋊C412yes—
149C22×C4×C71C22⋊C412yes—
150C22×C4×C71C22⋊C412yes—
151C22×C4×C71C22×C412yes—
152C22×C4×C71C22×C412yes—
153C22×C4×C71C22×C412yes—
154C22×C4×C71C22×C412yes—
155C22×C4×C71C22×C411yescharacteristic
156C22×C4×C71C2×D411yescharacteristic
157C22×C4×C71C2×D411yescharacteristic
158C22×C4×C71C2×D412yes—
159C22×C4×C71C2×D412yes—
160C22×C4×C71C2×D411yescharacteristic
161C22×C4×C71C2×D411yescharacteristic
162C22×C4×C71C2×D412yes—
163C22×C4×C71C2×D412yes—
164C22×C4×C71C2411yescharacteristic
165C4⋊C4×C71—24no—
166C4⋊C4×C71—24no—
167C4⋊C4×C71—22no—
168C4⋊C4×C71—22no—
169C4⋊C4×C71—22no—
170C4⋊C4×C71—22no—
171C22⋊C4×C71—24no—
172C22⋊C4×C71—24no—
173C22⋊C4×C71—24no—
174C22⋊C4×C71—24no—
175C22⋊C4×C71—22no—
176C22⋊C4×C71—22no—
177C22⋊C4×C71—22no—
178C22⋊C4×C71—22no—
179C23×C71—22no—
180C23×C71—22no—
181C23×C71—22no—
182C23×C71—22no—
183C23×C71—24no—
184C23×C71—24no—
185C23×C71—22no—
186C23×C71—24no—
187C23×C71—22no—
188C23×C71—24no—
189C23×C71—22no—
190C23×C71—22no—
191C23×C71—24no—
192C23×C71—42no—
193C23×C71—42no—
194C23×C71—416no—
195C23×C71—42no—
196C23×C71—42no—
197C23×C71—42no—
198C23×C71—42no—
199C23×C71—42no—
200C23×C71—42no—
201C23×C71—42no—
202C23×C71—42no—
203C23×C71—88no—
204C23×C71#9914yes—
205C23×C71C2×C22⋊C412yes—
206C23×C71C2×C22⋊C411yescharacteristic
207C23×C71C2×C22⋊C412yes—
208C23×C71C2×C22⋊C411yescharacteristic
209C23×C71#13114yes—
210C23×C71C23×C411yescharacteristic
211D4×C71—48no—
212D4×C71—48no—
213C2×C4×C71—24no—
214C2×C4×C71—24no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—24no—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
220C2×C4×C71—24no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—24no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—41no—
229C2×C4×C71—42no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—42no—
233C2×C4×C71—42no—
234C2×C4×C71—41no—
235C2×C4×C71—42no—
236C2×C4×C71—42no—
237C2×C4×C71—42no—
238C2×C4×C71—42no—
239C2×C4×C71—42no—
240C2×C4×C71—42no—
241C2×C4×C71—42no—
242C2×C4×C71—42no—
243C2×C4×C71C2×C22⋊C412yes—
244C2×C4×C71C2×C22⋊C412yes—
245C2×C4×C71C2×C22⋊C411yescharacteristic
246C2×C4×C71C2×C22⋊C411yescharacteristic
247C2×C4×C71C22×D411yescharacteristic
248C2×C4×C71C22×D411yescharacteristic
249C71⋊C8—44no—
250C71⋊C8—44no—
251C22×C71—22no—
252C22×C71—21no—
253C22×C71—21no—
254C22×C71—21no—
255C22×C71—21no—
256C22×C71—22no—
257C22×C71—28no—
258C22×C71—28no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—22no—
262C22×C71—48no—
263C22×C71—48no—
264C22×C71—44no—
265C22×C71—42no—
266C22×C71—42no—
267C22×C71—42no—
268C22×C71—84no—
269C22×C71—84no—
270C22×C71—84no—
271C22×C71#40812yes—
272C22×C71#40812yes—
273C22×C71C2×#9914yes—
274C22×C71C2×#9912yes—
275C22×C71#41312yes—
276C22×C71C22×C22⋊C411yescharacteristic
277C22×C71C2×#13112yes—
278C4×C71—21no—
279C4×C71—21no—
280C4×C71—21no—
281C4×C71—22no—
282C4×C71—21no—
283C4×C71—22no—
284C4×C71—42no—
285C4×C71—42no—
286C4×C71—42no—
287C2×#1468—711nomaximal, Sylow
288C2×C71—22no—
289C2×C71—22no—
290C2×C71—24no—
291C2×C71—42no—
292C2×C71—82no—
293C2×C71#146814yescomplemented
294C2×C71C2×#40811yescharacteristic
295C2×C71C22×#9911yescharacteristic
296C2×C71C2×#41311yescharacteristic
297C22×#123D7111yescharacteristic
298C2×#407—711no—
299C22×C22⋊C8—711no—
300#1468—718no—
301C2×#324—712no—
302C2×#323—712no—
303C71C2×#146811yescharacteristic, complemented, Sylow
304C23×D4Dic7111yescharacteristic
305C24×C4C2×D7111yescharacteristic
306C22×M16—1421no—
307C23×C8—1421no—
308C2×#123—21no—
309C2×#123—21no—
310C2×#123—22no—
311C2×#123—21no—
312C2×#123—21no—
313C2×#123—22no—
314C2×#123Dic7114yes—
315C2×#123C2×D7114yes—
316C22×C4⋊C4Dic7111yescharacteristic
317C22×C4⋊C4C2×D7111yescharacteristic
318C22×C22⋊C4Dic7111yescharacteristic
319C22×C22⋊C4Dic7111yescharacteristic
320C22×C22⋊C4C2×D7111yescharacteristic
321#407—714no—
322#407—1422no—
323C2×C22⋊C8—712no—
324C2×C22⋊C8—714no—
325C2×C22⋊C8—712no—
326C2×C22⋊C8—1422no—
327C2×C22⋊C8—1421no—
328C2×C22⋊C8—1421no—
329#324—718no—
330#323—718no—
331C25C2×Dic7111yescharacteristic
332C25C2×Dic7111yescharacteristic
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—44no—
337C22×D4C2×Dic7114yes—
338C23×C4—44no—
339C23×C4C2×Dic7111yescharacteristic
340C23×C4C2×Dic7111yescharacteristic
341C23×C4C2×Dic7111yescharacteristic
342C23×C4C2×Dic7111yescharacteristic
343C23×C4C2×Dic7112yes—
344C23×C4C22⋊D7111yescharacteristic
345C23×C4C22⋊D7111yescharacteristic
346C23×C4C22⋊D7112yes—
347C23×C4C22⋊D7111yescharacteristic
348C23×C4C22⋊D7111yescharacteristic
349C23×C4C22⋊D7112yes—
350C23×C4C22×D7111yescharacteristic
351C23×C4C22×D7112yes—
352C2×M16—2844no—
353C2×M16—1424no—
354C22×C8—2844no—
355C22×C8—1424no—
356C22×C8—1422no—
357C22×C8—1422no—
358#123—24no—
359#123—24no—
360#123—24no—
361#123—24no—
362#123—24no—
363#123—24no—
364#123—24no—
365#123—24no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4C2×Dic7114yes—
374C2×C4⋊C4C2×Dic7112yes—
375C2×C4⋊C4C22×D7112yes—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4C2×Dic7114yes—
387C2×C22⋊C4C2×Dic7114yes—
388C2×C22⋊C4C2×Dic7112yes—
389C2×C22⋊C4C22×D7112yes—
390C22⋊C8—714no—
391C22⋊C8—714no—
392C22⋊C8—1422no—
393C22⋊C8—1424no—
394C22⋊C8—1422no—
395C22⋊C8—1424no—
396C24—24no—
397C24—24no—
398C24—21no—
399C24—21no—
400C24—24no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22⋊Dic7111yescharacteristic
405C24C22⋊Dic7112yes—
406C24C22⋊Dic7111yescharacteristic
407C24C22×Dic7112yes—
408C24C22×Dic7111yescharacteristic
409C24C22×Dic7112yes—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—48no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4C22⋊Dic7112yes—
436C22×C4C22⋊Dic7112yes—
437C22×C4C22⋊Dic7111yescharacteristic
438C22×C4C22⋊Dic7111yescharacteristic
439C22×C4C22⋊Dic7112yes—
440C22×C4C22⋊Dic7112yes—
441C22×C4C22⋊Dic7112yes—
442C22×C4C22×Dic7112yes—
443C22×C4C22×Dic7112yes—
444C22×C4C22×Dic7112yes—
445C22×C4C22×Dic7112yes—
446C22×C4C22×Dic7111yescharacteristic
447C22×C4C2×C22⋊D7112yes—
448C22×C4C2×C22⋊D7112yes—
449C22×C4C2×C22⋊D7111yescharacteristic
450C22×C4C2×C22⋊D7111yescharacteristic
451C22×C4C2×C22⋊D7111yescharacteristic
452C22×C4C2×C22⋊D7111yescharacteristic
453C22×C4C2×C22⋊D7112yes—
454C22×C4C2×C22⋊D7112yes—
455C22×C4C23×D7111yescharacteristic
456M16—2848no—
457C2×C8—2848no—
458C2×C8—2844no—
459C2×C8—2844no—
460C2×C8—1424no—
461C2×C8—1424no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—42no—
490C23—42no—
491C23—416no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—88no—
501C23#1455604314yes—
502C23C2×C22⋊Dic7112yes—
503C23C2×C22⋊Dic7111yescharacteristic
504C23C2×C22⋊Dic7112yes—
505C23C2×C22⋊Dic7111yescharacteristic
506C23#1455613514yes—
507C23C23×Dic7111yescharacteristic
508D4—48no—
509D4—48no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4C2×C22⋊Dic7111yescharacteristic
541C2×C4C2×C22⋊Dic7111yescharacteristic
542C2×C4C2×C22⋊Dic7112yes—
543C2×C4C2×C22⋊Dic7112yes—
544C2×C4C22×C22⋊D7111yescharacteristic
545C2×C4C22×C22⋊D7111yescharacteristic
546C8—2844no—
547C8—2844no—
548C22—28no—
549C22—28no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22#1798596612yes—
569C22#1798596612yes—
570C22C2×#1455604314yes—
571C22C2×#1455604312yes—
572C22#1798598312yes—
573C22C22×C22⋊Dic7111yescharacteristic
574C22C2×#1455613512yes—
575C4—21no—
576C4—21no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—42no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—82no—
589C2#2258668814yesdirect product factor
590C2C2×#1798596611yescharacteristic
591C2C22×#1455604311yescharacteristic
592C2C2×#1798598311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC23×C71C23×C4
FrattiniC22×C4C23×D71
FittingC22×#123×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC23×D4×C71C4
cosocleC22×C4×C71C24
hypercenterC22×#123D71
hypocenterC71C2×#1468
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#17985963C2semidirect
2. C22×C22⋊(C71⋊C8)C2semidirect
3. #22586688C2direct (N × H)
4. C2×#17985725C2semidirect
5. C2×#17985723C2semidirect
10. #17985963C22semidirect
16. C2×C22⋊(C71⋊C8)C22semidirect
17. #17985725C22semidirect
18. #17985723C22semidirect
293. C2×C71#1468semidirect
303. C71C2×#1468semidirect
589. C2#22586688direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179859632
C22×C22⋊(C71⋊C8)2
#225866882
C2×#179857252
C2×#179857232
C22×#123×C712
C2×#146871