Finite Group Databasebeta

← C2×#22588799 #31999240

Subgroups — C2×#22588799 #31999240

Order 18176.

2066 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556087C211yesmaximal, characteristic, complemented
2C22×#14556043C212yesmaximal
3C2×#17985862C211yesmaximal, characteristic, complemented
4#22588799C218yesmaximal, direct product factor
5C2×#17985766C212yesmaximal, complemented
6C22×#123×C71C211yesmaximal, characteristic
7C2×#14556087—28no—
8C2×#14556087—22no—
9C2×#14556087C2214yescomplemented
10C22×C4⋊Dic71—21no—
11C22×C22⋊Dic71C2212yes—
12C22×C22⋊Dic71C2212yes—
13C22×C4×Dic71—21no—
14C2×#14556043—28no—
15C2×#14556043C2218yes—
16#17985862—22no—
17#17985862C2214yescomplemented
18C2×C71⋊#95—22no—
19#17985766C2218yescomplemented
20C23×D4×C71C2211yescharacteristic
21C24×C4×C71C2211yescharacteristic
22C2×#123×C71—22no—
23C2×#123×C71—22no—
24C2×#123×C71—21no—
25C2×#123×C71—21no—
26C2×#123×C71—22no—
27C2×#123×C71C414yescomplemented
28C2×#123×C71C2214yes—
29C22×C4⋊C4×C71C412yescomplemented
30C22×C22⋊C4×C71C412yescomplemented
31C22×C22⋊C4×C71C2211yescharacteristic
32C23×Dic71—22no—
33C23×Dic71—22no—
34#14556087—216no—
35#14556087—216no—
36C2×C4⋊Dic71—28no—
37C2×C4⋊Dic71—24no—
38C2×C22⋊Dic71—24no—
39C2×C22⋊Dic71—22no—
40C2×C22⋊Dic71—24no—
41C2×C22⋊Dic71—22no—
42C2×C22⋊Dic71—24no—
43C2×C22⋊Dic71D418yes—
44C2×C22⋊Dic71C2314yes—
45C2×C22⋊Dic71C2314yes—
46C2×C4×Dic71—28no—
47C2×C4×Dic71—24no—
48#14556043—216no—
49C71⋊#95—28no—
50C25×C71C2×C411yescharacteristic
51C25×C71C2311yescharacteristic
52C22×D4×C71—21no—
53C22×D4×C71—21no—
54C22×D4×C71—21no—
55C22×D4×C71—21no—
56C22×D4×C71—24no—
57C22×D4×C71—24no—
58C22×D4×C71C2×C412yes—
59C22×D4×C71C2312yes—
60C23×C4×C71—24no—
61C23×C4×C71—24no—
62C23×C4×C71C2×C411yescharacteristic
63C23×C4×C71C2×C412yes—
64C23×C4×C71C2×C412yes—
65C23×C4×C71C2×C412yes—
66C23×C4×C71D412yes—
67C23×C4×C71D412yes—
68C23×C4×C71D411yescharacteristic
69C23×C4×C71D411yescharacteristic
70C23×C4×C71D412yes—
71C23×C4×C71C2312yes—
72#123×C71—28no—
73#123×C71—24no—
74#123×C71—24no—
75#123×C71—28no—
76#123×C71—24no—
77#123×C71—24no—
78C2×C4⋊C4×C71—24no—
79C2×C4⋊C4×C71—22no—
80C2×C4⋊C4×C71—22no—
81C2×C4⋊C4×C71C2×C418yescomplemented
82C2×C22⋊C4×C71—21no—
83C2×C22⋊C4×C71—21no—
84C2×C22⋊C4×C71—21no—
85C2×C22⋊C4×C71—21no—
86C2×C22⋊C4×C71—24no—
87C2×C22⋊C4×C71—22no—
88C2×C22⋊C4×C71—22no—
89C2×C22⋊C4×C71C2×C412yes—
90C2×C22⋊C4×C71C2×C418yescomplemented
91C2×C22⋊C4×C71C2312yes—
92C22×Dic71—28no—
93C22×Dic71—24no—
94C22×Dic71—24no—
95C22×Dic71—48no—
96C22×Dic71—48no—
97C22×Dic71—44no—
98C4⋊Dic71—216no—
99C22⋊Dic71—28no—
100C22⋊Dic71—24no—
101C22⋊Dic71—24no—
102C22⋊Dic71—24no—
103C22⋊Dic71—44no—
104C22⋊Dic71C2×D418yes—
105C4×Dic71—216no—
106C24×C71—21no—
107C24×C71—21no—
108C24×C71—24no—
109C24×C71—44no—
110C24×C71—44no—
111C24×C71C22×C411yescharacteristic
112C24×C71C22×C412yes—
113C24×C71C22×C411yescharacteristic
114C24×C71C2×D414yes—
115C24×C71C2×D411yescharacteristic
116C24×C71C2×D411yescharacteristic
117C24×C71C2×D411yescharacteristic
118C24×C71C2×D411yescharacteristic
119C24×C71#4114yes—
120C24×C71C2411yescharacteristic
121C2×D4×C71—22no—
122C2×D4×C71—22no—
123C2×D4×C71—28no—
124C2×D4×C71—22no—
125C2×D4×C71—22no—
126C2×D4×C71—24no—
127C2×D4×C71—24no—
128C2×D4×C71—42no—
129C2×D4×C71—42no—
130C2×D4×C71—44no—
131C2×D4×C71—44no—
132C2×D4×C71—42no—
133C2×D4×C71—42no—
134C22×C4×C71—28no—
135C22×C4×C71—28no—
136C22×C4×C71—22no—
137C22×C4×C71—22no—
138C22×C4×C71—24no—
139C22×C4×C71—28no—
140C22×C4×C71—28no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—24no—
144C22×C4×C71—24no—
145C22×C4×C71—24no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—21no—
149C22×C4×C71—21no—
150C22×C4×C71—24no—
151C22×C4×C71—44no—
152C22×C4×C71C22⋊C414yes—
153C22×C4×C71C22⋊C412yes—
154C22×C4×C71C22⋊C412yes—
155C22×C4×C71C22⋊C414yes—
156C22×C4×C71C22⋊C412yes—
157C22×C4×C71C22⋊C412yes—
158C22×C4×C71C22×C412yes—
159C22×C4×C71C22×C414yes—
160C22×C4×C71C22×C414yes—
161C22×C4×C71C2×D414yes—
162C22×C4×C71C2×D412yes—
163C22×C4×C71C2×D412yes—
164C4⋊C4×C71—28no—
165C4⋊C4×C71—28no—
166C22⋊C4×C71—22no—
167C22⋊C4×C71—22no—
168C22⋊C4×C71—22no—
169C22⋊C4×C71—22no—
170C22⋊C4×C71—28no—
171C22⋊C4×C71—28no—
172C23×C71—24no—
173C23×C71—24no—
174C23×C71—24no—
175C23×C71—22no—
176C23×C71—24no—
177C23×C71—42no—
178C23×C71—48no—
179C23×C71—42no—
180C23×C71—42no—
181C23×C71—48no—
182C23×C71—42no—
183C23×C71—44no—
184C23×C71—44no—
185C23×C71—42no—
186C23×C71—42no—
187C23×C71—44no—
188C23×C71—88no—
189C23×C71C2×C22⋊C411yescharacteristic
190C23×C71C2×C22⋊C411yescharacteristic
191C23×C71C2×C22⋊C411yescharacteristic
192C23×C71C2×C22⋊C411yescharacteristic
193C23×C71C4×D414yes—
194C23×C71C4×D414yes—
195C23×C71#12012yes—
196C23×C71#12012yes—
197C23×C71#12114yes—
198C23×C71#12114yes—
199C23×C71#12312yes—
200C23×C71#12312yes—
201C23×C71C23×C411yescharacteristic
202C23×C71C22×D412yes—
203C23×C71C22×D411yescharacteristic
204C23×C71C22×D411yescharacteristic
205C23×C71C2×#4112yes—
206D4×C71—42no—
207D4×C71—44no—
208D4×C71—42no—
209D4×C71—44no—
210D4×C71—44no—
211C2×C4×C71—28no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—28no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—28no—
218C2×C4×C71—24no—
219C2×C4×C71—24no—
220C2×C4×C71—28no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—22no—
225C2×C4×C71—24no—
226C2×C4×C71—44no—
227C2×C4×C71—42no—
228C2×C4×C71—42no—
229C2×C4×C71—44no—
230C2×C4×C71—44no—
231C2×C4×C71C2×C22⋊C414yes—
232C2×C4×C71C2×C22⋊C414yes—
233C2×Dic71—28no—
234C2×Dic71—48no—
235C2×Dic71—416no—
236C2×Dic71—48no—
237C2×Dic71—44no—
238C2×Dic71—88no—
239C22×C71—22no—
240C22×C71—22no—
241C22×C71—22no—
242C22×C71—22no—
243C22×C71—22no—
244C22×C71—22no—
245C22×C71—24no—
246C22×C71—28no—
247C22×C71—28no—
248C22×C71—24no—
249C22×C71—44no—
250C22×C71—42no—
251C22×C71—44no—
252C22×C71—42no—
253C22×C71—42no—
254C22×C71—42no—
255C22×C71—44no—
256C22×C71—42no—
257C22×C71—84no—
258C22×C71—84no—
259C22×C71—84no—
260C22×C71#38612yes—
261C22×C71#38612yes—
262C22×C71#38612yes—
263C22×C71#38612yes—
264C22×C71#41014yes—
265C22×C71C22×C22⋊C411yescharacteristic
266C22×C71C2×C4×D412yes—
267C22×C71C2×#12011yescharacteristic
268C22×C71C2×#12112yes—
269C22×C71C2×#12311yescharacteristic
270C4×C71—24no—
271C4×C71—24no—
272C4×C71—44no—
273C4×C71—42no—
274Dic71—48no—
275Dic71—84no—
276C2×#1899—711nomaximal, Sylow
277C2×C71—21no—
278C2×C71—22no—
279C2×C71—22no—
280C2×C71—22no—
281C2×C71—21no—
282C2×C71—22no—
283C2×C71—22no—
284C2×C71—82no—
285C2×C71#189914yescomplemented
286C2×C71C2×#38611yescharacteristic
287C2×C71C2×#41012yes—
288C22×#123D7111yescharacteristic
289C22×#117—711no—
290C22×#99—712no—
291C2×#381—711no—
292#1899—718no—
293C2×#342—712no—
294C71C2×#189911yescharacteristic, complemented, Sylow
295C23×D4C2×D7111yescharacteristic
296C24×C4C2×D7111yescharacteristic
297C2×#123—21no—
298C2×#123—22no—
299C2×#123—22no—
300C2×#123—22no—
301C2×#123—21no—
302C2×#123Dic7114yescomplemented
303C2×#123C2×D7114yes—
304C2×#117—714no—
305C2×#117—1428no—
306C2×#117—1422no—
307C22×C4⋊C4—1421no—
308C22×C4⋊C4Dic7112yescomplemented
309C22×C22⋊C4—712no—
310C22×C22⋊C4—712no—
311C22×C22⋊C4Dic7112yescomplemented
312C22×C22⋊C4C2×D7111yescharacteristic
313C22×C42—1421no—
314C2×#99—718no—
315C2×#99—1428no—
316#381—714no—
317#381—1422no—
318C2×#95—1422no—
319#342—718no—
320C25C2×Dic7111yescharacteristic
321C25C22×D7111yescharacteristic
322C22×D4—24no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—24no—
328C22×D4C2×Dic7112yes—
329C22×D4C22×D7112yes—
330C23×C4—24no—
331C23×C4—24no—
332C23×C4—1422no—
333C23×C4—1422no—
334C23×C4C2×Dic7111yescharacteristic
335C23×C4C2×Dic7112yes—
336C23×C4C2×Dic7112yes—
337C23×C4C2×Dic7112yes—
338C23×C4C22⋊D7111yescharacteristic
339C23×C4C22⋊D7112yes—
340C23×C4C22⋊D7112yes—
341C23×C4C22⋊D7112yes—
342C23×C4C22⋊D7111yescharacteristic
343C23×C4C22×D7112yes—
344#123—24no—
345#123—28no—
346#123—24no—
347#123—28no—
348#123—24no—
349#123—24no—
350#117—14216no—
351#117—14216no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—1428no—
356C2×C4⋊C4—1424no—
357C2×C4⋊C4C2×Dic7118yescomplemented
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—718no—
366C2×C22⋊C4—714no—
367C2×C22⋊C4—714no—
368C2×C22⋊C4—1422no—
369C2×C22⋊C4—1424no—
370C2×C22⋊C4—1424no—
371C2×C22⋊C4—1424no—
372C2×C22⋊C4—1422no—
373C2×C22⋊C4C2×Dic7112yes—
374C2×C22⋊C4C2×Dic7118yescomplemented
375C2×C22⋊C4C22×D7112yes—
376C2×C42—1428no—
377C2×C42—1424no—
378#99—14216no—
379#95—1428no—
380C24—21no—
381C24—21no—
382C24—24no—
383C24—44no—
384C24—44no—
385C24C22×Dic7111yescharacteristic
386C24C22×Dic7112yes—
387C24C22×Dic7111yescharacteristic
388C24D4×D7114yes—
389C24C2×C22⋊D7111yescharacteristic
390C24C2×C22⋊D7111yescharacteristic
391C24C2×C22⋊D7111yescharacteristic
392C24C2×C22⋊D7111yescharacteristic
393C24#1176586414yes—
394C24C23×D7111yescharacteristic
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—28no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—42no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—2848no—
427C22×C4—2848no—
428C22×C4—2844no—
429C22×C4—1424no—
430C22×C4—1424no—
431C22×C4—1428no—
432C22×C4C22⋊Dic7114yes—
433C22×C4C22⋊Dic7112yes—
434C22×C4C22⋊Dic7114yes—
435C22×C4C22⋊Dic7112yes—
436C22×C4C22⋊Dic7112yes—
437C22×C4C22⋊Dic7112yes—
438C22×C4C22×Dic7112yes—
439C22×C4C22×Dic7114yes—
440C22×C4C22×Dic7114yes—
441C22×C4C2×C22⋊D7114yes—
442C22×C4C2×C22⋊D7112yes—
443C22×C4C2×C22⋊D7112yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—14216no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—2844no—
454C22⋊C4—718no—
455C22⋊C4—1424no—
456C22⋊C4—1424no—
457C22⋊C4—1424no—
458C22⋊C4—1428no—
459C42—14216no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—42no—
466C23—48no—
467C23—42no—
468C23—42no—
469C23—48no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—88no—
477C23C2×C22⋊Dic7111yescharacteristic
478C23C2×C22⋊Dic7111yescharacteristic
479C23C2×C22⋊Dic7111yescharacteristic
480C23C2×C22⋊Dic7111yescharacteristic
481C23D4×Dic7114yes—
482C23D4×Dic7114yes—
483C23#1455609612yes—
484C23#1455609612yes—
485C23#1455610014yes—
486C23#1455610014yes—
487C23#1455610912yes—
488C23#1455610912yes—
489C23C23×Dic7111yescharacteristic
490C23C2×D4×D7112yes—
491C23C22×C22⋊D7111yescharacteristic
492C23C22×C22⋊D7111yescharacteristic
493C23C2×#1176586412yes—
494D4—44no—
495D4—44no—
496D4—42no—
497D4—44no—
498D4—42no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—28416no—
520C2×C4—2848no—
521C2×C4—2848no—
522C2×C4—2844no—
523C2×C4—5688no—
524C2×C4—1428no—
525C2×C4C2×C22⋊Dic7114yes—
526C2×C4C2×C22⋊Dic7114yes—
527C22—24no—
528C22—28no—
529C22—28no—
530C22—24no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—44no—
542C22—42no—
543C22—42no—
544C22—44no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22#1798587912yes—
549C22#1798587912yes—
550C22#1798587912yes—
551C22#1798587912yes—
552C22#1798597314yes—
553C22C22×C22⋊Dic7111yescharacteristic
554C22C2×D4×Dic7112yes—
555C22C2×#1455609611yescharacteristic
556C22C2×#1455610012yes—
557C22C2×#1455610911yescharacteristic
558C4—24no—
559C4—24no—
560C4—42no—
561C4—44no—
562C4—2848no—
563C4—5684no—
564C2—21no—
565C2—22no—
566C2—22no—
567C2—22no—
568C2—22no—
569C2—22no—
570C2—21no—
571C2—82no—
572C2#2258879914yesdirect product factor
573C2C2×#1798587911yescharacteristic
574C2C2×#1798597312yes—
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC23×C71C23×C4
FrattiniC24C23×D71
FittingC22×#123×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC23×D4×C71C22
cosocleC24×C71C24
hypercenterC22×#123D71
hypocenterC71C2×#1899
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. C22×#14556087C2semidirect
3. C2×#17985862C2semidirect
4. #22588799C2direct (N × H)
5. C2×#17985766C2semidirect
9. C2×#14556087C22semidirect
17. #17985862C22semidirect
19. #17985766C22semidirect
27. C2×#123×C71C4semidirect
29. C22×C4⋊C4×C71C4semidirect
30. C22×C22⋊C4×C71C4semidirect
81. C2×C4⋊C4×C71C2×C4semidirect
90. C2×C22⋊C4×C71C2×C4semidirect
285. C2×C71#1899semidirect
294. C71C2×#1899semidirect
302. C2×#123Dic71semidirect
308. C22×C4⋊C4Dic71semidirect
311. C22×C22⋊C4Dic71semidirect
357. C2×C4⋊C4C2×Dic71semidirect
374. C2×C22⋊C4C2×Dic71semidirect
572. C2#22588799direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145560872
C22×#145560432
C2×#179858622
#225887992
C2×#179857662
C22×#123×C712
C2×#189971