Finite Group Databasebeta

← C2×#14532482 #17202150

Subgroups — C2×#14532482 #17202150

Order 4352.

3550 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587632C212yesmaximal
2C22×C4⋊C4×D17C211yesmaximal, characteristic
3#14532482C2116yesmaximal, direct product factor
4C2×#11762742C212yesmaximal, complemented
5C2×#11762728C212yesmaximal, complemented
6C2×#11762708C211yesmaximal, characteristic, complemented
7C2×#11762707C211yesmaximal, characteristic, complemented
8C2×#11762680C212yesmaximal
9C2×#11762678C211yesmaximal, characteristic, complemented
10C2×#11762666C211yesmaximal, characteristic
11C2×C4×#63227C211yesmaximal, characteristic
12C22×C17×C4⋊Q8C211yesmaximal, characteristic, complemented
13C23×C4×D17C2212yes—
14C23×C4×D17C2211yescharacteristic
15C2×#10587632—216no—
16C2×#10587632C22116yes—
17C22×C4⋊Dic17C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C22×C4⋊Dic17C2212yes—
20C22×#63225C2212yes—
21C22×#63225C2212yes—
22C22×#63225C2212yes—
23C22×#63225C2212yes—
24C2×C4⋊C4×D17—28no—
25C2×C4⋊C4×D17C2218yes—
26C22×#63227C2212yes—
27C22×#63227C2211yescharacteristic
28C22×#63227C2211yescharacteristic
29C22×C4×Dic17C2211yescharacteristic
30#11762742C22116yescomplemented
31#11762728C22116yescomplemented
32#11762708C2218yescomplemented
33#11762707C2218yescomplemented
34#11762680C22116yes—
35#11762678C2218yescomplemented
36#11762666C2218yes—
37C4×#63227C2218yes—
38C2×#10587502C2211yescharacteristic
39C2×#10587502C2211yescharacteristic
40C2×#10587502C2212yes—
41C2×#10587502C2212yes—
42C2×#10587502C2212yes—
43C23×Q8×C17C2212yes—
44C2×C17×C4⋊Q8—28no—
45C2×C17×C4⋊Q8C2218yescomplemented
46C22×C4⋊C4×C17C2211yescharacteristic
47C22×C4⋊C4×C17C2212yes—
48C22×C4⋊C4×C17C2211yescharacteristic
49C22×C42×C17C2211yescharacteristic
50C24×D17C2311yescharacteristic
51C22×Q8×C17—28no—
52C22×Q8×C17—28no—
53C22×Q8×C17—44no—
54C22×Q8×C17C2318yes—
55C23×C4×C17C2312yes—
56C23×C4×C17C2311yescharacteristic
57C23×C4×C17C2312yes—
58C23×C4×C17C2311yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C23×Dic17C2312yes—
61C23×Dic17C2312yes—
62C23×Dic17C2311yescharacteristic
63C23×Dic17C2311yescharacteristic
64C23×Dic17C2311yescharacteristic
65C22×C4×D17—24no—
66C22×C4×D17—24no—
67C22×C4×D17—28no—
68C22×C4×D17—28no—
69C22×C4×D17—44no—
70C22×C4×D17Q818yescomplemented
71C22×C4×D17C2318yes—
72C22×C4×D17C2314yes—
73C17×C4⋊Q8—232no—
74C2×C4⋊C4×C17—28no—
75C2×C4⋊C4×C17—24no—
76C2×C4⋊C4×C17D418yes—
77C2×C4⋊C4×C17C2318yes—
78C2×C4⋊C4×C17C2314yes—
79C2×C4⋊C4×C17C2314yes—
80C2×C42×C17—24no—
81C2×C42×C17C2314yes—
82#10587632—264no—
83C2×#63227—28no—
84C2×#63227D418yes—
85C2×#63227Q818yescomplemented
86C2×#63227C2318yes—
87C2×#63227C2314yes—
88C2×#63227C2314yes—
89C2×C4⋊Dic17—24no—
90C2×C4⋊Dic17—24no—
91C2×C4⋊Dic17—28no—
92C2×C4⋊Dic17C2314yes—
93C2×C4⋊Dic17C2314yes—
94C2×C4⋊Dic17C2318yes—
95C2×#63225—28no—
96C2×#63225—28no—
97C2×#63225—28no—
98C2×#63225—44no—
99C2×#63225C2318yes—
100C2×#63225C2318yes—
101C2×#63225C2318yes—
102C2×#63225C2318yes—
103C2×C4×Dic17—24no—
104C2×C4×Dic17C2314yes—
105C4⋊C4×D17—232no—
106#10587502C2318yes—
107#10587502C2318yes—
108#10587502C2314yes—
109#10587502C2318yes—
110#10587502C2314yes—
111C24×C17C2411yescharacteristic
112C23×D17—22no—
113C23×D17—22no—
114C23×D17—22no—
115C23×D17—22no—
116C23×D17C2×D414yes—
117C23×D17C2×Q814yes—
118C23×D17C2×Q814yes—
119C23×D17C2412yes—
120C2×Q8×C17—216no—
121C2×Q8×C17—216no—
122C2×Q8×C17—48no—
123C2×Q8×C17—432no—
124C22×C4×C17—24no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—24no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—24no—
131C22×C4×C17—24no—
132C22×C4×C17—24no—
133C22×C4×C17C2×D414yes—
134C22×C4×C17C2×D418yes—
135C22×C4×C17C2×D414yes—
136C22×C4×C17C2×Q814yes—
137C22×C4×C17C2×Q814yes—
138C22×C4×C17#4114yes—
139C22×C4×C17C2414yes—
140C22×C4×C17C2414yes—
141C22×C4×C17C2412yes—
142C22×C4×C17C2412yes—
143C22×C4×C17C2412yes—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—24no—
151C22×Dic17—24no—
152C22×Dic17—24no—
153C22×Dic17—24no—
154C22×Dic17—24no—
155C22×Dic17—24no—
156C22×Dic17C2×D414yes—
157C22×Dic17C2×Q814yes—
158C22×Dic17C2×Q814yes—
159C22×Dic17C2414yes—
160C22×Dic17C2414yes—
161C22×Dic17C2412yes—
162C22×Dic17C2412yes—
163C22×Dic17C2412yes—
164C2×C4×D17—232no—
165C2×C4×D17—28no—
166C2×C4×D17—28no—
167C2×C4×D17—216no—
168C2×C4×D17—216no—
169C2×C4×D17—48no—
170C2×C4×D17—432no—
171C2×C4×D17C2×Q8116yescomplemented
172C4⋊C4×C17—216no—
173C4⋊C4×C17—28no—
174C4⋊C4×C17C2×D4116yes—
175C42×C17—28no—
176#63227—216no—
177#63227C2×D4116yes—
178#63227C2×Q8116yescomplemented
179C4⋊Dic17—28no—
180C4⋊Dic17—28no—
181C4⋊Dic17—216no—
182#63225—216no—
183#63225—216no—
184#63225—216no—
185#63225—48no—
186C4×Dic17—28no—
187C2×#2440—171nomaximal, Sylow
188C23×C17C22×D412yes—
189C23×C17C22×D412yes—
190C23×C17C22×Q812yes—
191C23×C17C22×Q812yes—
192C23×C17C2×#4112yes—
193C23×C17H(5,𝔽2)12yes—
194C23×C172−1+412yes—
195C23×C17C2511yescharacteristic
196C22×D17—28no—
197C22×D17—28no—
198C22×D17—28no—
199C22×D17—28no—
200C22×D17—28no—
201C22×D17—28no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—22no—
205C22×D17—22no—
206C22×D17C4⋊Q8116yescomplemented
207C22×D17C22×D414yes—
208C22×D17C22×Q814yes—
209C22×D17C22×Q814yes—
210Q8×C17—432no—
211C2×C4×C17—216no—
212C2×C4×C17—28no—
213C2×C4×C17—28no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—24no—
217C2×C4×C17—24no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—48no—
224C2×C4×C17C22⋊Q8116yescomplemented
225C2×C4×C17C22×D414yes—
226C2×C4×C17C22×D418yes—
227C2×C4×C17C22×D414yes—
228C2×C4×C17C22×Q814yes—
229C2×C4×C17C22×Q814yes—
230C2×C4×C17C2×#4114yes—
231C2×Dic17—28no—
232C2×Dic17—28no—
233C2×Dic17—28no—
234C2×Dic17—22no—
235C2×Dic17—22no—
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—22no—
239C2×Dic17—22no—
240C2×Dic17—24no—
241C2×Dic17—24no—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—48no—
247C2×Dic17—48no—
248C2×Dic17C22×D414yes—
249C2×Dic17C22×Q814yes—
250C2×Dic17C22×Q814yes—
251C4×D17—232no—
252C4×D17—432no—
253C22×C4⋊Q8D1711yescharacteristic, complemented
254C22×C22⋊Q8—172no—
255C23×C4⋊C4—171no—
256#2440—1716no—
257C2×#398—172no—
258C2×#395—172no—
259C2×#389—171no—
260C2×#389—171no—
261C2×#385—172no—
262C2×#384—171no—
263C2×#380—171no—
264C2×C4×C22⋊C4—171no—
265C22×C17C2×C22⋊Q814yes—
266C22×C17C2×C4⋊Q814yes—
267C22×C17#54414yes—
268C22×C17D4214yes—
269C22×C17#54714yes—
270C22×C17D4×Q814yes—
271C22×C17#55414yes—
272C22×C17C23×D411yescharacteristic
273C22×C17C23×D411yescharacteristic
274C22×C17C23×Q811yescharacteristic
275C22×C17C23×Q811yescharacteristic
276C22×C17C22×#4111yescharacteristic
277C22×C17C2×H(5,𝔽2)11yescharacteristic
278C22×C17C2×2−1+411yescharacteristic
279C2×D17—24no—
280C2×D17—24no—
281C2×D17—24no—
282C2×D17—232no—
283C2×D17—24no—
284C2×D17—24no—
285C2×D17—24no—
286C2×D17C2×C4⋊Q818yescomplemented
287C4×C17—28no—
288C4×C17—24no—
289C4×C17—24no—
290C4×C17—44no—
291C4×C17C2×C22⋊Q818yescomplemented
292Dic17—24no—
293Dic17—24no—
294Dic17—24no—
295Dic17—44no—
296Dic17—44no—
297C23×Q8C2×D1712yes—
298C24×C4—171no—
299C24×C4—172no—
300C2×C4⋊Q8—28no—
301C2×C4⋊Q8C2×D1718yescomplemented
302C2×C22⋊Q8—1716no—
303C2×C22⋊Q8—3416no—
304C22×C4⋊C4—172no—
305C22×C4⋊C4—172no—
306C22×C4⋊C4—172no—
307C22×C4⋊C4—172no—
308C22×C4⋊C4—171no—
309C22×C4⋊C4—171no—
310C22×C4⋊C4—172no—
311C22×C4⋊C4—178no—
312C22×C4⋊C4—348no—
313C22×C4⋊C4C2×D1712yes—
314C22×C4⋊C4C2×D1711yescharacteristic
315C22×C4⋊C4C2×D1711yescharacteristic
316C22×C22⋊C4—171no—
317C22×C22⋊C4—171no—
318C22×C22⋊C4—172no—
319C22×C42—171no—
320C22×C42C2×D1711yescharacteristic
321#398—1716no—
322#395—1716no—
323#389—178no—
324#389—178no—
325#385—1716no—
326#384—178no—
327#380—178no—
328C4×C22⋊C4—178no—
329C2×#95—171no—
330C2×#95—172no—
331C2×#95—172no—
332C2×#95—171no—
333C2×#95—172no—
334C2×C17#244018yescomplemented
335C2×C17C22×C22⋊Q811yescharacteristic
336C2×C17C22×C4⋊Q811yescharacteristic
337C2×C17C2×#54411yescharacteristic
338C2×C17C2×D4211yescharacteristic
339C2×C17C2×#54711yescharacteristic
340C2×C17C2×D4×Q811yescharacteristic
341C2×C17C2×#55411yescharacteristic
342D17—28no—
343C25—171no—
344C22×Q8—28no—
345C22×Q8—28no—
346C22×Q8—44no—
347C22×Q8C22×D1718yes—
348C23×C4—171no—
349C23×C4—171no—
350C23×C4—171no—
351C23×C4—178no—
352C23×C4—174no—
353C23×C4—172no—
354C23×C4—172no—
355C23×C4—178no—
356C23×C4—348no—
357C23×C4—344no—
358C23×C4—344no—
359C23×C4—348no—
360C23×C4—684no—
361C23×C4C22×D1712yes—
362C23×C4C22×D1711yescharacteristic
363C23×C4C22×D1711yescharacteristic
364C23×C4C22×D1712yes—
365C23×C4C22×D1711yescharacteristic
366C4⋊Q8—232no—
367C22⋊Q8—3464no—
368C2×C4⋊C4—28no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—174no—
371C2×C4⋊C4—178no—
372C2×C4⋊C4—178no—
373C2×C4⋊C4—178no—
374C2×C4⋊C4—178no—
375C2×C4⋊C4—174no—
376C2×C4⋊C4—178no—
377C2×C4⋊C4—348no—
378C2×C4⋊C4—348no—
379C2×C4⋊C4—348no—
380C2×C4⋊C4—344no—
381C2×C4⋊C4—348no—
382C2×C4⋊C4—344no—
383C2×C4⋊C4—3432no—
384C2×C4⋊C4—684no—
385C2×C4⋊C4C22⋊D1718yes—
386C2×C4⋊C4C22×D1714yes—
387C2×C4⋊C4C22×D1718yes—
388C2×C4⋊C4C22×D1714yes—
389C2×C22⋊C4—178no—
390C2×C22⋊C4—178no—
391C2×C22⋊C4—174no—
392C2×C22⋊C4—174no—
393C2×C22⋊C4—178no—
394C2×C22⋊C4—348no—
395C2×C42—24no—
396C2×C42—174no—
397C2×C42—344no—
398C2×C42C22×D1714yes—
399#95—174no—
400#95—174no—
401#95—178no—
402#95—178no—
403#95—178no—
404C17C2×#244011yescharacteristic, complemented, Sylow
405C24—174no—
406C24—174no—
407C24—174no—
408C24—172no—
409C24—342no—
410C24—342no—
411C24—342no—
412C24—342no—
413C24C23×D1711yescharacteristic
414C2×Q8—216no—
415C2×Q8—216no—
416C2×Q8—48no—
417C2×Q8—432no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—174no—
428C22×C4—174no—
429C22×C4—174no—
430C22×C4—172no—
431C22×C4—172no—
432C22×C4—172no—
433C22×C4—1716no—
434C22×C4—174no—
435C22×C4—174no—
436C22×C4—3416no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—348no—
444C22×C4—348no—
445C22×C4—3432no—
446C22×C4—344no—
447C22×C4—344no—
448C22×C4—344no—
449C22×C4—344no—
450C22×C4—344no—
451C22×C4—344no—
452C22×C4—3416no—
453C22×C4—688no—
454C22×C4—6832no—
455C22×C4D4×D1714yes—
456C22×C4Q8×D1714yes—
457C22×C4Q8×D1714yes—
458C22×C4#6325614yes—
459C22×C4C2×C22⋊D1718yes—
460C22×C4C2×C22⋊D1714yes—
461C22×C4C23×D1714yes—
462C22×C4C23×D1712yes—
463C22×C4C23×D1712yes—
464C22×C4C23×D1714yes—
465C22×C4C23×D1712yes—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—3416no—
469C4⋊C4—3416no—
470C4⋊C4—3416no—
471C4⋊C4—348no—
472C4⋊C4—3416no—
473C4⋊C4—348no—
474C4⋊C4—688no—
475C4⋊C4C2×C22⋊D17116yes—
476C22⋊C4—1716no—
477C22⋊C4—1716no—
478C22⋊C4—3416no—
479C42—28no—
480C42—348no—
481C23—1716no—
482C23—174no—
483C23—174no—
484C23—174no—
485C23—348no—
486C23—348no—
487C23—348no—
488C23—348no—
489C23—342no—
490C23—342no—
491C23—342no—
492C23—342no—
493C23—348no—
494C23—348no—
495C23C2×D4×D1712yes—
496C23#1058769112yes—
497C23C2×Q8×D1712yes—
498C23C2×Q8×D1712yes—
499C23C2×#6325612yes—
500C23#1058769412yes—
501C23C22×C22⋊D1712yes—
502C23C24×D1711yescharacteristic
503Q8—432no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—216no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—48no—
517C2×C4—174no—
518C2×C4—174no—
519C2×C4—174no—
520C2×C4—348no—
521C2×C4—348no—
522C2×C4—348no—
523C2×C4—342no—
524C2×C4—342no—
525C2×C4—342no—
526C2×C4—342no—
527C2×C4—342no—
528C2×C4—342no—
529C2×C4—3432no—
530C2×C4—344no—
531C2×C4—344no—
532C2×C4—344no—
533C2×C4—344no—
534C2×C4—344no—
535C2×C4—344no—
536C2×C4—6832no—
537C2×C4—688no—
538C2×C4—688no—
539C2×C4#10587632116yescomplemented
540C2×C4C2×D4×D1714yes—
541C2×C4C2×Q8×D1714yes—
542C2×C4C2×Q8×D1714yes—
543C2×C4C2×#6325614yes—
544C2×C4C22×C22⋊D1718yes—
545C2×C4C22×C22⋊D1714yes—
546C22—178no—
547C22—344no—
548C22—3432no—
549C22—344no—
550C22—344no—
551C22—344no—
552C22—344no—
553C22—344no—
554C22C2×#1058763214yes—
555C22C4⋊Q8×D1714yes—
556C22#1176343614yes—
557C22D4×C22⋊D1714yes—
558C22#1176346214yes—
559C22Q8×C22⋊D1714yes—
560C22#1176352014yes—
561C22C22×D4×D1711yescharacteristic
562C22C23×C22⋊D1711yescharacteristic
563C22C22×Q8×D1711yescharacteristic
564C22C22×Q8×D1711yescharacteristic
565C22C22×#6325611yescharacteristic
566C22C2×#1058769111yescharacteristic
567C22C2×#1058769411yescharacteristic
568C4—28no—
569C4—24no—
570C4—24no—
571C4—44no—
572C4—344no—
573C4—344no—
574C4—344no—
575C4—684no—
576C4—684no—
577C4C2×#1058763218yescomplemented
578C2—348no—
579C2#1453248218yesdirect product factor
580C2C22×#1058763211yescharacteristic
581C2C2×C4⋊Q8×D1711yescharacteristic
582C2C2×#1176343611yescharacteristic
583C2C2×D4×C22⋊D1711yescharacteristic
584C2C2×#1176346211yescharacteristic
585C2C2×Q8×C22⋊D1711yescharacteristic
586C2C2×#1176352011yescharacteristic
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC22×C17×C4⋊Q8C2
socleC24×C17C24
prime radicalC24×D17C23
hypercenterC22×C4⋊Q8D17
hypocenterC17C2×#2440
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14532482C2direct (N × H)
4. C2×#11762742C2semidirect
5. C2×#11762728C2semidirect
6. C2×#11762708C2semidirect
7. C2×#11762707C2semidirect
9. C2×#11762678C2semidirect
12. C22×C17×C4⋊Q8C2semidirect
30. #11762742C22semidirect
31. #11762728C22semidirect
32. #11762708C22semidirect
33. #11762707C22semidirect
35. #11762678C22semidirect
45. C2×C17×C4⋊Q8C22semidirect
70. C22×C4×D17Q8semidirect
85. C2×#63227Q8semidirect
171. C2×C4×D17C2×Q8semidirect
178. #63227C2×Q8semidirect
206. C22×D17C4⋊Q8semidirect
224. C2×C4×C17C22⋊Q8semidirect
253. C22×C4⋊Q8D17semidirect
286. C2×D17C2×C4⋊Q8semidirect
291. C4×C17C2×C22⋊Q8semidirect
301. C2×C4⋊Q8C2×D17semidirect
334. C2×C17#2440semidirect
404. C17C2×#2440semidirect
539. C2×C4#10587632semidirect
577. C4C2×#10587632semidirect
579. C2#14532482direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105876322
C22×C4⋊C4×D172
#145324822
C2×#117627422
C2×#117627282
C2×#117627082
C2×#117627072
C2×#117626802
C2×#117626782
C2×#117626662
C2×C4×#632272
C22×C17×C4⋊Q82
C2×#244017