Finite Group Databasebeta

← C2×#22594134 #32164482

Subgroups — C2×#22594134 #32164482

Order 18176.

3110 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556103C211yesmaximal, characteristic
2C22×#14556085C211yesmaximal, characteristic, complemented
3#22594134C2116yesmaximal, direct product factor
4C2×#17985934C212yesmaximal, complemented
5C2×#17985928C212yesmaximal, complemented
6C2×#17985917C212yesmaximal, complemented
7C2×#17985916C212yesmaximal, complemented
8C2×#17985901C211yesmaximal, characteristic, complemented
9C2×#17985900C211yesmaximal, characteristic, complemented
10C2×#17985895C211yesmaximal, characteristic, complemented
11C2×#17985856C211yesmaximal, characteristic, complemented
12C2×#389×C71C211yesmaximal, characteristic, complemented
13C23×D284C2211yescharacteristic
14C23×D284C2212yes—
15C23×C4×D71C2211yescharacteristic
16C2×#14556103—28no—
17C2×#14556103C2218yes—
18C2×#14556085—28no—
19C2×#14556085C2218yescomplemented
20C22×C4⋊Dic71C2211yescharacteristic
21C22×Dic71⋊C4C2211yescharacteristic
22C22×Dic71⋊C4C2211yescharacteristic
23C22×Dic71⋊C4C2212yes—
24C22×(C2×D71)⋊C4C2211yescharacteristic
25C22×(C2×D71)⋊C4C2211yescharacteristic
26C22×(C2×D71)⋊C4C2212yes—
27C22×(C2×D71)⋊C4C2212yes—
28C22×(C2×D71)⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2212yes—
31C22×(C2×D71)⋊C4C2212yes—
32C22×C4×Dic71C2211yescharacteristic
33#17985934C22116yescomplemented
34#17985928C22116yescomplemented
35#17985917C22116yescomplemented
36#17985916C22116yescomplemented
37#17985901C2218yescomplemented
38#17985900C2218yescomplemented
39#17985895C2218yescomplemented
40#17985856C2218yescomplemented
41C2×C71⋊#95C2212yes—
42C2×C71⋊#95C2212yes—
43C22×C4⋊C4×C71C2211yescharacteristic
44C22×C4⋊C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2212yes—
46C22×C42×C71C2211yescharacteristic
47#389×C71C2218yescomplemented
48C2×#95×C71C2212yes—
49C24×D71C2312yes—
50C24×D71C2311yescharacteristic
51C22×D284—28no—
52C22×D284—24no—
53C22×D284—24no—
54C22×D284—24no—
55C22×D284—44no—
56C22×D284—44no—
57C22×D284C2314yes—
58C22×D284C2318yes—
59C23×Dic71C2311yescharacteristic
60C23×Dic71C2312yes—
61C23×Dic71C2311yescharacteristic
62C23×Dic71C2311yescharacteristic
63C22×C4×D71—24no—
64C22×C4×D71—24no—
65C22×C4×D71D418yes—
66C22×C4×D71C2314yes—
67#14556103—232no—
68#14556085—232no—
69C2×C4⋊Dic71—24no—
70C2×C4⋊Dic71C2314yes—
71C2×Dic71⋊C4—24no—
72C2×Dic71⋊C4—24no—
73C2×Dic71⋊C4—44no—
74C2×Dic71⋊C4C2314yes—
75C2×Dic71⋊C4C2314yes—
76C2×Dic71⋊C4C2318yes—
77C2×(C2×D71)⋊C4—24no—
78C2×(C2×D71)⋊C4—24no—
79C2×(C2×D71)⋊C4—28no—
80C2×(C2×D71)⋊C4—44no—
81C2×(C2×D71)⋊C4—44no—
82C2×(C2×D71)⋊C4—44no—
83C2×(C2×D71)⋊C4—44no—
84C2×(C2×D71)⋊C4—44no—
85C2×(C2×D71)⋊C4C2314yes—
86C2×(C2×D71)⋊C4C2314yes—
87C2×(C2×D71)⋊C4C2318yes—
88C2×(C2×D71)⋊C4C2318yes—
89C2×(C2×D71)⋊C4C2318yes—
90C2×(C2×D71)⋊C4C2318yes—
91C2×(C2×D71)⋊C4C2318yes—
92C2×(C2×D71)⋊C4C2318yes—
93C2×C4×Dic71—24no—
94C2×C4×Dic71C2314yes—
95C71⋊#95C2318yes—
96C71⋊#95C2318yes—
97C23×C4×C71C2312yes—
98C23×C4×C71C2312yes—
99C23×C4×C71C2311yescharacteristic
100C23×C4×C71C2311yescharacteristic
101C23×C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71—24no—
103C2×C4⋊C4×C71—24no—
104C2×C4⋊C4×C71—44no—
105C2×C4⋊C4×C71C2314yes—
106C2×C4⋊C4×C71C2314yes—
107C2×C4⋊C4×C71C2318yes—
108C2×C42×C71—24no—
109C2×C42×C71C2314yes—
110#95×C71C2318yes—
111C23×D71—22no—
112C23×D71—22no—
113C23×D71—22no—
114C23×D71—22no—
115C23×D71—22no—
116C23×D71—22no—
117C23×D71—24no—
118C23×D71—24no—
119C23×D71—24no—
120C23×D71—24no—
121C23×D71—24no—
122C23×D71—24no—
123C23×D71—24no—
124C23×D71C2×D414yes—
125C23×D71C2414yes—
126C23×D71C2412yes—
127C2×D284—216no—
128C2×D284—28no—
129C2×D284—28no—
130C2×D284—28no—
131C2×D284—48no—
132C2×D284—48no—
133C2×D284—432no—
134C2×D284—416no—
135C22×Dic71—22no—
136C22×Dic71—22no—
137C22×Dic71—22no—
138C22×Dic71—22no—
139C22×Dic71—24no—
140C22×Dic71—22no—
141C22×Dic71—22no—
142C22×Dic71—24no—
143C22×Dic71—24no—
144C22×Dic71—22no—
145C22×Dic71—22no—
146C22×Dic71C2×D414yes—
147C22×Dic71C2412yes—
148C22×Dic71C2414yes—
149C22×Dic71C2412yes—
150C22×Dic71C2412yes—
151C2×C4×D71—232no—
152C2×C4×D71—28no—
153C2×C4×D71—28no—
154C2×C4×D71C2×D4116yes—
155C4⋊Dic71—28no—
156Dic71⋊C4—28no—
157Dic71⋊C4—28no—
158Dic71⋊C4—48no—
159(C2×D71)⋊C4—28no—
160(C2×D71)⋊C4—28no—
161(C2×D71)⋊C4—216no—
162(C2×D71)⋊C4—48no—
163(C2×D71)⋊C4—48no—
164(C2×D71)⋊C4—48no—
165(C2×D71)⋊C4—48no—
166(C2×D71)⋊C4—48no—
167C4×Dic71—28no—
168C24×C71C2411yescharacteristic
169C22×C4×C71—24no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—24no—
173C22×C4×C71—24no—
174C22×C4×C71—24no—
175C22×C4×C71—22no—
176C22×C4×C71—24no—
177C22×C4×C71—22no—
178C22×C4×C71—22no—
179C22×C4×C71—24no—
180C22×C4×C71—22no—
181C22×C4×C71C2×D414yes—
182C22×C4×C71#4114yes—
183C22×C4×C71#4114yes—
184C22×C4×C71C2414yes—
185C22×C4×C71C2412yes—
186C22×C4×C71C2414yes—
187C22×C4×C71C2412yes—
188C22×C4×C71C2412yes—
189C4⋊C4×C71—28no—
190C4⋊C4×C71—28no—
191C4⋊C4×C71—48no—
192C42×C71—28no—
193C23×C71C22×D412yes—
194C23×C71C2×#4112yes—
195C23×C71C2×#4112yes—
196C23×C71H(5,𝔽2)12yes—
197C23×C71H(5,𝔽2)12yes—
198C23×C71H(5,𝔽2)12yes—
199C23×C712−1+412yes—
200C23×C71C2511yescharacteristic
201C22×D71—28no—
202C22×D71—28no—
203C22×D71—28no—
204C22×D71—22no—
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—22no—
209C22×D71—22no—
210C22×D71—24no—
211C22×D71—24no—
212C22×D71—24no—
213C22×D71—24no—
214C22×D71—24no—
215C22×D71—24no—
216C22×D71—24no—
217C22×D71—44no—
218C22×D71—44no—
219C22×D71—44no—
220C22×D71—44no—
221C22×D71—48no—
222C22×D71—48no—
223C22×D71—48no—
224C22×D71—48no—
225C22×D71—48no—
226C22×D71—48no—
227C22×D71—48no—
228C22×D71C22×D414yes—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—24no—
232C2×C4×C71—24no—
233C2×C4×C71—22no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—22no—
237C2×C4×C71—22no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—22no—
241C2×C4×C71—44no—
242C2×C4×C71—48no—
243C2×C4×C71—44no—
244C2×C4×C71—48no—
245C2×C4×C71#123116yes—
246C2×C4×C71C22×D414yes—
247C2×C4×C71C2×#4114yes—
248C2×C4×C71C2×#4114yes—
249C2×Dic71—28no—
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—24no—
255C2×Dic71—22no—
256C2×Dic71—22no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—22no—
260C2×Dic71—22no—
261C2×Dic71—44no—
262C2×Dic71—44no—
263C2×Dic71—48no—
264C2×Dic71C22×D414yes—
265D284—432no—
266D284—416no—
267C4×D71—232no—
268C22×C71C2×#12314yes—
269C22×C71#53514yes—
270C22×C71#53714yes—
271C22×C71#54014yes—
272C22×C71#54014yes—
273C22×C71#54214yes—
274C22×C71#54214yes—
275C22×C71C23×D411yescharacteristic
276C22×C71C22×#4111yescharacteristic
277C22×C71C22×#4111yescharacteristic
278C22×C71C2×H(5,𝔽2)11yescharacteristic
279C22×C71C2×H(5,𝔽2)11yescharacteristic
280C22×C71C2×H(5,𝔽2)11yescharacteristic
281C22×C71C2×2−1+411yescharacteristic
282C2×D71—24no—
283C2×D71—24no—
284C2×D71—24no—
285C2×D71—416no—
286C2×D71—42no—
287C2×D71—42no—
288C2×D71—42no—
289C2×D71—42no—
290C2×D71—44no—
291C2×D71—44no—
292C2×D71—44no—
293C2×D71—44no—
294C2×D71—44no—
295C2×D71—44no—
296C2×D71—44no—
297C2×D71—816no—
298C4×C71—44no—
299C4×C71—42no—
300C4×C71—42no—
301C4×C71—44no—
302C4×C71C2×#12318yes—
303Dic71—24no—
304Dic71—42no—
305Dic71—42no—
306Dic71—44no—
307C2×#2624—711nomaximal, Sylow
308C2×C71#262418yescomplemented
309C2×C71C22×#12311yescharacteristic
310C2×C71C2×#53511yescharacteristic
311C2×C71C2×#53711yescharacteristic
312C2×C71C2×#54011yescharacteristic
313C2×C71C2×#54011yescharacteristic
314C2×C71C2×#54211yescharacteristic
315C2×C71C2×#54211yescharacteristic
316D71—44no—
317D71—84no—
318C22×#121—711no—
319C22×#117—711no—
320#2624—7116no—
321C2×#399—712no—
322C2×#397—712no—
323C2×#394—712no—
324C2×#394—712no—
325C2×#390—711no—
326C2×#390—711no—
327C2×#389—711no—
328C2×#389D7111yescharacteristic, complemented
329C2×#380—711no—
330C71C2×#262411yescharacteristic, complemented, Sylow
331C23×D4—712no—
332C23×D4—711no—
333C24×C4—711no—
334C2×#121—718no—
335C2×#121—1428no—
336C2×#117—718no—
337C2×#117—1428no—
338C22×C4⋊C4—712no—
339C22×C4⋊C4—711no—
340C22×C4⋊C4—711no—
341C22×C4⋊C4—711no—
342C22×C4⋊C4C2×D7111yescharacteristic
343C22×C4⋊C4C2×D7111yescharacteristic
344C22×C4⋊C4C2×D7112yes—
345C22×C22⋊C4—712no—
346C22×C22⋊C4—712no—
347C22×C22⋊C4—712no—
348C22×C22⋊C4—712no—
349C22×C22⋊C4—712no—
350C22×C22⋊C4—711no—
351C22×C22⋊C4—711no—
352C22×C22⋊C4—712no—
353C22×C42—711no—
354C22×C42C2×D7111yescharacteristic
355#399—7116no—
356#397—7116no—
357#394—7116no—
358#394—7116no—
359#390—718no—
360#390—718no—
361#389—718no—
362#389C2×D7118yescomplemented
363#380—718no—
364C2×#95—712no—
365C2×#95—712no—
366C2×#95C2×D7112yes—
367C25—711no—
368C25—712no—
369C22×D4—2844no—
370C22×D4—2844no—
371C22×D4—718no—
372C22×D4—714no—
373C22×D4—1428no—
374C22×D4—1424no—
375C22×D4—1424no—
376C22×D4—1424no—
377C23×C4—711no—
378C23×C4—718no—
379C23×C4—714no—
380C23×C4—711no—
381C23×C4—712no—
382C23×C4—711no—
383C23×C4—1424no—
384C23×C4—1424no—
385C23×C4C22×D7111yescharacteristic
386C23×C4C22×D7111yescharacteristic
387C23×C4C22×D7112yes—
388C23×C4C22×D7112yes—
389C23×C4C22×D7111yescharacteristic
390#121—14232no—
391#117—14232no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—44no—
395C2×C4⋊C4—2844no—
396C2×C4⋊C4—718no—
397C2×C4⋊C4—714no—
398C2×C4⋊C4—714no—
399C2×C4⋊C4—714no—
400C2×C4⋊C4—1424no—
401C2×C4⋊C4—1424no—
402C2×C4⋊C4—1424no—
403C2×C4⋊C4C22×D7114yes—
404C2×C4⋊C4C22×D7114yes—
405C2×C4⋊C4C22×D7118yes—
406C2×C22⋊C4—2844no—
407C2×C22⋊C4—2844no—
408C2×C22⋊C4—2844no—
409C2×C22⋊C4—2844no—
410C2×C22⋊C4—2844no—
411C2×C22⋊C4—718no—
412C2×C22⋊C4—718no—
413C2×C22⋊C4—718no—
414C2×C22⋊C4—718no—
415C2×C22⋊C4—718no—
416C2×C22⋊C4—714no—
417C2×C22⋊C4—714no—
418C2×C22⋊C4—718no—
419C2×C22⋊C4—1428no—
420C2×C22⋊C4—1424no—
421C2×C22⋊C4—1424no—
422C2×C42—24no—
423C2×C42—714no—
424C2×C42—1424no—
425C2×C42C22×D7114yes—
426#95—718no—
427#95—718no—
428#95C22×D7118yes—
429C24—714no—
430C24—712no—
431C24—714no—
432C24—1424no—
433C24—1424no—
434C24—1424no—
435C24—1424no—
436C24—1424no—
437C24—1424no—
438C24—1424no—
439C24—1422no—
440C24—1422no—
441C24—1422no—
442C24—1422no—
443C24—1422no—
444C24—1422no—
445C24C23×D7111yescharacteristic
446C2×D4—2848no—
447C2×D4—2848no—
448C2×D4—28432no—
449C2×D4—28416no—
450C2×D4—14216no—
451C2×D4—1428no—
452C2×D4—1428no—
453C2×D4—1428no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—714no—
467C22×C4—712no—
468C22×C4—7116no—
469C22×C4—712no—
470C22×C4—714no—
471C22×C4—712no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4—1424no—
475C22×C4—1422no—
476C22×C4—1422no—
477C22×C4—1424no—
478C22×C4—1424no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1422no—
483C22×C4—1428no—
484C22×C4—1428no—
485C22×C4—14232no—
486C22×C4D4×D7114yes—
487C22×C4#1176586514yes—
488C22×C4#1176586614yes—
489C22×C4C23×D7114yes—
490C22×C4C23×D7114yes—
491C22×C4C23×D7112yes—
492C22×C4C23×D7112yes—
493C22×C4C23×D7112yes—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—48no—
497C4⋊C4—2848no—
498C4⋊C4—1428no—
499C4⋊C4—1428no—
500C4⋊C4—1428no—
501C22⋊C4—2848no—
502C22⋊C4—2848no—
503C22⋊C4—2848no—
504C22⋊C4—2848no—
505C22⋊C4—2848no—
506C22⋊C4—14216no—
507C22⋊C4—1428no—
508C22⋊C4—1428no—
509C42—28no—
510C42—1428no—
511C23—2844no—
512C23—2844no—
513C23—2844no—
514C23—2844no—
515C23—2848no—
516C23—2848no—
517C23—2848no—
518C23—2848no—
519C23—2848no—
520C23—2848no—
521C23—2848no—
522C23—714no—
523C23—1424no—
524C23—1424no—
525C23—1424no—
526C23—1424no—
527C23—1424no—
528C23—1424no—
529C23—1424no—
530C23—1428no—
531C23—1422no—
532C23—1422no—
533C23—1422no—
534C23—1422no—
535C23—1422no—
536C23—1422no—
537C23—1428no—
538C23—1428no—
539C23C2×D4×D7112yes—
540C23C2×#1176586512yes—
541C23C2×#1176586612yes—
542C23#1455617312yes—
543C23#1455617412yes—
544C23#1455617412yes—
545C23#1455617612yes—
546C23C24×D7111yescharacteristic
547D4—28432no—
548D4—28416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—2844no—
566C2×C4—2844no—
567C2×C4—2848no—
568C2×C4—714no—
569C2×C4—1422no—
570C2×C4—1422no—
571C2×C4—1424no—
572C2×C4—1422no—
573C2×C4—1422no—
574C2×C4—1424no—
575C2×C4—1424no—
576C2×C4—1422no—
577C2×C4—1422no—
578C2×C4—1422no—
579C2×C4—1422no—
580C2×C4—1428no—
581C2×C4—14232no—
582C2×C4#14556113116yes—
583C2×C4C2×D4×D7114yes—
584C2×C4C2×#1176586514yes—
585C2×C4C2×#1176586614yes—
586C22—28416no—
587C22—2842no—
588C22—2842no—
589C22—2842no—
590C22—2842no—
591C22—2844no—
592C22—2844no—
593C22—2844no—
594C22—2844no—
595C22—2844no—
596C22—2844no—
597C22—2844no—
598C22—56816no—
599C22—1424no—
600C22—1424no—
601C22—1424no—
602C22C2×#1455611314yes—
603C22#1798656514yes—
604C22#1798657914yes—
605C22#1798660014yes—
606C22#1798660514yes—
607C22#1798662114yes—
608C22#1798662314yes—
609C22C22×D4×D7111yescharacteristic
610C22C22×#1176586511yescharacteristic
611C22C22×#1176586611yescharacteristic
612C22C2×#1455617311yescharacteristic
613C22C2×#1455617411yescharacteristic
614C22C2×#1455617411yescharacteristic
615C22C2×#1455617611yescharacteristic
616C4—44no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—2842no—
621C4—2842no—
622C4—2844no—
623C4—1424no—
624C4C2×#1455611318yes—
625C2—2844no—
626C2—5684no—
627C2#2259413418yesdirect product factor
628C2C22×#1455611311yescharacteristic
629C2C2×#1798656511yescharacteristic
630C2C2×#1798657911yescharacteristic
631C2C2×#1798660011yescharacteristic
632C2C2×#1798660511yescharacteristic
633C2C2×#1798662111yescharacteristic
634C2C2×#1798662311yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#389×C71C2
socleC24×C71C24
prime radicalC22×#14556103C2
hypercenterC2×#389D71
hypocenterC71C2×#2624
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×#14556085C2semidirect
3. #22594134C2direct (N × H)
4. C2×#17985934C2semidirect
5. C2×#17985928C2semidirect
6. C2×#17985917C2semidirect
7. C2×#17985916C2semidirect
8. C2×#17985901C2semidirect
9. C2×#17985900C2semidirect
10. C2×#17985895C2semidirect
11. C2×#17985856C2semidirect
12. C2×#389×C71C2semidirect
19. C2×#14556085C22semidirect
33. #17985934C22semidirect
34. #17985928C22semidirect
35. #17985917C22semidirect
36. #17985916C22semidirect
37. #17985901C22semidirect
38. #17985900C22semidirect
39. #17985895C22semidirect
40. #17985856C22semidirect
47. #389×C71C22semidirect
308. C2×C71#2624semidirect
328. C2×#389D71semidirect
330. C71C2×#2624semidirect
362. #389C2×D71semidirect
627. C2#22594134direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561032
C22×#145560852
#225941342
C2×#179859342
C2×#179859282
C2×#179859172
C2×#179859162
C2×#179859012
C2×#179859002
C2×#179858952
C2×#179858562
C2×#389×C712
C2×#262471