Finite Group Databasebeta

← C2×#22588649 #31998655

Subgroups — C2×#22588649 #31998655

Order 18176.

1966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556104C211yesmaximal, characteristic
2C22×#14556086C211yesmaximal, characteristic
3C22×#14556052C211yesmaximal, characteristic, complemented
4C22×(C2×D71)⋊C8C211yesmaximal, characteristic
5#22588649C218yesmaximal, direct product factor
6C2×#17985762C211yesmaximal, characteristic, complemented
7C2×#17985761C211yesmaximal, characteristic, complemented
8C22×#103×C71C211yesmaximal, characteristic, complemented
9C23×C4×D71C2211yescharacteristic
10C23×C71⋊C8C2211yescharacteristic
11C2×#14556104—22no—
12C2×#14556104—24no—
13C2×#14556104C418yescomplemented
14C2×#14556104C2214yes—
15C2×#14556086—24no—
16C2×#14556086—24no—
17C2×#14556086—22no—
18C2×#14556086C2214yes—
19C22×C4⋊Dic71C2211yescharacteristic
20C22×C4⋊Dic71C2211yescharacteristic
21C22×Dic71⋊C4—21no—
22C22×(C2×D71)⋊C4—21no—
23C22×(C2×D71)⋊C4—21no—
24C22×C4×Dic71—21no—
25C2×#14556052—24no—
26C2×#14556052C2214yescomplemented
27C2×(C2×D71)⋊C8—24no—
28C2×(C2×D71)⋊C8C2214yes—
29#17985762C2214yescomplemented
30#17985761C2214yescomplemented
31C23×Q8×C71C2211yescharacteristic
32C23×C8×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C2×#103×C71—24no—
35C2×#103×C71C2214yescomplemented
36C24×D71D411yescharacteristic
37C23×Dic71—21no—
38C23×Dic71—21no—
39C23×Dic71D411yescharacteristic
40C22×C4×D71—22no—
41C22×C4×D71—22no—
42C22×C4×D71—22no—
43C22×C4×D71—22no—
44C22×C4×D71C2×C414yes—
45C22×C4×D71D412yes—
46C22×C4×D71D414yes—
47C22×C4×D71C2312yes—
48C22×C71⋊C8—22no—
49C22×C71⋊C8—22no—
50C22×C71⋊C8—22no—
51C22×C71⋊C8C2312yes—
52#14556104—216no—
53#14556104—28no—
54#14556104C2×C4116yescomplemented
55#14556086—216no—
56#14556086—28no—
57#14556086—28no—
58C2×C4⋊Dic71—21no—
59C2×C4⋊Dic71—22no—
60C2×C4⋊Dic71—21no—
61C2×C4⋊Dic71—22no—
62C2×C4⋊Dic71C2×C414yes—
63C2×C4⋊Dic71D414yes—
64C2×C4⋊Dic71C2312yes—
65C2×C4⋊Dic71C2312yes—
66C2×Dic71⋊C4—28no—
67C2×Dic71⋊C4—24no—
68C2×(C2×D71)⋊C4—28no—
69C2×(C2×D71)⋊C4—28no—
70C2×(C2×D71)⋊C4—24no—
71C2×(C2×D71)⋊C4—24no—
72C2×C4×Dic71—28no—
73C2×C4×Dic71—24no—
74#14556052—28no—
75(C2×D71)⋊C8—28no—
76C22×Q8×C71—22no—
77C22×Q8×C71—21no—
78C22×Q8×C71—22no—
79C22×Q8×C71—22no—
80C22×Q8×C71—22no—
81C22×Q8×C71—22no—
82C22×Q8×C71C2×C414yes—
83C22×Q8×C71C2312yes—
84C23×C4×C71—21no—
85C23×C4×C71—21no—
86C23×C4×C71C2311yescharacteristic
87C22×C8×C71—22no—
88C22×C8×C71—22no—
89C22×C8×C71—22no—
90C22×C8×C71C2312yes—
91C2×C4⋊C4×C71—21no—
92C2×C4⋊C4×C71—22no—
93C2×C4⋊C4×C71D414yes—
94C2×C4⋊C4×C71C2312yes—
95#103×C71—28no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71—22no—
102C23×D71C22⋊C414yes—
103C23×D71C2×D412yes—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—24no—
107C22×Dic71—24no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—42no—
111C22×Dic71—42no—
112C22×Dic71—42no—
113C22×Dic71—42no—
114C22×Dic71C22⋊C414yes—
115C22×Dic71C2×D412yes—
116C2×C4×D71—28no—
117C2×C4×D71—28no—
118C2×C4×D71—24no—
119C2×C4×D71—28no—
120C2×C4×D71—28no—
121C2×C4×D71—24no—
122C2×C4×D71—22no—
123C2×C4×D71—22no—
124C2×C4×D71—22no—
125C2×C4×D71—22no—
126C2×C4×D71C22⋊C418yes—
127C2×C4×D71C22×C414yes—
128C2×C4×D71C2×D414yes—
129C2×C71⋊C8—22no—
130C2×C71⋊C8—22no—
131C2×C71⋊C8—22no—
132C2×C71⋊C8—44no—
133C4⋊Dic71—24no—
134C4⋊Dic71—24no—
135C4⋊Dic71—22no—
136C4⋊Dic71—22no—
137C4⋊Dic71C22×C414yes—
138C4⋊Dic71C2×D414yes—
139Dic71⋊C4—216no—
140(C2×D71)⋊C4—216no—
141(C2×D71)⋊C4—216no—
142C4×Dic71—216no—
143C24×C71C2×D411yescharacteristic
144C2×Q8×C71—24no—
145C2×Q8×C71—22no—
146C2×Q8×C71—28no—
147C2×Q8×C71—28no—
148C2×Q8×C71—22no—
149C2×Q8×C71—22no—
150C2×Q8×C71—22no—
151C2×Q8×C71—22no—
152C2×Q8×C71—42no—
153C2×Q8×C71—44no—
154C2×Q8×C71—44no—
155C2×Q8×C71—42no—
156C2×Q8×C71C22×C414yes—
157C22×C4×C71—24no—
158C22×C4×C71—24no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—42no—
162C22×C4×C71—42no—
163C22×C4×C71—42no—
164C22×C4×C71—42no—
165C22×C4×C71C22×C412yes—
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D412yes—
168C22×C4×C71C2×D412yes—
169C22×C4×C71C2×D412yes—
170C22×C4×C71#4112yes—
171C22×C4×C71#4112yes—
172C22×C4×C71C2411yescharacteristic
173C2×C8×C71—22no—
174C2×C8×C71—22no—
175C2×C8×C71—22no—
176C2×C8×C71—44no—
177C4⋊C4×C71—24no—
178C4⋊C4×C71—22no—
179C4⋊C4×C71C2×D414yes—
180C23×C71C2×C22⋊C412yes—
181C23×C71#12012yes—
182C23×C71#12312yes—
183C23×C71#13512yes—
184C23×C71#13512yes—
185C23×C71#13712yes—
186C23×C71#13712yes—
187C23×C71C22×D411yescharacteristic
188C22×D71—22no—
189C22×D71—22no—
190C22×D71—22no—
191C22×D71—22no—
192C22×D71—22no—
193C22×D71—22no—
194C22×D71—28no—
195C22×D71—28no—
196C22×D71—28no—
197C22×D71—44no—
198C22×D71—44no—
199C22×D71—44no—
200C22×D71—44no—
201C22×D71C2×C22⋊C414yes—
202Q8×C71—24no—
203Q8×C71—24no—
204Q8×C71—42no—
205Q8×C71—42no—
206Q8×C71—48no—
207C2×C4×C71—24no—
208C2×C4×C71—24no—
209C2×C4×C71—42no—
210C2×C4×C71—42no—
211C2×C4×C71—42no—
212C2×C4×C71—42no—
213C2×C4×C71—48no—
214C2×C4×C71—48no—
215C2×C4×C71C2×C22⋊C412yes—
216C2×C4×C71C4×D414yes—
217C2×C4×C71C4×D414yes—
218C2×C4×C71#12012yes—
219C2×C4×C71#12114yes—
220C2×C4×C71#12114yes—
221C2×C4×C71#12312yes—
222C2×C4×C71C23×C411yescharacteristic
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C22×D411yescharacteristic
225C2×C4×C71C22×D411yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×C4×C71C2×#4111yescharacteristic
228C2×Dic71—22no—
229C2×Dic71—22no—
230C2×Dic71—28no—
231C2×Dic71—24no—
232C2×Dic71—24no—
233C2×Dic71—42no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—48no—
238C2×Dic71—48no—
239C2×Dic71C2×C22⋊C414yes—
240C4×D71—24no—
241C4×D71—24no—
242C4×D71—24no—
243C4×D71—24no—
244C4×D71—216no—
245C8×C71—42no—
246C71⋊C8—42no—
247C22×C71#38614yes—
248C22×C71#41614yes—
249C22×C71#41914yes—
250C22×C71#45214yes—
251C22×C71#45214yes—
252C22×C71#48514yes—
253C22×C71#48514yes—
254C22×C71C22×C22⋊C411yescharacteristic
255C22×C71C2×#12011yescharacteristic
256C22×C71C2×#12311yescharacteristic
257C22×C71C2×#13511yescharacteristic
258C22×C71C2×#13511yescharacteristic
259C22×C71C2×#13711yescharacteristic
260C22×C71C2×#13711yescharacteristic
261C2×D71—24no—
262C2×D71—24no—
263C2×D71—24no—
264C2×D71—42no—
265C2×D71—42no—
266C2×D71—416no—
267C2×D71—42no—
268C2×D71—42no—
269C4×C71—44no—
270C4×C71—44no—
271C4×C71#38614yes—
272C4×C71C2×C4×D411yescharacteristic
273C4×C71C2×C4×D411yescharacteristic
274C4×C71C2×#12111yescharacteristic
275C4×C71C2×#12111yescharacteristic
276Dic71—24no—
277Dic71—44no—
278Dic71—44no—
279C2×#1873—711nomaximal, Sylow
280C2×C71#187318yescomplemented
281C2×C71C2×#38611yescharacteristic
282C2×C71C2×#41611yescharacteristic
283C2×C71C2×#41911yescharacteristic
284C2×C71C2×#45211yescharacteristic
285C2×C71C2×#45211yescharacteristic
286C2×C71C2×#48511yescharacteristic
287C2×C71C2×#48511yescharacteristic
288D71—44no—
289C22×C22⋊Q8—711no—
290C22×#117—711no—
291C22×#103—711no—
292C22×#103D7111yescharacteristic, complemented
293C22×C22⋊C8—711no—
294#1873—718no—
295C2×#340—711no—
296C2×#340—711no—
297C71C2×#187311yescharacteristic, complemented, Sylow
298C23×Q8C2×D7111yescharacteristic
299C24×C4—711no—
300C23×C8—711no—
301C23×C8C2×D7111yescharacteristic
302C2×C22⋊Q8—718no—
303C2×C22⋊Q8—714no—
304C2×C22⋊Q8—1422no—
305C2×C22⋊Q8—1424no—
306C2×#117—714no—
307C2×#117—1424no—
308C2×#117—1422no—
309C2×#117—1424no—
310C22×C4⋊C4—711no—
311C22×C4⋊C4—711no—
312C22×C4⋊C4—1421no—
313C22×C4⋊C4C2×D7111yescharacteristic
314C22×C22⋊C4—1421no—
315C22×C22⋊C4—1421no—
316C22×C42—1421no—
317C2×#103—24no—
318C2×#103—714no—
319C2×#103—1424no—
320C2×#103C2×D7114yescomplemented
321C2×C22⋊C8—714no—
322C2×C22⋊C8—1424no—
323#340—714no—
324#340—714no—
325C25—711no—
326C22×Q8—21no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8—22no—
332C22×Q8C4×D7114yes—
333C22×Q8C22×D7112yes—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—711no—
337C23×C4—714no—
338C23×C4—712no—
339C23×C4—714no—
340C23×C4—712no—
341C23×C4—1421no—
342C23×C4—1421no—
343C23×C4—1422no—
344C23×C4—1422no—
345C23×C4—1422no—
346C23×C4—1422no—
347C23×C4C22×D7111yescharacteristic
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351C22×C8—712no—
352C22×C8—1422no—
353C22×C8—1422no—
354C22×C8—1422no—
355C22×C8C22×D7112yes—
356C22⋊Q8—7116no—
357C22⋊Q8—1428no—
358C22⋊Q8—14216no—
359#117—1428no—
360#117—14216no—
361#117—1428no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—714no—
365C2×C4⋊C4—714no—
366C2×C4⋊C4—712no—
367C2×C4⋊C4—712no—
368C2×C4⋊C4—1424no—
369C2×C4⋊C4—1428no—
370C2×C4⋊C4—1421no—
371C2×C4⋊C4—1422no—
372C2×C4⋊C4—1421no—
373C2×C4⋊C4—1422no—
374C2×C4⋊C4C22⋊D7114yes—
375C2×C4⋊C4C22×D7112yes—
376C2×C22⋊C4—1424no—
377C2×C22⋊C4—1424no—
378C2×C22⋊C4—1428no—
379C2×C22⋊C4—1428no—
380C2×C42—1424no—
381C2×C42—1428no—
382#103—28no—
383#103—1428no—
384C22⋊C8—1428no—
385C24—712no—
386C24—714no—
387C24—1422no—
388C24—1422no—
389C24—1422no—
390C24—1422no—
391C24—1422no—
392C24—1422no—
393C24D4×D7111yescharacteristic
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8C2×C4×D7114yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—2842no—
416C22×C4—2842no—
417C22×C4—2842no—
418C22×C4—2842no—
419C22×C4—712no—
420C22×C4—714no—
421C22×C4—714no—
422C22×C4—714no—
423C22×C4—718no—
424C22×C4—1422no—
425C22×C4—1422no—
426C22×C4—1424no—
427C22×C4—1424no—
428C22×C4—1422no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1422no—
432C22×C4—1422no—
433C22×C4—1422no—
434C22×C4—1428no—
435C22×C4—1428no—
436C22×C4—1424no—
437C22×C4—1428no—
438C22×C4—1428no—
439C22×C4—1424no—
440C22×C4C2×C4×D7112yes—
441C22×C4D4×D7112yes—
442C22×C4D4×D7112yes—
443C22×C4D4×D7111yescharacteristic
444C22×C4C2×C22⋊D7112yes—
445C22×C4#1176586412yes—
446C22×C4#1176586512yes—
447C22×C4C23×D7111yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—2844no—
453C2×C8—1422no—
454C2×C8—1422no—
455C2×C8—1422no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—714no—
459C4⋊C4—714no—
460C4⋊C4—14216no—
461C4⋊C4—1422no—
462C4⋊C4—1422no—
463C4⋊C4—1424no—
464C4⋊C4—1424no—
465C4⋊C4C2×C22⋊D7114yes—
466C22⋊C4—14216no—
467C22⋊C4—14216no—
468C42—14216no—
469C23—2844no—
470C23—2844no—
471C23—2844no—
472C23—2844no—
473C23—714no—
474C23—1428no—
475C23—1422no—
476C23—1422no—
477C23—1428no—
478C23—1428no—
479C23—1422no—
480C23—1422no—
481C23—1422no—
482C23—1422no—
483C23C22⋊C4×D7112yes—
484C23#1455609612yes—
485C23#1455611012yes—
486C23#1455614912yes—
487C23#1455615112yes—
488C23#1455615712yes—
489C23#1455616012yes—
490C23C2×D4×D7111yescharacteristic
491Q8—24no—
492Q8—24no—
493Q8—48no—
494Q8—42no—
495Q8—42no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—2848no—
505C2×C4—2848no—
506C2×C4—2842no—
507C2×C4—2842no—
508C2×C4—2842no—
509C2×C4—2842no—
510C2×C4—714no—
511C2×C4—1424no—
512C2×C4—1424no—
513C2×C4—1428no—
514C2×C4—1422no—
515C2×C4—1422no—
516C2×C4—14216no—
517C2×C4—1424no—
518C2×C4—1424no—
519C2×C4—1424no—
520C2×C4—1424no—
521C2×C4C22⋊C4×D7112yes—
522C2×C4(C22⋊D71)⋊C414yes—
523C2×C4C4×C22⋊D7114yes—
524C2×C4#1455609612yes—
525C2×C4#1455609914yes—
526C2×C4#1455610114yes—
527C2×C4#1455611012yes—
528C2×C4C22×C4×D7111yescharacteristic
529C2×C4C2×D4×D7111yescharacteristic
530C2×C4C2×D4×D7111yescharacteristic
531C2×C4C22×C22⋊D7111yescharacteristic
532C2×C4C2×#1176586411yescharacteristic
533C2×C4C2×#1176586511yescharacteristic
534C8—42no—
535C8—2842no—
536C22—2842no—
537C22—2842no—
538C22—2842no—
539C22—2842no—
540C22—28416no—
541C22—1424no—
542C22—1424no—
543C22—1424no—
544C22#1798587514yes—
545C22#1798599414yes—
546C22#1798600614yes—
547C22#1798615814yes—
548C22#1798616014yes—
549C22#1798633814yes—
550C22#1798633914yes—
551C22C2×C22⋊C4×D7111yescharacteristic
552C22C2×#1455609611yescharacteristic
553C22C2×#1455611011yescharacteristic
554C22C2×#1455614911yescharacteristic
555C22C2×#1455615111yescharacteristic
556C22C2×#1455615711yescharacteristic
557C22C2×#1455616011yescharacteristic
558C4—44no—
559C4—44no—
560C4—2844no—
561C4—2844no—
562C4—1424no—
563C4#1798587514yes—
564C4C2×(C22⋊D71)⋊C411yescharacteristic
565C4C2×C4×C22⋊D7111yescharacteristic
566C4C2×#1455609911yescharacteristic
567C4C2×#1455610111yescharacteristic
568C2—2844no—
569C2#2258864918yesdirect product factor
570C2C2×#1798587511yescharacteristic
571C2C2×#1798599411yescharacteristic
572C2C2×#1798600611yescharacteristic
573C2C2×#1798615811yescharacteristic
574C2C2×#1798616011yescharacteristic
575C2C2×#1798633811yescharacteristic
576C2C2×#1798633911yescharacteristic
subgroupG/H
centerC24D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC22×#103×C71C2
socleC24×C71C2×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenterC22×#103D71
hypocenterC71C2×#1873
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#14556052C2semidirect
5. #22588649C2direct (N × H)
6. C2×#17985762C2semidirect
7. C2×#17985761C2semidirect
8. C22×#103×C71C2semidirect
13. C2×#14556104C4semidirect
26. C2×#14556052C22semidirect
29. #17985762C22semidirect
30. #17985761C22semidirect
35. C2×#103×C71C22semidirect
54. #14556104C2×C4semidirect
280. C2×C71#1873semidirect
292. C22×#103D71semidirect
297. C71C2×#1873semidirect
320. C2×#103C2×D71semidirect
569. C2#22588649direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561042
C22×#145560862
C22×#145560522
C22×(C2×D71)⋊C82
#225886492
C2×#179857622
C2×#179857612
C22×#103×C712
C2×#187371