Finite Group Databasebeta

← C2×#22588058 #31994854

Subgroups — C2×#22588058 #31994854

Order 18176.

2574 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765848C211yesmaximal, characteristic
2C22×#14556086C211yesmaximal, characteristic
3C22×C4⋊C4×D71C211yesmaximal, characteristic
4#22588058C218yesmaximal, direct product factor
5C2×#17985762C211yesmaximal, characteristic, complemented
6C2×#17985761C211yesmaximal, characteristic, complemented
7C2×#17985761C211yesmaximal, characteristic, complemented
8C2×#340×C71C211yesmaximal, characteristic, complemented
9C23×C4×D71—21no—
10C23×C4×D71C2211yescharacteristic
11C22×#11765848—24no—
12C22×#11765848—24no—
13C22×#11765848C418yescomplemented
14C22×#11765848C2214yes—
15C23×C71⋊C8C2211yescharacteristic
16C2×#14556086—22no—
17C2×#14556086—24no—
18C2×#14556086C418yescomplemented
19C2×#14556086C2214yes—
20C22×C4⋊Dic71C2211yescharacteristic
21C22×C4⋊Dic71C2211yescharacteristic
22C22×Dic71⋊C4—21no—
23C2×C4⋊C4×D71—24no—
24C2×C4⋊C4×D71—22no—
25C2×C4⋊C4×D71C418yescomplemented
26C2×C4⋊C4×D71C2214yes—
27C22×(C2×D71)⋊C4—21no—
28C22×C4×Dic71—21no—
29#17985762C2214yescomplemented
30#17985761C2214yescomplemented
31#17985761C2214yescomplemented
32C23×C8×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C22×C4⋊C4×C71C2211yescharacteristic
35#340×C71C2214yescomplemented
36C24×D71D411yescharacteristic
37C23×Dic71—21no—
38C23×Dic71—21no—
39C23×Dic71D411yescharacteristic
40C22×C4×D71—28no—
41C22×C4×D71—24no—
42C22×C4×D71—44no—
43C22×C4×D71—44no—
44C22×C4×D71C2×C414yes—
45C22×C4×D71C2×C414yes—
46C22×C4×D71C2×C414yes—
47C22×C4×D71D412yes—
48C22×C4×D71D414yes—
49C22×C4×D71D414yes—
50C22×C4×D71Q814yes—
51C22×C4×D71C2312yes—
52C2×#11765848—232no—
53C2×#11765848—28no—
54C2×#11765848—28no—
55C2×#11765848C2×C4116yescomplemented
56C22×C71⋊C8—22no—
57C22×C71⋊C8—22no—
58C22×C71⋊C8C2×C414yescomplemented
59C22×C71⋊C8C2312yes—
60#14556086—216no—
61#14556086—28no—
62#14556086C2×C4116yescomplemented
63C2×C4⋊Dic71—21no—
64C2×C4⋊Dic71—22no—
65C2×C4⋊Dic71—21no—
66C2×C4⋊Dic71—22no—
67C2×C4⋊Dic71C2×C414yes—
68C2×C4⋊Dic71C2×C414yescomplemented
69C2×C4⋊Dic71C2312yes—
70C2×C4⋊Dic71C2312yes—
71C2×Dic71⋊C4—28no—
72C2×Dic71⋊C4—24no—
73C4⋊C4×D71—216no—
74C4⋊C4×D71—28no—
75C4⋊C4×D71C2×C4116yescomplemented
76C2×(C2×D71)⋊C4—28no—
77C2×(C2×D71)⋊C4—24no—
78C2×C4×Dic71—28no—
79C2×C4×Dic71—24no—
80C23×C4×C71—21no—
81C23×C4×C71—21no—
82C23×C4×C71C2311yescharacteristic
83C22×C8×C71—22no—
84C22×C8×C71—22no—
85C22×C8×C71C2×C414yescomplemented
86C22×C8×C71C2312yes—
87C2×C4⋊C4×C71—21no—
88C2×C4⋊C4×C71—22no—
89C2×C4⋊C4×C71—21no—
90C2×C4⋊C4×C71—22no—
91C2×C4⋊C4×C71C2×C414yes—
92C2×C4⋊C4×C71C2×C414yescomplemented
93C2×C4⋊C4×C71C2312yes—
94C2×C4⋊C4×C71C2312yes—
95C23×D71—22no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71C22⋊C414yes—
100C23×D71C22⋊C414yes—
101C23×D71C4⋊C414yes—
102C23×D71C2×D412yes—
103C22×Dic71—24no—
104C22×Dic71—24no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—42no—
108C22×Dic71—42no—
109C22×Dic71—42no—
110C22×Dic71—42no—
111C22×Dic71C22⋊C414yes—
112C22×Dic71C22⋊C414yes—
113C22×Dic71C4⋊C414yes—
114C22×Dic71C2×D412yes—
115C2×C4×D71—216no—
116C2×C4×D71—48no—
117C2×C4×D71—48no—
118C2×C4×D71—432no—
119C2×C4×D71C42116yes—
120C2×C4×D71C22⋊C418yes—
121C2×C4×D71C22⋊C418yes—
122C2×C4×D71C22⋊C4116yes—
123C2×C4×D71C4⋊C418yes—
124C2×C4×D71C4⋊C4116yes—
125C2×C4×D71C4⋊C4116yes—
126C2×C4×D71C22×C414yes—
127C2×C4×D71C22×C414yes—
128C2×C4×D71C22×C414yes—
129C2×C4×D71C2×D414yes—
130C2×C4×D71C2×D414yes—
131C2×C4×D71C2×Q814yes—
132#11765848—232no—
133C2×C71⋊C8—28no—
134C2×C71⋊C8—22no—
135C2×C71⋊C8—22no—
136C2×C71⋊C8C22×C414yescomplemented
137C4⋊Dic71—24no—
138C4⋊Dic71—24no—
139C4⋊Dic71—22no—
140C4⋊Dic71—22no—
141C4⋊Dic71C22×C414yes—
142C4⋊Dic71C22×C414yescomplemented
143Dic71⋊C4—216no—
144(C2×D71)⋊C4—216no—
145C4×Dic71—216no—
146C24×C71C2×D411yescharacteristic
147C22×C4×C71—24no—
148C22×C4×C71—24no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—42no—
152C22×C4×C71—42no—
153C22×C4×C71—42no—
154C22×C4×C71—42no—
155C22×C4×C71C22×C412yes—
156C22×C4×C71C22×C412yes—
157C22×C4×C71C22×C412yes—
158C22×C4×C71C2×D411yescharacteristic
159C22×C4×C71C2×D412yes—
160C22×C4×C71C2×D412yes—
161C22×C4×C71C2×Q812yes—
162C22×C4×C71C2411yescharacteristic
163C2×C8×C71—28no—
164C2×C8×C71—22no—
165C2×C8×C71—22no—
166C2×C8×C71C22×C414yescomplemented
167C4⋊C4×C71—24no—
168C4⋊C4×C71—24no—
169C4⋊C4×C71—22no—
170C4⋊C4×C71—22no—
171C4⋊C4×C71C22×C414yes—
172C4⋊C4×C71C22×C414yescomplemented
173C23×C71C2×C22⋊C412yes—
174C23×C71C2×C22⋊C412yes—
175C23×C71C2×C4⋊C412yes—
176C23×C71AGL(1,ℤ/8)12yes—
177C23×C71AGL(1,ℤ/8)12yes—
178C23×C71#13712yes—
179C23×C71#13712yes—
180C23×C71C22×D411yescharacteristic
181C22×D71—22no—
182C22×D71—22no—
183C22×D71—22no—
184C22×D71—22no—
185C22×D71—28no—
186C22×D71—28no—
187C22×D71—28no—
188C22×D71—28no—
189C22×D71—28no—
190C22×D71—28no—
191C22×D71#95116yes—
192C22×D71C2×C22⋊C414yes—
193C22×D71C2×C22⋊C414yes—
194C22×D71C2×C4⋊C414yes—
195C2×C4×C71—24no—
196C2×C4×C71—24no—
197C2×C4×C71—42no—
198C2×C4×C71—42no—
199C2×C4×C71—42no—
200C2×C4×C71—42no—
201C2×C4×C71—48no—
202C2×C4×C71—48no—
203C2×C4×C71C2×C4214yes—
204C2×C4×C71C2×C22⋊C414yes—
205C2×C4×C71C2×C22⋊C412yes—
206C2×C4×C71C2×C22⋊C412yes—
207C2×C4×C71C2×C4⋊C414yes—
208C2×C4×C71C2×C4⋊C414yes—
209C2×C4×C71C2×C4⋊C412yes—
210C2×C4×C71C23×C411yescharacteristic
211C2×C4×C71C23×C411yescharacteristic
212C2×C4×C71C23×C411yescharacteristic
213C2×C4×C71C22×D411yescharacteristic
214C2×C4×C71C22×D411yescharacteristic
215C2×C4×C71C22×Q811yescharacteristic
216C2×Dic71—24no—
217C2×Dic71—24no—
218C2×Dic71—42no—
219C2×Dic71—42no—
220C2×Dic71—42no—
221C2×Dic71—42no—
222C2×Dic71—48no—
223C2×Dic71—48no—
224C2×Dic71#95116yes—
225C2×Dic71C2×C22⋊C414yes—
226C2×Dic71C2×C22⋊C414yes—
227C2×Dic71C2×C4⋊C414yes—
228C4×D71—432no—
229C4×D71#95132yes—
230C4×D71C2×C4218yes—
231C4×D71C2×C22⋊C418yes—
232C4×D71C2×C4⋊C418yes—
233C4×D71C2×C4⋊C418yes—
234C8×C71—24no—
235C71⋊C8—24no—
236C22×C71C2×#9514yes—
237C22×C71#41714yes—
238C22×C71#41714yes—
239C22×C71#41814yes—
240C22×C71#41914yes—
241C22×C71#42814yes—
242C22×C71#42814yes—
243C22×C71C22×C22⋊C411yescharacteristic
244C22×C71C22×C22⋊C411yescharacteristic
245C22×C71C22×C4⋊C411yescharacteristic
246C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
247C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C71C2×#13711yescharacteristic
249C22×C71C2×#13711yescharacteristic
250C2×D71—24no—
251C2×D71—24no—
252C2×D71—24no—
253C2×D71—24no—
254C2×D71—24no—
255C2×D71—24no—
256C2×D71—232no—
257C2×D71C2×#9518yes—
258C4×C71—44no—
259C4×C71—44no—
260C4×C71C2×#9514yes—
261C4×C71C22×C4211yescharacteristic
262C4×C71C22×C22⋊C411yescharacteristic
263C4×C71C22×C4⋊C411yescharacteristic
264C4×C71C22×C4⋊C411yescharacteristic
265Dic71—44no—
266Dic71—44no—
267Dic71C2×#9518yes—
268C2×#1736—711nomaximal, Sylow
269C2×C71#173618yescomplemented
270C2×C71C22×#9511yescharacteristic
271C2×C71C2×#41811yescharacteristic
272C2×C71C2×#41911yescharacteristic
273C2×C71C2×#41711yescharacteristic
274C2×C71C2×#41711yescharacteristic
275C2×C71C2×#42811yescharacteristic
276C2×C71C2×#42811yescharacteristic
277D71—28no—
278C23×M16—711no—
279C22×#117—711no—
280C23×C4⋊C4—711no—
281#1736—718no—
282C2×#340—711no—
283C2×#340—711no—
284C2×#340—711no—
285C2×#340D7111yescharacteristic, complemented
286C71C2×#173611yescharacteristic, complemented, Sylow
287C24×C4—711no—
288C24×C4—1421no—
289C22×M16—718no—
290C22×M16—714no—
291C22×M16—1424no—
292C22×M16—1424no—
293C23×C8—711no—
294C23×C8C2×D7111yescharacteristic
295C2×#117—718no—
296C2×#117—714no—
297C2×#117—1422no—
298C2×#117—1424no—
299C22×C4⋊C4—711no—
300C22×C4⋊C4—711no—
301C22×C4⋊C4—718no—
302C22×C4⋊C4—714no—
303C22×C4⋊C4—1422no—
304C22×C4⋊C4—1424no—
305C22×C4⋊C4—1421no—
306C22×C4⋊C4C2×D7111yescharacteristic
307C22×C4⋊C4C2×D7111yescharacteristic
308C22×C22⋊C4—1421no—
309C22×C42—1421no—
310#340—714no—
311#340—714no—
312#340—714no—
313#340C2×D7114yescomplemented
314C25—711no—
315C23×C4—21no—
316C23×C4—21no—
317C23×C4—2844no—
318C23×C4—2844no—
319C23×C4—711no—
320C23×C4—714no—
321C23×C4—714no—
322C23×C4—714no—
323C23×C4—712no—
324C23×C4—714no—
325C23×C4—714no—
326C23×C4—714no—
327C23×C4—712no—
328C23×C4—1424no—
329C23×C4—1428no—
330C23×C4—1421no—
331C23×C4—1421no—
332C23×C4C22×D7111yescharacteristic
333C2×M16—7116no—
334C2×M16—1428no—
335C2×M16—1428no—
336C2×M16—14232no—
337C22×C8—22no—
338C22×C8—22no—
339C22×C8—714no—
340C22×C8—712no—
341C22×C8—1422no—
342C22×C8—1422no—
343C22×C8C4×D7114yescomplemented
344C22×C8C22×D7112yes—
345#117—7116no—
346#117—1428no—
347#117—14216no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—714no—
353C2×C4⋊C4—714no—
354C2×C4⋊C4—712no—
355C2×C4⋊C4—712no—
356C2×C4⋊C4—7116no—
357C2×C4⋊C4—1428no—
358C2×C4⋊C4—14216no—
359C2×C4⋊C4—1424no—
360C2×C4⋊C4—1428no—
361C2×C4⋊C4—1422no—
362C2×C4⋊C4—1421no—
363C2×C4⋊C4—1422no—
364C2×C4⋊C4—1421no—
365C2×C4⋊C4C4×D7114yes—
366C2×C4⋊C4C4×D7114yescomplemented
367C2×C4⋊C4C22×D7112yes—
368C2×C4⋊C4C22×D7112yes—
369C2×C22⋊C4—1424no—
370C2×C22⋊C4—1428no—
371C2×C42—1424no—
372C2×C42—1428no—
373C24—712no—
374C24—714no—
375C24—714no—
376C24—714no—
377C24—1422no—
378C24—1422no—
379C24—1422no—
380C24—1422no—
381C24D4×D7111yescharacteristic
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—28432no—
391C22×C4—2848no—
392C22×C4—2848no—
393C22×C4—2842no—
394C22×C4—2842no—
395C22×C4—2842no—
396C22×C4—2842no—
397C22×C4—712no—
398C22×C4—714no—
399C22×C4—714no—
400C22×C4—714no—
401C22×C4—714no—
402C22×C4—714no—
403C22×C4—714no—
404C22×C4—714no—
405C22×C4—714no—
406C22×C4—714no—
407C22×C4—7116no—
408C22×C4—718no—
409C22×C4—718no—
410C22×C4—7116no—
411C22×C4—7116no—
412C22×C4—7116no—
413C22×C4—718no—
414C22×C4—14216no—
415C22×C4—1422no—
416C22×C4—1422no—
417C22×C4—1424no—
418C22×C4—1424no—
419C22×C4C2×C4×D7112yes—
420C22×C4C2×C4×D7112yes—
421C22×C4C2×C4×D7112yes—
422C22×C4D4×D7112yes—
423C22×C4D4×D7112yes—
424C22×C4D4×D7111yescharacteristic
425C22×C4Q8×D7112yes—
426C22×C4C23×D7111yescharacteristic
427M16—14232no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—28no—
431C2×C8—714no—
432C2×C8—1422no—
433C2×C8—1422no—
434C2×C8—1428no—
435C2×C8C2×C4×D7114yescomplemented
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—714no—
441C4⋊C4—714no—
442C4⋊C4—14216no—
443C4⋊C4—1422no—
444C4⋊C4—1422no—
445C4⋊C4—1424no—
446C4⋊C4—1424no—
447C4⋊C4C2×C4×D7114yes—
448C4⋊C4C2×C4×D7114yescomplemented
449C22⋊C4—14216no—
450C42—14216no—
451C23—7116no—
452C23—714no—
453C23—714no—
454C23—714no—
455C23—1428no—
456C23—1428no—
457C23—1428no—
458C23—1428no—
459C23—1422no—
460C23—1422no—
461C23—1422no—
462C23—1422no—
463C23—1428no—
464C23—1428no—
465C23C22⋊C4×D7112yes—
466C23C22⋊C4×D7112yes—
467C23C4⋊C4×D7112yes—
468C23#1455615212yes—
469C23#1455615512yes—
470C23#1455615712yes—
471C23#1455616012yes—
472C23C2×D4×D7111yescharacteristic
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—28432no—
482C2×C4—2848no—
483C2×C4—2848no—
484C2×C4—2842no—
485C2×C4—2842no—
486C2×C4—2842no—
487C2×C4—2842no—
488C2×C4—7116no—
489C2×C4—714no—
490C2×C4—714no—
491C2×C4—714no—
492C2×C4—7132no—
493C2×C4—718no—
494C2×C4—718no—
495C2×C4—718no—
496C2×C4—718no—
497C2×C4—1424no—
498C2×C4—1424no—
499C2×C4C42×D7114yes—
500C2×C4C22⋊C4×D7112yes—
501C2×C4C22⋊C4×D7112yes—
502C2×C4C22⋊C4×D7114yes—
503C2×C4C4⋊C4×D7112yes—
504C2×C4C4⋊C4×D7114yes—
505C2×C4C4⋊C4×D7114yes—
506C2×C4C22×C4×D7111yescharacteristic
507C2×C4C22×C4×D7111yescharacteristic
508C2×C4C22×C4×D7111yescharacteristic
509C2×C4C2×D4×D7111yescharacteristic
510C2×C4C2×D4×D7111yescharacteristic
511C2×C4C2×Q8×D7111yescharacteristic
512C8—24no—
513C8—1424no—
514C22—718no—
515C22—1424no—
516C22—1424no—
517C22—14232no—
518C22—1424no—
519C22—1424no—
520C22—1424no—
521C22—1424no—
522C22#95×D7114yes—
523C22#1798599714yes—
524C22#1798599914yes—
525C22#1798600214yes—
526C22#1798600614yes—
527C22#1798603814yes—
528C22#1798604014yes—
529C22C2×C22⋊C4×D7111yescharacteristic
530C22C2×C22⋊C4×D7111yescharacteristic
531C22C2×C4⋊C4×D7111yescharacteristic
532C22C2×#1455615211yescharacteristic
533C22C2×#1455615511yescharacteristic
534C22C2×#1455615711yescharacteristic
535C22C2×#1455616011yescharacteristic
536C4—44no—
537C4—44no—
538C4—2844no—
539C4—2844no—
540C4—718no—
541C4#95×D7114yes—
542C4C2×C42×D7111yescharacteristic
543C4C2×C22⋊C4×D7111yescharacteristic
544C4C2×C4⋊C4×D7111yescharacteristic
545C4C2×C4⋊C4×D7111yescharacteristic
546C2—1428no—
547C2#2258805818yesdirect product factor
548C2C2×#95×D7111yescharacteristic
549C2C2×#1798600211yescharacteristic
550C2C2×#1798600611yescharacteristic
551C2C2×#1798599711yescharacteristic
552C2C2×#1798599911yescharacteristic
553C2C2×#1798603811yescharacteristic
554C2C2×#1798604011yescharacteristic
subgroupG/H
centerC24D4×D71
commutatorC4×C71C22×C42
FrattiniC22×C4C23×D71
FittingC2×#340×C71C2
socleC24×C71C2×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenterC2×#340D71
hypocenterC71C2×#1736
LCSG ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. #22588058C2direct (N × H)
5. C2×#17985762C2semidirect
6. C2×#17985761C2semidirect
7. C2×#17985761C2semidirect
8. C2×#340×C71C2semidirect
13. C22×#11765848C4semidirect
18. C2×#14556086C4semidirect
25. C2×C4⋊C4×D71C4semidirect
29. #17985762C22semidirect
30. #17985761C22semidirect
31. #17985761C22semidirect
35. #340×C71C22semidirect
55. C2×#11765848C2×C4semidirect
58. C22×C71⋊C8C2×C4semidirect
62. #14556086C2×C4semidirect
68. C2×C4⋊Dic71C2×C4semidirect
75. C4⋊C4×D71C2×C4semidirect
85. C22×C8×C71C2×C4semidirect
92. C2×C4⋊C4×C71C2×C4semidirect
136. C2×C71⋊C8C22×C4semidirect
142. C4⋊Dic71C22×C4semidirect
166. C2×C8×C71C22×C4semidirect
172. C4⋊C4×C71C22×C4semidirect
269. C2×C71#1736semidirect
285. C2×#340D71semidirect
286. C71C2×#1736semidirect
313. #340C2×D71semidirect
343. C22×C8C4×D71semidirect
366. C2×C4⋊C4C4×D71semidirect
435. C2×C8C2×C4×D71semidirect
448. C4⋊C4C2×C4×D71semidirect
547. C2#22588058direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658482
C22×#145560862
C22×C4⋊C4×D712
#225880582
C2×#179857622
C2×#179857612
C2×#340×C712
C2×#173671