Finite Group Databasebeta

← C2×#22586829 #31964544

Subgroups — C2×#22586829 #31964544

Order 18176.

1606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986510C211yesmaximal, characteristic
2C22×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
3#22586829C218yesmaximal, direct product factor
4C2×#17985735C212yesmaximal, complemented
5C2×#17985732C212yesmaximal, complemented
6C2×C4×C22⋊C4×C71C211yesmaximal, characteristic, complemented
7C22×#11765865C411yescharacteristic, complemented
8C23×C71⋊C8—21no—
9#17986510—22no—
10#17986510—22no—
11#17986510C414yescomplemented
12#17986510C2214yes—
13C2×#14556111—21no—
14C2×#14556107C411yescharacteristic, complemented
15C2×#14556107C2211yescharacteristic
16C2×#14556102C411yescharacteristic, complemented
17C2×#14556102C2211yescharacteristic
18C2×#14556097—21no—
19C2×C4×C22⋊D71—21no—
20C2×C4×C22⋊D71—21no—
21C2×#14556087C411yescharacteristic, complemented
22C2×C22⋊(C71⋊C8)—21no—
23C2×C22⋊(C71⋊C8)—22no—
24C2×C22⋊(C71⋊C8)C2212yes—
25C2×C22⋊(C71⋊C8)C2214yescomplemented
26#17985735C2218yescomplemented
27#17985732C2218yescomplemented
28C24×C4×C71C2211yescharacteristic
29C22×C22⋊C4×C71—21no—
30C22×C42×C71—21no—
31C4×C22⋊C4×C71—22no—
32C4×C22⋊C4×C71C2214yescomplemented
33C2×#95×C71C2212yes—
34C2×#11765865—22no—
35C2×#11765865—44no—
36C2×#11765865C2×C414yescomplemented
37C22×Dic142C2×C411yescharacteristic
38C22×D284C2×C411yescharacteristic
39C22×C22⋊D71—21no—
40C22×C22⋊D71—21no—
41C22×C22⋊D71—21no—
42C22×C4×D71—21no—
43C22×C71⋊C8—22no—
44C22×C71⋊C8—24no—
45C22×C71⋊C8—44no—
46#14556111—24no—
47#14556111—24no—
48#14556107—22no—
49#14556107—21no—
50#14556107—21no—
51#14556107C2×C412yes—
52#14556107C2×C414yescomplemented
53#14556107C2312yes—
54#14556102—22no—
55#14556102—21no—
56#14556102—21no—
57#14556102C2×C412yes—
58#14556102C2×C414yescomplemented
59#14556102C2312yes—
60#14556097—24no—
61#14556097—24no—
62C4×C22⋊D71—24no—
63C4×C22⋊D71—24no—
64C4×C22⋊D71—24no—
65C4×C22⋊D71—24no—
66#14556087—22no—
67#14556087C2×C414yescomplemented
68C2×C4⋊Dic71C2×C411yescharacteristic
69C2×C4⋊Dic71C2×C411yescharacteristic
70C2×Dic71⋊C4—21no—
71C2×Dic71⋊C4—21no—
72C2×C22⋊Dic71—21no—
73C2×C22⋊Dic71—21no—
74C2×C22⋊Dic71—21no—
75C2×(C2×D71)⋊C4—21no—
76C2×(C2×D71)⋊C4—21no—
77C2×C4×Dic71—21no—
78C22⋊(C71⋊C8)—22no—
79C22⋊(C71⋊C8)—24no—
80C22⋊(C71⋊C8)C2314yes—
81C25×C71D411yescharacteristic
82C23×C4×C71—22no—
83C23×C4×C71—44no—
84C23×C4×C71C2×C411yescharacteristic
85C23×C4×C71C2×C411yescharacteristic
86C23×C4×C71C2×C412yes—
87C23×C4×C71D411yescharacteristic
88C23×C4×C71D411yescharacteristic
89C23×C4×C71D412yes—
90C23×C4×C71D411yescharacteristic
91C23×C4×C71D412yes—
92C23×C4×C71C2311yescharacteristic
93C23×C4×C71C2312yes—
94C2×C22⋊C4×C71—28no—
95C2×C22⋊C4×C71—24no—
96C2×C42×C71—24no—
97C2×C42×C71—44no—
98#95×C71—22no—
99#95×C71C2314yes—
100C23×D71—21no—
101#11765865—48no—
102#11765865—48no—
103C2×Dic142—21no—
104C2×Dic142—42no—
105C2×Dic142C22×C412yes—
106C2×D284—21no—
107C2×D284—42no—
108C2×D284C22×C412yes—
109C2×C22⋊D71—24no—
110C2×C22⋊D71—22no—
111C2×C22⋊D71—24no—
112C2×C22⋊D71—22no—
113C2×C22⋊D71—44no—
114C2×C22⋊D71—44no—
115C2×C22⋊D71—44no—
116C22×Dic71—21no—
117C22×Dic71—21no—
118C22×Dic71—21no—
119C2×C4×D71—24no—
120C2×C4×D71—44no—
121C2×C71⋊C8—24no—
122C2×C71⋊C8—44no—
123C2×C71⋊C8—48no—
124C4⋊Dic71—21no—
125C4⋊Dic71—21no—
126C4⋊Dic71C22×C412yes—
127C4⋊Dic71C22×C412yes—
128Dic71⋊C4—22no—
129Dic71⋊C4—24no—
130Dic71⋊C4—22no—
131C22⋊Dic71—24no—
132C22⋊Dic71—22no—
133C22⋊Dic71—24no—
134C22⋊Dic71—22no—
135(C2×D71)⋊C4—22no—
136(C2×D71)⋊C4—24no—
137(C2×D71)⋊C4—22no—
138C4×Dic71—24no—
139C24×C71—24no—
140C24×C71—24no—
141C24×C71—42no—
142C24×C71C22⋊C411yescharacteristic
143C24×C71C22⋊C412yes—
144C24×C71C22⋊C411yescharacteristic
145C24×C71C2×D412yes—
146C24×C71C2×D411yescharacteristic
147C22×C4×C71—24no—
148C22×C4×C71—24no—
149C22×C4×C71—24no—
150C22×C4×C71—28no—
151C22×C4×C71—24no—
152C22×C4×C71—44no—
153C22×C4×C71—44no—
154C22×C4×C71—44no—
155C22×C4×C71—44no—
156C22×C4×C71—42no—
157C22×C4×C71—42no—
158C22×C4×C71—42no—
159C22×C4×C71—44no—
160C22×C4×C71—44no—
161C22×C4×C71C22⋊C412yes—
162C22×C4×C71C22⋊C411yescharacteristic
163C22×C4×C71C22⋊C412yes—
164C22×C4×C71C22⋊C412yes—
165C22×C4×C71C22⋊C412yes—
166C22×C4×C71C22⋊C412yes—
167C22×C4×C71C22⋊C411yescharacteristic
168C22×C4×C71C22×C411yescharacteristic
169C22×C4×C71C22×C412yes—
170C22×C4×C71C22×C411yescharacteristic
171C22×C4×C71C22×C412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D412yes—
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D412yes—
178C22×C4×C71C2411yescharacteristic
179C22⋊C4×C71—216no—
180C42×C71—48no—
181C23×C71—22no—
182C23×C71—22no—
183C23×C71—22no—
184C23×C71—22no—
185C23×C71—24no—
186C23×C71—22no—
187C23×C71—24no—
188C23×C71—22no—
189C23×C71—24no—
190C23×C71—24no—
191C23×C71—22no—
192C23×C71—42no—
193C23×C71—42no—
194C23×C71—416no—
195C23×C71#10414yes—
196C23×C71C2×C22⋊C412yes—
197C23×C71C2×C22⋊C411yescharacteristic
198C23×C71C2×C22⋊C412yes—
199C23×C71C2×C22⋊C411yescharacteristic
200C23×C71#13514yes—
201C23×C71C22×D411yescharacteristic
202C22×D71—22no—
203C22×D71—42no—
204C22×D71—42no—
205C22×D71—42no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—24no—
210C2×C4×C71—24no—
211C2×C4×C71—24no—
212C2×C4×C71—24no—
213C2×C4×C71—24no—
214C2×C4×C71—24no—
215C2×C4×C71—28no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—41no—
219C2×C4×C71—42no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—42no—
223C2×C4×C71—41no—
224C2×C4×C71—44no—
225C2×C4×C71—416no—
226C2×C4×C71—44no—
227C2×C4×C71#9914yes—
228C2×C4×C71C2×C22⋊C412yes—
229C2×C4×C71C2×C22⋊C411yescharacteristic
230C2×C4×C71C2×C22⋊C411yescharacteristic
231C2×C4×C71C2×C22⋊C412yes—
232C2×C4×C71C23×C411yescharacteristic
233C2×C4×C71C22×D411yescharacteristic
234C22⋊D71—48no—
235C22⋊D71—44no—
236C22⋊D71—44no—
237C22⋊D71—48no—
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—22no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244D284—42no—
245D284—42no—
246C4×D71—48no—
247Dic142—42no—
248Dic142—42no—
249C71⋊C8—44no—
250C22×C71—21no—
251C22×C71—21no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71—21no—
255C22×C71—21no—
256C22×C71—28no—
257C22×C71—28no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—48no—
262C22×C71—48no—
263C22×C71—44no—
264C22×C71C2×#9912yes—
265C22×C71#41612yes—
266C22×C71#41612yes—
267C22×C71C2×#10414yes—
268C22×C71C2×#10412yes—
269C22×C71C22×C22⋊C411yescharacteristic
270C22×C71C2×#13512yes—
271C2×D71—42no—
272C2×D71—42no—
273C2×D71—42no—
274C2×D71—84no—
275C4×C71—21no—
276C4×C71—21no—
277C4×C71—21no—
278C4×C71—22no—
279C4×C71—21no—
280C4×C71—48no—
281C4×C71C2×#9914yes—
282Dic71—42no—
283Dic71—42no—
284Dic71—42no—
285C2×#1501—711nomaximal, Sylow
286C2×C71—22no—
287C2×C71—22no—
288C2×C71—24no—
289C2×C71—42no—
290C2×C71#150114yescomplemented
291C2×C71C22×#9911yescharacteristic
292C2×C71C2×#41611yescharacteristic
293C2×C71C22×#10411yescharacteristic
294D71—82no—
295C2×#525—711no—
296C22×C22⋊C8—711no—
297C2×C4×C22⋊C4D7111yescharacteristic, complemented
298#1501—718no—
299C2×#328—712no—
300C2×#327—712no—
301C71C2×#150111yescharacteristic, complemented, Sylow
302C22×#41—711no—
303C24×C4C2×D7111yescharacteristic
304C23×C8—1421no—
305#525—714no—
306#525—714no—
307#525—1422no—
308#525—1422no—
309C2×#123—1421no—
310C2×C22⋊Q8—711no—
311C2×C22⋊Q8—711no—
312C2×#121—711no—
313C2×#121—711no—
314C2×#120—1421no—
315C2×C4×D4—1421no—
316C2×C4×D4—1421no—
317C2×#117—711no—
318C22×C22⋊C4—21no—
319C22×C42—21no—
320C2×C22⋊C8—712no—
321C2×C22⋊C8—714no—
322C2×C22⋊C8—1421no—
323C2×C22⋊C8—1422no—
324C4×C22⋊C4—22no—
325C4×C22⋊C4C2×D7114yescomplemented
326C2×#95C2×D7112yes—
327#328—718no—
328#327—718no—
329C25C22⋊D7111yescharacteristic
330C2×#41—2844no—
331C2×#41—714no—
332C2×#41—1422no—
333C22×Q8—711no—
334C22×D4—711no—
335C22×D4—1421no—
336C22×D4—1421no—
337C22×D4—1421no—
338C23×C4—22no—
339C23×C4—44no—
340C23×C4—1421no—
341C23×C4C4×D7111yescharacteristic
342C23×C4C4×D7111yescharacteristic
343C23×C4C4×D7112yes—
344C23×C4D28411yescharacteristic
345C23×C4D28411yescharacteristic
346C23×C4D28412yes—
347C23×C4C22⋊D7111yescharacteristic
348C23×C4C22⋊D7112yes—
349C23×C4C22×D7111yescharacteristic
350C23×C4C22×D7112yes—
351C22×C8—2844no—
352C22×C8—1422no—
353C22×C8—1424no—
354#123—1424no—
355#123—1424no—
356C22⋊Q8—712no—
357C22⋊Q8—714no—
358C22⋊Q8—712no—
359C22⋊Q8—1422no—
360C22⋊Q8—1421no—
361C22⋊Q8—1421no—
362#121—712no—
363#121—714no—
364#121—712no—
365#121—1422no—
366#121—1421no—
367#121—1421no—
368#120—1424no—
369#120—1424no—
370C4×D4—1424no—
371C4×D4—1424no—
372C4×D4—1424no—
373C4×D4—1424no—
374#117—714no—
375#117—1422no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—711no—
378C2×C4⋊C4—1421no—
379C2×C4⋊C4—1421no—
380C2×C22⋊C4—28no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—1421no—
383C2×C22⋊C4—1421no—
384C2×C22⋊C4—1421no—
385C2×C22⋊C4—1421no—
386C2×C22⋊C4—1421no—
387C2×C42—24no—
388C2×C42—44no—
389C2×C42—1421no—
390C22⋊C8—714no—
391C22⋊C8—1422no—
392C22⋊C8—1424no—
393#95—22no—
394#95C22×D7114yes—
395C24—24no—
396C24—24no—
397C24—42no—
398C24—1421no—
399C24(C2×D71)⋊C411yescharacteristic
400C24(C2×D71)⋊C412yes—
401C24(C2×D71)⋊C411yescharacteristic
402C24C2×C22⋊D7112yes—
403C24C2×C22⋊D7111yescharacteristic
404#41—2848no—
405#41—2848no—
406C2×Q8—2842no—
407C2×Q8—712no—
408C2×Q8—1421no—
409C2×D4—2842no—
410C2×D4—2844no—
411C2×D4—2844no—
412C2×D4—2844no—
413C2×D4—712no—
414C2×D4—1422no—
415C2×D4—1422no—
416C2×D4—1424no—
417C2×D4—1424no—
418C2×D4—1421no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—2844no—
434C22×C4—1421no—
435C22×C4—1421no—
436C22×C4—1421no—
437C22×C4—1424no—
438C22×C4(C2×D71)⋊C411yescharacteristic
439C22×C4(C2×D71)⋊C412yes—
440C22×C4(C2×D71)⋊C412yes—
441C22×C4(C2×D71)⋊C412yes—
442C22×C4(C2×D71)⋊C412yes—
443C22×C4(C2×D71)⋊C412yes—
444C22×C4(C2×D71)⋊C411yescharacteristic
445C22×C4C2×C4×D7111yescharacteristic
446C22×C4C2×C4×D7112yes—
447C22×C4C2×C4×D7111yescharacteristic
448C22×C4C2×C4×D7112yes—
449C22×C4C2×C22⋊D7112yes—
450C22×C4C2×C22⋊D7111yescharacteristic
451C22×C4C2×D28411yescharacteristic
452C22×C4C2×D28412yes—
453C22×C4C2×D28411yescharacteristic
454C22×C4C2×D28412yes—
455C22×C4C23×D7111yescharacteristic
456C2×C8—2844no—
457C2×C8—2848no—
458C2×C8—1424no—
459C4⋊C4—712no—
460C4⋊C4—712no—
461C4⋊C4—1422no—
462C4⋊C4—1422no—
463C4⋊C4—1424no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C22⋊C4—216no—
467C22⋊C4—1422no—
468C22⋊C4—1424no—
469C22⋊C4—1422no—
470C22⋊C4—1424no—
471C22⋊C4—1422no—
472C22⋊C4—1422no—
473C22⋊C4—1424no—
474C42—48no—
475C42—1424no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—42no—
488C23—42no—
489C23—416no—
490C23—2842no—
491C23—2842no—
492C23—2842no—
493C23—1422no—
494C23#1455605614yes—
495C23C2×(C2×D71)⋊C412yes—
496C23C2×(C2×D71)⋊C411yescharacteristic
497C23C2×(C2×D71)⋊C412yes—
498C23C2×(C2×D71)⋊C411yescharacteristic
499C23#1455614814yes—
500C23C22×C22⋊D7111yescharacteristic
501Q8—2842no—
502Q8—2842no—
503D4—2842no—
504D4—2842no—
505D4—2848no—
506D4—2844no—
507D4—2844no—
508D4—2848no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—416no—
529C2×C4—44no—
530C2×C4—2842no—
531C2×C4—2842no—
532C2×C4—2842no—
533C2×C4—2848no—
534C2×C4—1422no—
535C2×C4—1422no—
536C2×C4—1422no—
537C2×C4#1455604214yes—
538C2×C4C2×(C2×D71)⋊C412yes—
539C2×C4C2×(C2×D71)⋊C411yescharacteristic
540C2×C4C2×(C2×D71)⋊C411yescharacteristic
541C2×C4C2×(C2×D71)⋊C412yes—
542C2×C4C22×C4×D7111yescharacteristic
543C2×C4C22×D28411yescharacteristic
544C8—2844no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—48no—
557C22—48no—
558C22—44no—
559C22—2842no—
560C22—2842no—
561C22—2842no—
562C22—5684no—
563C22C2×#1455604212yes—
564C22#1798599312yes—
565C22#1798599312yes—
566C22C2×#1455605614yes—
567C22C2×#1455605612yes—
568C22C22×(C2×D71)⋊C411yescharacteristic
569C22C2×#1455614812yes—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—48no—
576C4—2842no—
577C4—2842no—
578C4—2842no—
579C4C2×#1455604214yes—
580C2—22no—
581C2—22no—
582C2—24no—
583C2—42no—
584C2—5682no—
585C2#2258682914yesdirect product factor
586C2C22×#1455604211yescharacteristic
587C2C2×#1798599311yescharacteristic
588C2C22×#1455605611yescharacteristic
subgroupG/H
centerC22×C4(C2×D71)⋊C4
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×C4×C22⋊C4×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#17986510C2
cosocleC22×C4×C71C24
hypercenterC2×C4×C22⋊C4D71
hypocenterC71C2×#1501
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊(C71⋊C8)C2semidirect
3. #22586829C2direct (N × H)
4. C2×#17985735C2semidirect
5. C2×#17985732C2semidirect
6. C2×C4×C22⋊C4×C71C2semidirect
7. C22×#11765865C4semidirect
11. #17986510C4semidirect
14. C2×#14556107C4semidirect
16. C2×#14556102C4semidirect
21. C2×#14556087C4semidirect
25. C2×C22⋊(C71⋊C8)C22semidirect
26. #17985735C22semidirect
27. #17985732C22semidirect
32. C4×C22⋊C4×C71C22semidirect
36. C2×#11765865C2×C4semidirect
52. #14556107C2×C4semidirect
58. #14556102C2×C4semidirect
67. #14556087C2×C4semidirect
290. C2×C71#1501semidirect
297. C2×C4×C22⋊C4D71semidirect
301. C71C2×#1501semidirect
325. C4×C22⋊C4C2×D71semidirect
585. C2#22586829direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179865102
C22×C22⋊(C71⋊C8)2
#225868292
C2×#179857352
C2×#179857322
C2×C4×C22⋊C4×C712
C2×#150171