Finite Group Databasebeta

← C2×#22593261 #32169223

Subgroups — C2×#22593261 #32169223

Order 18176.

3614 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556106C212yesmaximal, complemented
2C22×C4×Dic142C211yesmaximal, characteristic, complemented
3C22×#14556090C211yesmaximal, characteristic
4#22593261C2116yesmaximal, direct product factor
5C2×#17985923C212yesmaximal, complemented
6C2×#17985921C212yesmaximal, complemented
7C2×#17985901C211yesmaximal, characteristic
8C2×#17985900C211yesmaximal, characteristic
9C2×#17985895C211yesmaximal, characteristic, complemented
10C2×#17985868C211yesmaximal, characteristic, complemented
11C2×#17985858C212yesmaximal, complemented
12C2×#389×C71C211yesmaximal, characteristic, complemented
13C23×Dic142C2211yescharacteristic
14C23×D284C2211yescharacteristic
15C23×C4×D71C2212yes—
16C2×#14556106—216no—
17C2×#14556106C22116yescomplemented
18C2×C4×Dic142—28no—
19C2×C4×Dic142C2218yescomplemented
20C2×#14556090—28no—
21C2×#14556090C2218yes—
22C22×C4⋊Dic71C2211yescharacteristic
23C22×C4⋊Dic71C2211yescharacteristic
24C22×Dic71⋊C4C2212yes—
25C22×Dic71⋊C4C2211yescharacteristic
26C22×Dic71⋊C4C2211yescharacteristic
27C22×Dic71⋊C4C2212yes—
28C22×Dic71⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2212yes—
31C22×(C2×D71)⋊C4C2212yes—
32C22×(C2×D71)⋊C4C2212yes—
33C22×(C2×D71)⋊C4C2212yes—
34C22×C4×Dic71C2211yescharacteristic
35C22×C4×Dic71C2211yescharacteristic
36#17985923C22116yescomplemented
37#17985921C22116yescomplemented
38#17985901C2218yes—
39#17985900C2218yes—
40#17985895C2218yescomplemented
41#17985868C2218yescomplemented
42#17985858C22116yescomplemented
43C2×C71⋊#95C2212yes—
44C22×C4⋊C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2212yes—
47C22×C42×C71C2211yescharacteristic
48#389×C71C2218yescomplemented
49C2×#95×C71C2212yes—
50C24×D71C2312yes—
51C22×Dic142—24no—
52C22×Dic142—24no—
53C22×Dic142D418yescomplemented
54C22×Dic142C2314yes—
55C22×D284—24no—
56C22×D284—24no—
57C22×D284Q818yescomplemented
58C22×D284C2314yes—
59C23×Dic71C2311yescharacteristic
60C23×Dic71C2312yes—
61C23×Dic71C2311yescharacteristic
62C23×Dic71C2311yescharacteristic
63C23×Dic71C2311yescharacteristic
64C22×C4×D71—28no—
65C22×C4×D71—28no—
66C22×C4×D71—28no—
67C22×C4×D71C2318yes—
68#14556106—264no—
69C4×Dic142—232no—
70#14556090—232no—
71C2×C4⋊Dic71—24no—
72C2×C4⋊Dic71—24no—
73C2×C4⋊Dic71C2314yes—
74C2×C4⋊Dic71C2314yes—
75C2×Dic71⋊C4—24no—
76C2×Dic71⋊C4—24no—
77C2×Dic71⋊C4—28no—
78C2×Dic71⋊C4—28no—
79C2×Dic71⋊C4—44no—
80C2×Dic71⋊C4C2314yes—
81C2×Dic71⋊C4C2318yes—
82C2×Dic71⋊C4C2318yes—
83C2×Dic71⋊C4C2314yes—
84C2×Dic71⋊C4C2318yes—
85C2×(C2×D71)⋊C4—28no—
86C2×(C2×D71)⋊C4—28no—
87C2×(C2×D71)⋊C4—28no—
88C2×(C2×D71)⋊C4—44no—
89C2×(C2×D71)⋊C4—44no—
90C2×(C2×D71)⋊C4C2318yes—
91C2×(C2×D71)⋊C4C2318yes—
92C2×(C2×D71)⋊C4C2318yes—
93C2×(C2×D71)⋊C4C2318yes—
94C2×(C2×D71)⋊C4C2318yes—
95C2×C4×Dic71—24no—
96C2×C4×Dic71D418yes—
97C2×C4×Dic71C2314yes—
98C2×C4×Dic71C2314yes—
99C71⋊#95C2318yes—
100C23×C4×C71C2312yes—
101C23×C4×C71C2311yescharacteristic
102C23×C4×C71C2312yes—
103C23×C4×C71C2311yescharacteristic
104C23×C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71—24no—
106C2×C4⋊C4×C71—28no—
107C2×C4⋊C4×C71—24no—
108C2×C4⋊C4×C71C2314yes—
109C2×C4⋊C4×C71C2314yes—
110C2×C4⋊C4×C71C2318yes—
111C2×C42×C71—24no—
112C2×C42×C71C2314yes—
113#95×C71C2318yes—
114C23×D71—24no—
115C23×D71—24no—
116C23×D71—24no—
117C23×D71—24no—
118C23×D71—24no—
119C23×D71—24no—
120C23×D71C2×Q818yes—
121C23×D71C2414yes—
122C2×Dic142—232no—
123C2×Dic142—28no—
124C2×Dic142—28no—
125C2×Dic142C2×D4116yescomplemented
126C2×D284—232no—
127C2×D284—28no—
128C2×D284—28no—
129C2×D284C2×Q8116yescomplemented
130C22×Dic71—22no—
131C22×Dic71—22no—
132C22×Dic71—22no—
133C22×Dic71—22no—
134C22×Dic71—22no—
135C22×Dic71—24no—
136C22×Dic71—24no—
137C22×Dic71—22no—
138C22×Dic71—24no—
139C22×Dic71—22no—
140C22×Dic71C2×D414yes—
141C22×Dic71C2×D414yes—
142C22×Dic71C2×D414yes—
143C22×Dic71C2×D414yes—
144C22×Dic71#4114yes—
145C22×Dic71C2412yes—
146C22×Dic71C2414yes—
147C22×Dic71C2412yes—
148C22×Dic71C2412yes—
149C22×Dic71C2412yes—
150C2×C4×D71—216no—
151C2×C4×D71—216no—
152C2×C4×D71—216no—
153C2×C4×D71—432no—
154C4⋊Dic71—28no—
155C4⋊Dic71—28no—
156Dic71⋊C4—28no—
157Dic71⋊C4—28no—
158Dic71⋊C4—216no—
159Dic71⋊C4—216no—
160Dic71⋊C4—48no—
161(C2×D71)⋊C4—216no—
162(C2×D71)⋊C4—216no—
163(C2×D71)⋊C4—216no—
164(C2×D71)⋊C4—48no—
165(C2×D71)⋊C4—48no—
166C4×Dic71—28no—
167C4×Dic71C2×D4116yes—
168C24×C71C2411yescharacteristic
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71—24no—
172C22×C4×C71—24no—
173C22×C4×C71—22no—
174C22×C4×C71—24no—
175C22×C4×C71—24no—
176C22×C4×C71—24no—
177C22×C4×C71—22no—
178C22×C4×C71—24no—
179C22×C4×C71—22no—
180C22×C4×C71C2×D414yes—
181C22×C4×C71C2×D414yes—
182C22×C4×C71C2×Q814yes—
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—216no—
191C4⋊C4×C71—28no—
192C42×C71—28no—
193C23×C71C22×D412yes—
194C23×C71C22×D412yes—
195C23×C71C22×Q812yes—
196C23×C71C2×#4112yes—
197C23×C71C2×#4112yes—
198C23×C71H(5,𝔽2)12yes—
199C23×C712−1+412yes—
200C23×C71C2511yescharacteristic
201C22×D71—216no—
202C22×D71—216no—
203C22×D71—216no—
204C22×D71—24no—
205C22×D71—24no—
206C22×D71—24no—
207C22×D71—24no—
208C22×D71—24no—
209C22×D71—24no—
210C22×D71—48no—
211C22×D71—48no—
212C22×D71—48no—
213C22×D71—48no—
214C22×D71C22×Q818yes—
215C2×C4×C71—28no—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—22no—
225C2×C4×C71—24no—
226C2×C4×C71—22no—
227C2×C4×C71—48no—
228C2×C4×C71—44no—
229C2×C4×C71—48no—
230C2×C4×C71C22⋊Q8116yescomplemented
231C2×C4×C71C22×D414yes—
232C2×C4×C71C22×D414yes—
233C2×C4×C71C22×Q814yes—
234C2×C4×C71C2×#4114yes—
235C2×Dic71—28no—
236C2×Dic71—28no—
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—22no—
241C2×Dic71—22no—
242C2×Dic71—24no—
243C2×Dic71—24no—
244C2×Dic71—22no—
245C2×Dic71—24no—
246C2×Dic71—22no—
247C2×Dic71—44no—
248C2×Dic71—48no—
249C2×Dic71#121116yes—
250C2×Dic71C22×D414yes—
251C2×Dic71C22×D414yes—
252C2×Dic71C22×D414yes—
253C2×Dic71C22×D414yes—
254C2×Dic71C2×#4114yes—
255D284—232no—
256C4×D71—432no—
257Dic142—232no—
258C22×C71C2×#12114yes—
259C22×C71C2×C22⋊Q814yes—
260C22×C71#53714yes—
261C22×C71D4×Q814yes—
262C22×C71#55014yes—
263C22×C71#55414yes—
264C22×C71#55614yes—
265C22×C71C23×D411yescharacteristic
266C22×C71C23×D411yescharacteristic
267C22×C71C23×Q811yescharacteristic
268C22×C71C22×#4111yescharacteristic
269C22×C71C22×#4111yescharacteristic
270C22×C71C2×H(5,𝔽2)11yescharacteristic
271C22×C71C2×2−1+411yescharacteristic
272C2×D71—28no—
273C2×D71—28no—
274C2×D71—28no—
275C2×D71—44no—
276C2×D71—44no—
277C2×D71—44no—
278C2×D71—44no—
279C2×D71—432no—
280C4×C71—24no—
281C4×C71—44no—
282C4×C71—42no—
283C4×C71—44no—
284C4×C71C2×C22⋊Q818yescomplemented
285Dic71—24no—
286Dic71—24no—
287Dic71—42no—
288Dic71—44no—
289Dic71C2×#12118yes—
290C2×#2538—711nomaximal, Sylow
291C2×C71#253818yescomplemented
292C2×C71C22×#12111yescharacteristic
293C2×C71C22×C22⋊Q811yescharacteristic
294C2×C71C2×#53711yescharacteristic
295C2×C71C2×D4×Q811yescharacteristic
296C2×C71C2×#55011yescharacteristic
297C2×C71C2×#55411yescharacteristic
298C2×C71C2×#55611yescharacteristic
299D71—48no—
300C22×C22⋊Q8—712no—
301C22×C4×Q8—711no—
302C22×C4×D4—711no—
303#2538—7116no—
304C2×#396—712no—
305C2×#396—712no—
306C2×#390—711no—
307C2×#390—711no—
308C2×#389—711no—
309C2×#389D7111yescharacteristic, complemented
310C2×#384—711no—
311C2×#380—712no—
312C71C2×#253811yescharacteristic, complemented, Sylow
313C23×Q8—711no—
314C23×D4—711no—
315C24×C4—712no—
316C2×C22⋊Q8—7116no—
317C2×C22⋊Q8—14216no—
318C2×C4×Q8—718no—
319C2×C4×Q8—1428no—
320C2×C4×D4—718no—
321C2×C4×D4—1428no—
322C22×C4⋊C4—712no—
323C22×C4⋊C4—711no—
324C22×C4⋊C4—711no—
325C22×C4⋊C4—712no—
326C22×C4⋊C4—711no—
327C22×C4⋊C4—712no—
328C22×C4⋊C4—711no—
329C22×C4⋊C4C2×D7112yes—
330C22×C4⋊C4C2×D7111yescharacteristic
331C22×C4⋊C4C2×D7111yescharacteristic
332C22×C22⋊C4—712no—
333C22×C22⋊C4—712no—
334C22×C22⋊C4—712no—
335C22×C22⋊C4—712no—
336C22×C22⋊C4—712no—
337C22×C42—711no—
338C22×C42—711no—
339C22×C42C2×D7111yescharacteristic
340#396—7116no—
341#396—7116no—
342#390—718no—
343#390—718no—
344#389—718no—
345#389C2×D7118yescomplemented
346#384—718no—
347#380—7116no—
348C2×#95—712no—
349C2×#95C2×D7112yes—
350C25—712no—
351C22×Q8—718no—
352C22×Q8—714no—
353C22×Q8—1424no—
354C22×Q8—1424no—
355C22×D4—718no—
356C22×D4—714no—
357C22×D4—1424no—
358C22×D4—1424no—
359C23×C4—711no—
360C23×C4—711no—
361C23×C4—711no—
362C23×C4—712no—
363C23×C4—711no—
364C23×C4—718no—
365C23×C4—1428no—
366C23×C4—1428no—
367C23×C4—1428no—
368C23×C4C22×D7112yes—
369C23×C4C22×D7111yescharacteristic
370C23×C4C22×D7112yes—
371C23×C4C22×D7111yescharacteristic
372C23×C4C22×D7111yescharacteristic
373C22⋊Q8—14264no—
374C4×Q8—14232no—
375C4×D4—14232no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—28no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—2844no—
380C2×C4⋊C4—718no—
381C2×C4⋊C4—714no—
382C2×C4⋊C4—714no—
383C2×C4⋊C4—714no—
384C2×C4⋊C4—718no—
385C2×C4⋊C4—718no—
386C2×C4⋊C4—714no—
387C2×C4⋊C4—1424no—
388C2×C4⋊C4—1424no—
389C2×C4⋊C4—1424no—
390C2×C4⋊C4—1428no—
391C2×C4⋊C4—1428no—
392C2×C4⋊C4—1424no—
393C2×C4⋊C4C22×D7114yes—
394C2×C4⋊C4C22×D7118yes—
395C2×C4⋊C4C22×D7114yes—
396C2×C22⋊C4—2844no—
397C2×C22⋊C4—2844no—
398C2×C22⋊C4—718no—
399C2×C22⋊C4—718no—
400C2×C22⋊C4—718no—
401C2×C22⋊C4—718no—
402C2×C22⋊C4—718no—
403C2×C22⋊C4—1428no—
404C2×C22⋊C4—1428no—
405C2×C22⋊C4—1428no—
406C2×C42—24no—
407C2×C42—718no—
408C2×C42—714no—
409C2×C42—714no—
410C2×C42—1424no—
411C2×C42C22×D7114yes—
412#95—718no—
413#95C22×D7118yes—
414C24—718no—
415C24—714no—
416C24—1424no—
417C24—1424no—
418C24—1424no—
419C24—1424no—
420C24—1424no—
421C24—1424no—
422C24C23×D7111yescharacteristic
423C2×Q8—7116no—
424C2×Q8—1428no—
425C2×Q8—1428no—
426C2×Q8—14232no—
427C2×D4—7116no—
428C2×D4—1428no—
429C2×D4—1428no—
430C2×D4—14232no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—28432no—
443C22×C4—714no—
444C22×C4—714no—
445C22×C4—714no—
446C22×C4—712no—
447C22×C4—714no—
448C22×C4—714no—
449C22×C4—712no—
450C22×C4—712no—
451C22×C4—712no—
452C22×C4—714no—
453C22×C4—14216no—
454C22×C4—14216no—
455C22×C4—14216no—
456C22×C4—1422no—
457C22×C4—1424no—
458C22×C4—1424no—
459C22×C4—1422no—
460C22×C4—1424no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1422no—
464C22×C4—1422no—
465C22×C4—1422no—
466C22×C4D4×D7114yes—
467C22×C4D4×D7114yes—
468C22×C4Q8×D7114yes—
469C22×C4#1176586514yes—
470C22×C4C23×D7114yes—
471C22×C4C23×D7112yes—
472C22×C4C23×D7114yes—
473C22×C4C23×D7112yes—
474C22×C4C23×D7112yes—
475C4⋊C4—28no—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—2848no—
479C4⋊C4—1428no—
480C4⋊C4—1428no—
481C4⋊C4—14216no—
482C4⋊C4—14216no—
483C4⋊C4—1428no—
484C4⋊C4—1428no—
485C22⋊C4—2848no—
486C22⋊C4—2848no—
487C22⋊C4—14216no—
488C22⋊C4—14216no—
489C22⋊C4—14216no—
490C42—28no—
491C42—7116no—
492C42—1428no—
493C23—2848no—
494C23—2848no—
495C23—2848no—
496C23—2848no—
497C23—718no—
498C23—1424no—
499C23—1424no—
500C23—1424no—
501C23—1424no—
502C23—1424no—
503C23—1424no—
504C23—14216no—
505C23—14216no—
506C23—14216no—
507C23C2×D4×D7112yes—
508C23C2×D4×D7112yes—
509C23C2×Q8×D7112yes—
510C23#41×D7112yes—
511C23C2×#1176586512yes—
512C23#1455617412yes—
513C23#1455617612yes—
514C23C24×D7111yescharacteristic
515Q8—14232no—
516D4—14232no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—28432no—
533C2×C4—2844no—
534C2×C4—2848no—
535C2×C4—714no—
536C2×C4—714no—
537C2×C4—714no—
538C2×C4—7116no—
539C2×C4—714no—
540C2×C4—714no—
541C2×C4—1422no—
542C2×C4—1424no—
543C2×C4—1424no—
544C2×C4—1422no—
545C2×C4—1424no—
546C2×C4—1422no—
547C2×C4—1422no—
548C2×C4—1422no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1428no—
552C2×C4—1428no—
553C2×C4#14556106116yescomplemented
554C2×C4C2×D4×D7114yes—
555C2×C4C2×D4×D7114yes—
556C2×C4C2×Q8×D7114yes—
557C2×C4C2×#1176586514yes—
558C22—2844no—
559C22—2844no—
560C22—2844no—
561C22—2844no—
562C22—28432no—
563C22—1428no—
564C22—1428no—
565C22—1428no—
566C22#121×D7114yes—
567C22C2×#1455610614yes—
568C22#1798657914yes—
569C22#1798666414yes—
570C22#1798667014yes—
571C22#1798670514yes—
572C22#1798672114yes—
573C22C22×D4×D7111yescharacteristic
574C22C22×D4×D7111yescharacteristic
575C22C22×Q8×D7111yescharacteristic
576C22C2×#41×D7111yescharacteristic
577C22C22×#1176586511yescharacteristic
578C22C2×#1455617411yescharacteristic
579C22C2×#1455617611yescharacteristic
580C4—24no—
581C4—44no—
582C4—44no—
583C4—42no—
584C4—2844no—
585C4—2842no—
586C4—718no—
587C4—1424no—
588C4—1424no—
589C4C2×#1455610618yescomplemented
590C2—2848no—
591C2#2259326118yesdirect product factor
592C2C2×#121×D7111yescharacteristic
593C2C22×#1455610611yescharacteristic
594C2C2×#1798657911yescharacteristic
595C2C2×#1798666411yescharacteristic
596C2C2×#1798667011yescharacteristic
597C2C2×#1798670511yescharacteristic
598C2C2×#1798672111yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#389×C71C2
socleC24×C71C24
prime radicalC23×D284C22
hypercenterC2×#389D71
hypocenterC71C2×#2538
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#14556106C2semidirect
2. C22×C4×Dic142C2semidirect
4. #22593261C2direct (N × H)
5. C2×#17985923C2semidirect
6. C2×#17985921C2semidirect
9. C2×#17985895C2semidirect
10. C2×#17985868C2semidirect
11. C2×#17985858C2semidirect
12. C2×#389×C71C2semidirect
17. C2×#14556106C22semidirect
19. C2×C4×Dic142C22semidirect
36. #17985923C22semidirect
37. #17985921C22semidirect
40. #17985895C22semidirect
41. #17985868C22semidirect
42. #17985858C22semidirect
48. #389×C71C22semidirect
53. C22×Dic142D4semidirect
57. C22×D284Q8semidirect
125. C2×Dic142C2×D4semidirect
129. C2×D284C2×Q8semidirect
230. C2×C4×C71C22⋊Q8semidirect
284. C4×C71C2×C22⋊Q8semidirect
291. C2×C71#2538semidirect
309. C2×#389D71semidirect
312. C71C2×#2538semidirect
345. #389C2×D71semidirect
553. C2×C4#14556106semidirect
589. C4C2×#14556106semidirect
591. C2#22593261direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561062
C22×C4×Dic1422
C22×#145560902
#225932612
C2×#179859232
C2×#179859212
C2×#179859012
C2×#179859002
C2×#179858952
C2×#179858682
C2×#179858582
C2×#389×C712
C2×#253871