Finite Group Databasebeta

← C2×#22586689 #31963397

Subgroups — C2×#22586689 #31963397

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
1C22×#14556111C211yesmaximal, characteristic
2C22×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
3#22586689C218yesmaximal, direct product factor
4C2×#17985724C212yesmaximal, complemented
5C2×#17985722C212yesmaximal, complemented
6C2×#407×C71C211yesmaximal, characteristic, complemented
7C23×C22⋊D71C411yescharacteristic
8C23×C71⋊C8—21no—
9C2×#14556111—21no—
10C2×#14556111—21no—
11C2×#14556111—22no—
12C2×#14556111—21no—
13C2×#14556111—21no—
14C2×#14556111—22no—
15C2×#14556111C414yes—
16C2×#14556111C2214yes—
17C22×Dic71⋊C4C411yescharacteristic
18C22×Dic71⋊C4C2211yescharacteristic
19C22×C22⋊Dic71C411yescharacteristic
20C22×(C2×D71)⋊C4C411yescharacteristic
21C22×(C2×D71)⋊C4C2211yescharacteristic
22C2×C22⋊(C71⋊C8)—22no—
23C2×C22⋊(C71⋊C8)—21no—
24C2×C22⋊(C71⋊C8)C2212yes—
25C2×C22⋊(C71⋊C8)C2214yescomplemented
26#17985724C2218yescomplemented
27#17985722C2218yescomplemented
28C24×C4×C71C2211yescharacteristic
29C22×M16×C71—21no—
30#407×C71—22no—
31#407×C71C2214yescomplemented
32C2×C22⋊C8×C71—21no—
33C2×C22⋊C8×C71C2212yes—
34C24×D71C2×C411yescharacteristic
35C22×C22⋊D71—21no—
36C22×C22⋊D71—22no—
37C22×C22⋊D71—21no—
38C22×C22⋊D71—44no—
39C22×C22⋊D71C2×C414yes—
40C23×Dic71C2×C411yescharacteristic
41C23×Dic71C2×C411yescharacteristic
42C23×Dic71C2×C411yescharacteristic
43C22×C71⋊C8—22no—
44C22×C71⋊C8—24no—
45C22×C71⋊C8—44no—
46#14556111—24no—
47#14556111—24no—
48#14556111—24no—
49#14556111—24no—
50#14556111—24no—
51#14556111—24no—
52#14556111—24no—
53#14556111—24no—
54C2×Dic71⋊C4—21no—
55C2×Dic71⋊C4—21no—
56C2×Dic71⋊C4—22no—
57C2×Dic71⋊C4—21no—
58C2×Dic71⋊C4—21no—
59C2×Dic71⋊C4—21no—
60C2×Dic71⋊C4—21no—
61C2×Dic71⋊C4C2×C414yes—
62C2×Dic71⋊C4C2×C412yes—
63C2×Dic71⋊C4C2312yes—
64C2×C22⋊Dic71—21no—
65C2×C22⋊Dic71—22no—
66C2×C22⋊Dic71—21no—
67C2×C22⋊Dic71C2×C414yes—
68C2×(C2×D71)⋊C4—21no—
69C2×(C2×D71)⋊C4—21no—
70C2×(C2×D71)⋊C4—22no—
71C2×(C2×D71)⋊C4—21no—
72C2×(C2×D71)⋊C4—21no—
73C2×(C2×D71)⋊C4—21no—
74C2×(C2×D71)⋊C4—21no—
75C2×(C2×D71)⋊C4C2×C414yes—
76C2×(C2×D71)⋊C4C2×C412yes—
77C2×(C2×D71)⋊C4C2312yes—
78C22⋊(C71⋊C8)—22no—
79C22⋊(C71⋊C8)—24no—
80C22⋊(C71⋊C8)C2314yes—
81C25×C71C2×C411yescharacteristic
82C23×C4×C71—44no—
83C23×C4×C71C2×C411yescharacteristic
84C23×C4×C71C2×C412yes—
85C23×C4×C71D411yescharacteristic
86C23×C4×C71D411yescharacteristic
87C23×C4×C71D411yescharacteristic
88C23×C4×C71D412yes—
89C23×C4×C71D411yescharacteristic
90C23×C4×C71D412yes—
91C23×C4×C71C2311yescharacteristic
92C23×C4×C71C2312yes—
93C2×M16×C71—24no—
94C2×M16×C71—44no—
95C22×C8×C71—22no—
96C22⋊C8×C71—24no—
97C22⋊C8×C71—22no—
98C22⋊C8×C71C2314yes—
99C23×D71—21no—
100C23×D71—21no—
101C23×D71—24no—
102C23×D71—42no—
103C23×D71—42no—
104C23×D71C22×C412yes—
105C2×C22⋊D71—24no—
106C2×C22⋊D71—24no—
107C2×C22⋊D71—48no—
108C2×C22⋊D71—44no—
109C2×C22⋊D71—44no—
110C2×C22⋊D71—48no—
111C22×Dic71—21no—
112C22×Dic71—21no—
113C22×Dic71—21no—
114C22×Dic71—21no—
115C22×Dic71—21no—
116C22×Dic71—21no—
117C22×Dic71—24no—
118C22×Dic71—42no—
119C22×Dic71—42no—
120C22×Dic71C22×C412yes—
121C22×Dic71C22×C412yes—
122C22×Dic71C22×C412yes—
123C2×C71⋊C8—24no—
124C2×C71⋊C8—44no—
125C2×C71⋊C8—48no—
126Dic71⋊C4—24no—
127Dic71⋊C4—24no—
128Dic71⋊C4—22no—
129Dic71⋊C4—22no—
130Dic71⋊C4—22no—
131Dic71⋊C4—22no—
132C22⋊Dic71—24no—
133C22⋊Dic71—24no—
134(C2×D71)⋊C4—24no—
135(C2×D71)⋊C4—24no—
136(C2×D71)⋊C4—22no—
137(C2×D71)⋊C4—22no—
138(C2×D71)⋊C4—22no—
139(C2×D71)⋊C4—22no—
140C24×C71—24no—
141C24×C71—24no—
142C24×C71—42no—
143C24×C71C22⋊C411yescharacteristic
144C24×C71C22⋊C412yes—
145C24×C71C22⋊C411yescharacteristic
146C24×C71C22×C412yes—
147C24×C71C22×C411yescharacteristic
148C22×C4×C71—24no—
149C22×C4×C71—24no—
150C22×C4×C71—42no—
151C22×C4×C71—44no—
152C22×C4×C71—44no—
153C22×C4×C71—42no—
154C22×C4×C71—42no—
155C22×C4×C71—44no—
156C22×C4×C71—42no—
157C22×C4×C71—44no—
158C22×C4×C71C22⋊C411yescharacteristic
159C22×C4×C71C22⋊C411yescharacteristic
160C22×C4×C71C22⋊C412yes—
161C22×C4×C71C22⋊C412yes—
162C22×C4×C71C22⋊C412yes—
163C22×C4×C71C22⋊C412yes—
164C22×C4×C71C22⋊C412yes—
165C22×C4×C71C22×C412yes—
166C22×C4×C71C22×C411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D412yes—
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2×D412yes—
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D412yes—
175C22×C4×C71C2411yescharacteristic
176M16×C71—48no—
177C2×C8×C71—24no—
178C2×C8×C71—44no—
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—42no—
191C23×C71—42no—
192C23×C71—416no—
193C23×C71#9914yes—
194C23×C71C2×C22⋊C412yes—
195C23×C71C2×C22⋊C411yescharacteristic
196C23×C71C2×C22⋊C412yes—
197C23×C71C2×C22⋊C411yescharacteristic
198C23×C71#13114yes—
199C23×C71C23×C411yescharacteristic
200C22×D71—22no—
201C22×D71—24no—
202C22×D71—42no—
203C22×D71—42no—
204C22×D71—42no—
205C22×D71—42no—
206C22×D71—42no—
207C22×D71—42no—
208C22×D71—42no—
209C22×D71—42no—
210C22×D71—88no—
211C2×C4×C71—22no—
212C2×C4×C71—24no—
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—42no—
220C2×C4×C71—42no—
221C2×C4×C71—42no—
222C2×C4×C71—41no—
223C2×C4×C71—42no—
224C2×C4×C71—41no—
225C2×C4×C71—42no—
226C2×C4×C71—41no—
227C2×C4×C71—42no—
228C2×C4×C71—41no—
229C2×C4×C71C2×C22⋊C412yes—
230C2×C4×C71C2×C22⋊C412yes—
231C2×C4×C71C2×C22⋊C411yescharacteristic
232C2×C4×C71C2×C22⋊C411yescharacteristic
233C2×C4×C71C22×D411yescharacteristic
234C2×C4×C71C22×D411yescharacteristic
235C22⋊D71—48no—
236C22⋊D71—48no—
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—24no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—42no—
246C2×Dic71—42no—
247C2×Dic71—42no—
248C2×Dic71—42no—
249C8×C71—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#40812yes—
266C22×C71#40812yes—
267C22×C71C2×#9914yes—
268C22×C71C2×#9912yes—
269C22×C71#41312yes—
270C22×C71C22×C22⋊C411yescharacteristic
271C22×C71C2×#13112yes—
272C2×D71—42no—
273C2×D71—42no—
274C2×D71—42no—
275C2×D71—84no—
276C2×D71—84no—
277C2×D71—84no—
278C4×C71—21no—
279C4×C71—21no—
280C4×C71—21no—
281C4×C71—22no—
282C4×C71—21no—
283C4×C71—22no—
284Dic71—42no—
285Dic71—42no—
286Dic71—42no—
287C2×#1468—711nomaximal, Sylow
288C2×C71—22no—
289C2×C71—22no—
290C2×C71—24no—
291C2×C71—42no—
292C2×C71#146814yescomplemented
293C2×C71C2×#40811yescharacteristic
294C2×C71C22×#9911yescharacteristic
295C2×C71C2×#41311yescharacteristic
296D71—82no—
297C22×#123—711no—
298C2×#407D7111yescharacteristic, complemented
299C22×C22⋊C8—711no—
300#1468—718no—
301C2×#324—712no—
302C2×#323—712no—
303C71C2×#146811yescharacteristic, complemented, Sylow
304C23×D4—711no—
305C24×C4C2×D7111yescharacteristic
306C22×M16—21no—
307C23×C8—1421no—
308C2×#123—714no—
309C2×#123—714no—
310C2×#123—1421no—
311C2×#123—1422no—
312C2×#123—1421no—
313C2×#123—1421no—
314C2×#123—1421no—
315C2×#123—1422no—
316C22×C4⋊C4—711no—
317C22×C4⋊C4—711no—
318C22×C22⋊C4—711no—
319C22×C22⋊C4—711no—
320C22×C22⋊C4—711no—
321#407—22no—
322#407C2×D7114yescomplemented
323C2×C22⋊C8—21no—
324C2×C22⋊C8—712no—
325C2×C22⋊C8—714no—
326C2×C22⋊C8—1422no—
327C2×C22⋊C8—1421no—
328C2×C22⋊C8C2×D7112yes—
329#324—718no—
330#323—718no—
331C25—711no—
332C25C4×D7111yescharacteristic
333C22×D4—2844no—
334C22×D4—714no—
335C22×D4—1422no—
336C22×D4—1421no—
337C22×D4—1421no—
338C23×C4—44no—
339C23×C4—711no—
340C23×C4—711no—
341C23×C4—711no—
342C23×C4C4×D7111yescharacteristic
343C23×C4C4×D7112yes—
344C23×C4D28411yescharacteristic
345C23×C4D28412yes—
346C23×C4D28411yescharacteristic
347C23×C4C22⋊D7111yescharacteristic
348C23×C4C22⋊D7111yescharacteristic
349C23×C4C22⋊D7112yes—
350C23×C4C22×D7111yescharacteristic
351C23×C4C22×D7112yes—
352C2×M16—24no—
353C2×M16—44no—
354C22×C8—22no—
355C22×C8—2844no—
356C22×C8—1424no—
357C22×C8—1422no—
358#123—1424no—
359#123—1424no—
360#123—1424no—
361#123—1424no—
362#123—1424no—
363#123—1424no—
364#123—1424no—
365#123—1424no—
366C2×C4⋊C4—714no—
367C2×C4⋊C4—712no—
368C2×C4⋊C4—712no—
369C2×C4⋊C4—1422no—
370C2×C4⋊C4—1421no—
371C2×C4⋊C4—1421no—
372C2×C4⋊C4—1421no—
373C2×C4⋊C4—1421no—
374C2×C4⋊C4—1421no—
375C2×C4⋊C4—1421no—
376C2×C22⋊C4—714no—
377C2×C22⋊C4—712no—
378C2×C22⋊C4—714no—
379C2×C22⋊C4—712no—
380C2×C22⋊C4—1422no—
381C2×C22⋊C4—1421no—
382C2×C22⋊C4—1421no—
383C2×C22⋊C4—1422no—
384C2×C22⋊C4—1421no—
385C2×C22⋊C4—1421no—
386C2×C22⋊C4—1421no—
387C2×C22⋊C4—1421no—
388C2×C22⋊C4—1421no—
389C2×C22⋊C4—1421no—
390C22⋊C8—24no—
391C22⋊C8—22no—
392C22⋊C8—714no—
393C22⋊C8—1422no—
394C22⋊C8—1424no—
395C22⋊C8C22×D7114yes—
396C24—24no—
397C24—24no—
398C24—42no—
399C24—2842no—
400C24—2842no—
401C24—712no—
402C24—1421no—
403C24—1421no—
404C24—1424no—
405C24(C2×D71)⋊C411yescharacteristic
406C24(C2×D71)⋊C412yes—
407C24(C2×D71)⋊C411yescharacteristic
408C24C2×C4×D7112yes—
409C24C2×C4×D7111yescharacteristic
410C2×D4—2848no—
411C2×D4—2844no—
412C2×D4—2844no—
413C2×D4—2848no—
414C2×D4—1424no—
415C2×D4—1424no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—2842no—
427C22×C4—2842no—
428C22×C4—712no—
429C22×C4—712no—
430C22×C4—712no—
431C22×C4—1421no—
432C22×C4—1421no—
433C22×C4—1421no—
434C22×C4—1421no—
435C22×C4—1421no—
436C22×C4—1421no—
437C22×C4—1424no—
438C22×C4(C2×D71)⋊C412yes—
439C22×C4(C2×D71)⋊C412yes—
440C22×C4(C2×D71)⋊C411yescharacteristic
441C22×C4(C2×D71)⋊C411yescharacteristic
442C22×C4(C2×D71)⋊C412yes—
443C22×C4(C2×D71)⋊C412yes—
444C22×C4(C2×D71)⋊C412yes—
445C22×C4C2×C4×D7112yes—
446C22×C4C2×C4×D7111yescharacteristic
447C22×C4C2×C22⋊D7112yes—
448C22×C4C2×C22⋊D7111yescharacteristic
449C22×C4C2×C22⋊D7111yescharacteristic
450C22×C4C2×C22⋊D7112yes—
451C22×C4C2×D28412yes—
452C22×C4C2×D28411yescharacteristic
453C22×C4C2×D28411yescharacteristic
454C22×C4C2×D28412yes—
455C22×C4C23×D7111yescharacteristic
456M16—48no—
457C2×C8—24no—
458C2×C8—44no—
459C2×C8—2848no—
460C2×C8—2844no—
461C2×C8—1424no—
462C4⋊C4—1422no—
463C4⋊C4—1422no—
464C4⋊C4—1424no—
465C4⋊C4—1424no—
466C4⋊C4—1422no—
467C4⋊C4—1422no—
468C22⋊C4—1424no—
469C22⋊C4—1424no—
470C22⋊C4—1422no—
471C22⋊C4—1422no—
472C22⋊C4—1424no—
473C22⋊C4—1424no—
474C22⋊C4—1422no—
475C22⋊C4—1422no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—416no—
490C23—2842no—
491C23—2842no—
492C23—2842no—
493C23—2842no—
494C23—2842no—
495C23—2842no—
496C23—2842no—
497C23—2842no—
498C23—5688no—
499C23—1422no—
500C23—1424no—
501C23#1455604214yes—
502C23C2×(C2×D71)⋊C412yes—
503C23C2×(C2×D71)⋊C411yescharacteristic
504C23C2×(C2×D71)⋊C412yes—
505C23C2×(C2×D71)⋊C411yescharacteristic
506C23#1455613614yes—
507C23C22×C4×D7111yescharacteristic
508D4—2848no—
509D4—2848no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—2842no—
529C2×C4—2842no—
530C2×C4—2842no—
531C2×C4—2842no—
532C2×C4—2842no—
533C2×C4—2842no—
534C2×C4—2842no—
535C2×C4—2842no—
536C2×C4—1422no—
537C2×C4—1422no—
538C2×C4—1422no—
539C2×C4—1424no—
540C2×C4C2×(C2×D71)⋊C411yescharacteristic
541C2×C4C2×(C2×D71)⋊C411yescharacteristic
542C2×C4C2×(C2×D71)⋊C412yes—
543C2×C4C2×(C2×D71)⋊C412yes—
544C2×C4C22×C22⋊D7111yescharacteristic
545C2×C4C22×D28411yescharacteristic
546C8—44no—
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—2842no—
563C22—2842no—
564C22—2842no—
565C22—5684no—
566C22—5684no—
567C22—5684no—
568C22#1798596712yes—
569C22#1798596712yes—
570C22C2×#1455604214yes—
571C22C2×#1455604212yes—
572C22#1798598112yes—
573C22C22×(C2×D71)⋊C411yescharacteristic
574C22C2×#1455613612yes—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—21no—
580C4—22no—
581C4—2842no—
582C4—2842no—
583C4—2842no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—5682no—
589C2#2258668914yesdirect product factor
590C2C2×#1798596711yescharacteristic
591C2C22×#1455604211yescharacteristic
592C2C2×#1798598111yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC23×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#407×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC23×C22⋊D71C4
cosocleC22×C4×C71C24
hypercenterC2×#407D71
hypocenterC71C2×#1468
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊(C71⋊C8)C2semidirect
3. #22586689C2direct (N × H)
4. C2×#17985724C2semidirect
5. C2×#17985722C2semidirect
6. C2×#407×C71C2semidirect
25. C2×C22⋊(C71⋊C8)C22semidirect
26. #17985724C22semidirect
27. #17985722C22semidirect
31. #407×C71C22semidirect
292. C2×C71#1468semidirect
298. C2×#407D71semidirect
303. C71C2×#1468semidirect
322. #407C2×D71semidirect
589. C2#22586689direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561112
C22×C22⋊(C71⋊C8)2
#225866892
C2×#179857242
C2×#179857222
C2×#407×C712
C2×#146871