Finite Group Databasebeta

← #30825908

Subgroups — #30825908

Order 17152.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22569358C211yesmaximal, characteristic
2#22569290C211yesmaximal, characteristic, complemented
3#22559088C211yesmaximal, characteristic, complemented
4#22559081C211yesmaximal, characteristic, complemented
5#22559060C211yesmaximal, characteristic, complemented
6#22559053C211yesmaximal, characteristic, complemented
7#22559028C211yesmaximal, characteristic, complemented
8SD16⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
9Q16×C67⋊C8C211yesmaximal, characteristic, complemented
10SD16×C67⋊C8C211yesmaximal, characteristic, complemented
11#22558998C211yesmaximal, characteristic, complemented
12#22558638C211yesmaximal, characteristic, complemented
13#22558617C211yesmaximal, characteristic
14#22558509C211yesmaximal, characteristic, complemented
15C67×#3108C211yesmaximal, characteristic
16#16863527C2211yescharacteristic
17Q8×C67⋊C8—21no—
18Q8×C67⋊C8—21no—
19Q8×C67⋊C8C2211yescharacteristic
20Q8×C67⋊C8C2211yescharacteristic
21Q8×C67⋊C8C2211yescharacteristic
22Q8×C67⋊C8C2211yescharacteristic
23Q8×C67⋊C8C2211yescharacteristic
24#16863474—21no—
25#16863474C2211yescharacteristic
26#16863468—21no—
27D4×C67⋊C8C2211yescharacteristic
28#16863457—21no—
29#16863449—21no—
30#16863449—21no—
31#16863449C2211yescharacteristic
32#16863449C2211yescharacteristic
33#16863449C2211yescharacteristic
34#16863423—21no—
35#16863420C2211yescharacteristic
36#16863149C2211yescharacteristic
37#16863149C2211yescharacteristic
38#16863147C2211yescharacteristic
39#16863147C2211yescharacteristic
40Q8⋊(C67⋊C8)C2211yescharacteristic
41Q8⋊(C67⋊C8)C2211yescharacteristic
42Q8⋊(C67⋊C8)C2211yescharacteristic
43Q8⋊(C67⋊C8)C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46D4⋊(C67⋊C8)C2211yescharacteristic
47D4⋊(C67⋊C8)C2211yescharacteristic
48#16863118C2211yescharacteristic
49#16863118C2211yescharacteristic
50#16863117C2211yescharacteristic
51C8×C67⋊C8C2211yescharacteristic
52#529×C67C2211yescharacteristic
53C2×C4×Q8×C67C2211yescharacteristic
54#483×C67C411yescharacteristic
55#480×C67C411yescharacteristic
56#479×C67C411yescharacteristic
57#479×C67C411yescharacteristic
58#464×C67C411yescharacteristic
59#464×C67C411yescharacteristic
60#451×C67C411yescharacteristic
61#448×C67C411yescharacteristic
62C4×Q16×C67C2211yescharacteristic
63C4×Q16×C67C2211yescharacteristic
64C4×SD16×C67C2211yescharacteristic
65C4×SD16×C67C2211yescharacteristic
66#431×C67C2211yescharacteristic
67C2×#11761612—21no—
68C2×#11761612—21no—
69C22×C67⋊C8—21no—
70C4⋊(C67⋊C8)—21no—
71C4⋊(C67⋊C8)—21no—
72C4⋊(C67⋊C8)—21no—
73C4⋊(C67⋊C8)—21no—
74C4⋊(C67⋊C8)—21no—
75C4⋊(C67⋊C8)—21no—
76C4⋊(C67⋊C8)—21no—
77C4⋊(C67⋊C8)—21no—
78C4⋊(C67⋊C8)D411yescharacteristic
79C4⋊(C67⋊C8)D411yescharacteristic
80C4⋊(C67⋊C8)C2311yescharacteristic
81C4⋊(C67⋊C8)C2311yescharacteristic
82C4⋊(C67⋊C8)C2311yescharacteristic
83C4⋊(C67⋊C8)C2311yescharacteristic
84C22⋊(C67⋊C8)—21no—
85C22⋊(C67⋊C8)—21no—
86C22⋊(C67⋊C8)—21no—
87C22⋊(C67⋊C8)—21no—
88C22⋊(C67⋊C8)D411yescharacteristic
89C22⋊(C67⋊C8)D411yescharacteristic
90#14523954—21no—
91C4×C67⋊C8—21no—
92C4×C67⋊C8—21no—
93C4×C67⋊C8—21no—
94C4×C67⋊C8—21no—
95C4×C67⋊C8—21no—
96C4×C67⋊C8—21no—
97C4×C67⋊C8C2311yescharacteristic
98C4×C67⋊C8C2311yescharacteristic
99C4×C67⋊C8C2311yescharacteristic
100C4×C67⋊C8C2311yescharacteristic
101C22×Q8×C67C2×C411yescharacteristic
102C2×Q16×C67C2×C411yescharacteristic
103C2×Q16×C67C2×C411yescharacteristic
104C2×SD16×C67C2×C411yescharacteristic
105C2×SD16×C67C2×C411yescharacteristic
106#126×C67—21no—
107#125×C67C2×C411yescharacteristic
108#124×C67C2×C411yescharacteristic
109#123×C67—21no—
110C22⋊Q8×C67C2×C411yescharacteristic
111#121×C67C2×C411yescharacteristic
112C4×Q8×C67—21no—
113C4×Q8×C67—21no—
114C4×Q8×C67C2×C411yescharacteristic
115C4×Q8×C67C2×C411yescharacteristic
116C4×Q8×C67C2311yescharacteristic
117C4×Q8×C67C2311yescharacteristic
118C4×Q8×C67C2311yescharacteristic
119C4×D4×C67—21no—
120C4×D4×C67C2311yescharacteristic
121#117×C67—21no—
122C2×C4⋊C4×C67—21no—
123C2×C4⋊C4×C67C2×C411yescharacteristic
124C2×C42×C67—21no—
125C2×C42×C67C2311yescharacteristic
126#107×C67C2×C411yescharacteristic
127#107×C67C2×C411yescharacteristic
128#106×C67C2×C411yescharacteristic
129#106×C67C2×C411yescharacteristic
130C4⋊C8×C67C2×C411yescharacteristic
131C4⋊C8×C67C2×C411yescharacteristic
132#103×C67C2×C411yescharacteristic
133#103×C67C2×C411yescharacteristic
134#103×C67C2×C411yescharacteristic
135#103×C67C2×C411yescharacteristic
136#103×C67C2×C411yescharacteristic
137#103×C67C2×C411yescharacteristic
138#102×C67C2×C411yescharacteristic
139#102×C67C2×C411yescharacteristic
140C22⋊C8×C67C2×C411yescharacteristic
141C22⋊C8×C67C2×C411yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144#11761612—81no—
145#11761612—81no—
146C2×C67⋊C8—21no—
147C2×C67⋊C8—21no—
148C2×C67⋊C8—21no—
149C2×C67⋊C8—21no—
150C2×C67⋊C8—21no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—81no—
153C2×C67⋊C8C2×D411yescharacteristic
154C2×C67⋊C8C2×D411yescharacteristic
155C2×C67⋊C8C2×D411yescharacteristic
156C2×C67⋊C8C2×D411yescharacteristic
157C2×Q8×C67—22no—
158C2×Q8×C67—22no—
159C2×Q8×C67—21no—
160C2×Q8×C67C22×C411yescharacteristic
161C2×Q8×C67C22×C411yescharacteristic
162C2×Q8×C67C22×C411yescharacteristic
163C2×D4×C67—21no—
164C2×D4×C67C22×C411yescharacteristic
165C22×C4×C67—21no—
166C22×C4×C67—21no—
167C22×C4×C67—21no—
168C22×C4×C67C22×C411yescharacteristic
169C22×C4×C67C2×D411yescharacteristic
170C22×C4×C67#4111yescharacteristic
171Q16×C67—22no—
172Q16×C67—22no—
173SD16×C67—22no—
174SD16×C67—22no—
175C2×C8×C67C22×C411yescharacteristic
176C2×C8×C67C22×C411yescharacteristic
177C2×C8×C67C22×C411yescharacteristic
178C2×C8×C67C22×C411yescharacteristic
179C4⋊C4×C67—21no—
180C4⋊C4×C67—21no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—21no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—21no—
188C4⋊C4×C67C22×C411yescharacteristic
189C4⋊C4×C67C22×C411yescharacteristic
190C4⋊C4×C67C22×C411yescharacteristic
191C4⋊C4×C67C22×C411yescharacteristic
192C22⋊C4×C67—21no—
193C22⋊C4×C67—21no—
194C22⋊C4×C67—21no—
195C22⋊C4×C67—21no—
196C22⋊C4×C67—21no—
197C42×C67—21no—
198C42×C67—21no—
199C42×C67—21no—
200C42×C67—21no—
201C42×C67—21no—
202C42×C67C22×C411yescharacteristic
203C42×C67C2×D411yescharacteristic
204C42×C67#4111yescharacteristic
205C42×C67C2411yescharacteristic
206C23×C67—21no—
207C23×C67C4×D411yescharacteristic
208Q8×C67—24no—
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67#13112yes—
214Q8×C67#13112yes—
215D4×C67—21no—
216D4×C67—21no—
217D4×C67—81no—
218C2×C4×C67—24no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—41no—
231C2×C4×C67—41no—
232C2×C4×C67—41no—
233C2×C4×C67—81no—
234C2×C4×C67C4×D411yescharacteristic
235C2×C4×C67C4×D411yescharacteristic
236C2×C4×C67C4×D411yescharacteristic
237C2×C4×C67C23×C411yescharacteristic
238C2×C4×C67C22×D411yescharacteristic
239C2×C4×C67C2×#4111yescharacteristic
240C8×C67—21no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C67⋊C8—21no—
245C67⋊C8—21no—
246C67⋊C8—21no—
247C67⋊C8—21no—
248C67⋊C8—41no—
249C67⋊C8—41no—
250C67⋊C8—41no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C22×C67—21no—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67C2×C4×D411yescharacteristic
261C4×C67—22no—
262C4×C67—22no—
263C4×C67—21no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—21no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67C2×#13111yescharacteristic
273C4×C67C2×#13111yescharacteristic
274C4×C67#57711yescharacteristic
275C4×C67#57811yescharacteristic
276#17919—671nomaximal, Sylow
277C2×C67—41no—
278C2×C67—81no—
279C2×C67#292911yescharacteristic
280C2×C67#294711yescharacteristic
281C2×C67#295711yescharacteristic
282#3108D6711yescharacteristic
283#2993—671no—
284#2982—671no—
285#1588—671no—
286#1587—671no—
287#1584—671no—
288#1583—671no—
289#1579—671no—
290SD16⋊C8—671no—
291C8×Q16—671no—
292C8×SD16—671no—
293#1574—671no—
294#1487—671no—
295#1482—671no—
296#1454—671no—
297C67#1791911yescharacteristic, complemented, Sylow
298#529C2×D6711yescharacteristic
299C2×C4×Q8C2×D6711yescharacteristic
300#483Dic6711yescharacteristic
301#480Dic6711yescharacteristic
302#479Dic6711yescharacteristic
303#479Dic6711yescharacteristic
304#464Dic6711yescharacteristic
305#464Dic6711yescharacteristic
306#451Dic6711yescharacteristic
307#448Dic6711yescharacteristic
308#446—671no—
309C8×Q8—671no—
310C8×Q8—671no—
311C8×Q8—671no—
312C8×Q8—671no—
313C8×Q8—671no—
314C8×Q8—1341no—
315C8×Q8—1341no—
316C4×Q16C2×D6711yescharacteristic
317C4×Q16C2×D6711yescharacteristic
318C4×SD16C2×D6711yescharacteristic
319C4×SD16C2×D6711yescharacteristic
320#436—671no—
321#436—1341no—
322#435—1341no—
323C8×D4—671no—
324#433—1341no—
325#431—671no—
326#431—671no—
327#431—671no—
328#431—1341no—
329#431—1341no—
330#431C2×D6711yescharacteristic
331#424—1341no—
332#423—671no—
333#335—671no—
334#335—671no—
335#334—671no—
336#334—671no—
337Q8⋊C8—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343D4⋊C8—671no—
344D4⋊C8—671no—
345#322—671no—
346#322—671no—
347#322—671no—
348C82—671no—
349C22×Q8C2×Dic6711yescharacteristic
350C2×Q16C2×Dic6711yescharacteristic
351C2×Q16C2×Dic6711yescharacteristic
352C2×SD16C2×Dic6711yescharacteristic
353C2×SD16C2×Dic6711yescharacteristic
354C2×M16—1341no—
355C2×M16—1341no—
356C22×C8—1341no—
357#126—21no—
358#125C2×Dic6711yescharacteristic
359#124C2×Dic6711yescharacteristic
360#123—21no—
361C22⋊Q8C2×Dic6711yescharacteristic
362#121C2×Dic6711yescharacteristic
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8C2×Dic6711yescharacteristic
366C4×Q8C2×Dic6711yescharacteristic
367C4×Q8C22×D6711yescharacteristic
368C4×Q8C22×D6711yescharacteristic
369C4×Q8C22×D6711yescharacteristic
370C4×D4—21no—
371C4×D4C22×D6711yescharacteristic
372#117—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C2×Dic6711yescharacteristic
375C2×C42—21no—
376C2×C42C22×D6711yescharacteristic
377#107C2×Dic6711yescharacteristic
378#107C2×Dic6711yescharacteristic
379#106C2×Dic6711yescharacteristic
380#106C2×Dic6711yescharacteristic
381C4⋊C8—671no—
382C4⋊C8—671no—
383C4⋊C8—671no—
384C4⋊C8—671no—
385C4⋊C8—671no—
386C4⋊C8—671no—
387C4⋊C8—1341no—
388C4⋊C8—1341no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8—1341no—
395C4⋊C8C2×Dic6711yescharacteristic
396C4⋊C8C2×Dic6711yescharacteristic
397#103C2×Dic6711yescharacteristic
398#103C2×Dic6711yescharacteristic
399#103C2×Dic6711yescharacteristic
400#103C2×Dic6711yescharacteristic
401#103C2×Dic6711yescharacteristic
402#103C2×Dic6711yescharacteristic
403#102C2×Dic6711yescharacteristic
404#102C2×Dic6711yescharacteristic
405C22⋊C8—671no—
406C22⋊C8—671no—
407C22⋊C8—1341no—
408C22⋊C8—1341no—
409C22⋊C8—1341no—
410C22⋊C8—1341no—
411C22⋊C8C2×Dic6711yescharacteristic
412C22⋊C8C2×Dic6711yescharacteristic
413#97—1341no—
414C4×C8—671no—
415C4×C8—671no—
416C4×C8—671no—
417C4×C8—671no—
418C4×C8—1341no—
419C4×C8—1341no—
420C4×C8—1341no—
421C4×C8—1341no—
422C4×C8—1341no—
423C4×C8—1341no—
424C4×C8C22×D6711yescharacteristic
425C4×C8C22×D6711yescharacteristic
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8C22×Dic6711yescharacteristic
430C2×Q8C22×Dic6711yescharacteristic
431C2×Q8C22×Dic6711yescharacteristic
432C2×D4—21no—
433C2×D4C22×Dic6711yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C22×Dic6711yescharacteristic
438C22×C4D4×D6711yescharacteristic
439C22×C4#1176162711yescharacteristic
440Q16—22no—
441Q16—22no—
442SD16—22no—
443SD16—22no—
444M16—5361no—
445M16—5361no—
446C2×C8—5361no—
447C2×C8—671no—
448C2×C8—671no—
449C2×C8—671no—
450C2×C8—671no—
451C2×C8—1341no—
452C2×C8—1341no—
453C2×C8—1341no—
454C2×C8—1341no—
455C2×C8—1341no—
456C2×C8—1341no—
457C2×C8C22×Dic6711yescharacteristic
458C2×C8C22×Dic6711yescharacteristic
459C2×C8C22×Dic6711yescharacteristic
460C2×C8C22×Dic6711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4C22×Dic6711yescharacteristic
471C4⋊C4C22×Dic6711yescharacteristic
472C4⋊C4C22×Dic6711yescharacteristic
473C4⋊C4C22×Dic6711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42C22×Dic6711yescharacteristic
485C42D4×D6711yescharacteristic
486C42#1176162711yescharacteristic
487C42C23×D6711yescharacteristic
488C23—21no—
489C23D4×Dic6711yescharacteristic
490Q8—24no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8#1452405012yes—
496Q8#1452405012yes—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4D4×Dic6711yescharacteristic
517C2×C4D4×Dic6711yescharacteristic
518C2×C4D4×Dic6711yescharacteristic
519C2×C4C23×Dic6711yescharacteristic
520C2×C4C2×D4×D6711yescharacteristic
521C2×C4C2×#1176162711yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—2681no—
527C8—2681no—
528C8—2681no—
529C8—2681no—
530C8—2681no—
531C8—2681no—
532C8—1341no—
533C8—1341no—
534C8—1341no—
535C8—1341no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×D4×Dic6711yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4C2×#1452405011yescharacteristic
555C4C2×#1452405011yescharacteristic
556C4#1686420511yescharacteristic
557C4#1686421211yescharacteristic
558C2—41no—
559C2—81no—
560C2#2256890411yescharacteristic
561C2#2256905311yescharacteristic
562C2#2256912611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#3108C2
socleC22×C67C2×C4×D4
prime radical#121×C67C2×C4
cosocleC42×C67C24
hypercenter#3108D67
hypocenterC67#17919
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3108

Complements

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

NH ≅ G/Ntype
2. #22569290C2semidirect
3. #22559088C2semidirect
4. #22559081C2semidirect
5. #22559060C2semidirect
6. #22559053C2semidirect
7. #22559028C2semidirect
8. SD16⋊(C67⋊C8)C2semidirect
9. Q16×C67⋊C8C2semidirect
10. SD16×C67⋊C8C2semidirect
11. #22558998C2semidirect
12. #22558638C2semidirect
14. #22558509C2semidirect
297. C67#17919semidirect

Maximal subgroups

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

subgroupindex
#225693582
#225692902
#225590882
#225590812
#225590602
#225590532
#225590282
SD16⋊(C67⋊C8)2
Q16×C67⋊C82
SD16×C67⋊C82
#225589982
#225586382
#225586172
#225585092
C67×#31082
#1791967