Finite Group Databasebeta

← #30865382

Subgroups — #30865382

Order 17152.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863457C211yesmaximal, characteristic
2#22562818C211yesmaximal, characteristic
3#22561183C211yesmaximal, characteristic, complemented
4#22561179C211yesmaximal, characteristic, complemented
5#22561175C211yesmaximal, characteristic, complemented
6#22561171C211yesmaximal, characteristic, complemented
7#22561167C211yesmaximal, characteristic, complemented
8#22561167C211yesmaximal, characteristic, complemented
9#22561162C211yesmaximal, characteristic, complemented
10#22561158C211yesmaximal, characteristic, complemented
11#22560304C211yesmaximal, characteristic
12#22560304C211yesmaximal, characteristic
13#22560299C211yesmaximal, characteristic
14#22560295C211yesmaximal, characteristic
15C2×#544×C67C211yesmaximal, characteristic
16#16863457—24no—
17C2×C4⋊(C67⋊C8)C2211yescharacteristic
18C2×C4⋊(C67⋊C8)C2211yescharacteristic
19C2×C22⋊(C67⋊C8)C2211yescharacteristic
20C2×C22⋊(C67⋊C8)C2211yescharacteristic
21C2×#14523954C2211yescharacteristic
22C2×C4×C67⋊C8C2211yescharacteristic
23#16863289—21no—
24#16863289—21no—
25#16863282—21no—
26#16863282—21no—
27#16863266C2211yescharacteristic
28#16863262C2211yescharacteristic
29#16863256C2211yescharacteristic
30#16863256C2211yescharacteristic
31C4×C4⋊Dic67C2211yescharacteristic
32C4×C4⋊Dic67C2211yescharacteristic
33#16863158C2211yescharacteristic
34#16863158C2211yescharacteristic
35#16863158C2211yescharacteristic
36#16863158C2211yescharacteristic
37#16863158C2211yescharacteristic
38#16863158C2211yescharacteristic
39#16863158C2211yescharacteristic
40#16863158C2211yescharacteristic
41#16863158C2211yescharacteristic
42#16863158C2211yescharacteristic
43#16863158C2211yescharacteristic
44#16863158C2211yescharacteristic
45#16863158C2211yescharacteristic
46#16863158C2211yescharacteristic
47#16863158C2211yescharacteristic
48#16863158C2211yescharacteristic
49#544×C67—24no—
50#544×C67C418yescomplemented
51C2×C4⋊Q8×C67—21no—
52C2×C4⋊Q8×C67C2211yescharacteristic
53C2×C4⋊Q8×C67C2211yescharacteristic
54C2×#125×C67—21no—
55C2×#125×C67C2211yescharacteristic
56C2×#125×C67C2211yescharacteristic
57C2×C22⋊Q8×C67—21no—
58C2×C22⋊Q8×C67—21no—
59C2×#117×C67C2211yescharacteristic
60C2×#117×C67C2211yescharacteristic
61C22×C4⋊C4×C67C2211yescharacteristic
62C22×C67⋊C8C2311yescharacteristic
63C22×C67⋊C8C2311yescharacteristic
64C22×C67⋊C8C2311yescharacteristic
65C22×C67⋊C8C2311yescharacteristic
66C2×C4⋊Dic67—21no—
67C2×C4⋊Dic67—21no—
68C2×C4⋊Dic67C2311yescharacteristic
69C2×C4⋊Dic67C2311yescharacteristic
70C2×C4⋊Dic67C2311yescharacteristic
71C2×C4⋊Dic67C2311yescharacteristic
72C2×C22⋊Dic67—21no—
73C2×C22⋊Dic67—21no—
74C2×C4×Dic67—21no—
75C2×C4×Dic67—21no—
76C4⋊(C67⋊C8)—22no—
77C4⋊(C67⋊C8)—22no—
78C22⋊(C67⋊C8)—22no—
79C22⋊(C67⋊C8)—22no—
80#14523954—22no—
81C4×C67⋊C8—22no—
82C67⋊#95—21no—
83C67⋊#95—21no—
84C67⋊#95—21no—
85C67⋊#95—21no—
86C67⋊#95—21no—
87C67⋊#95—21no—
88C22×Q8×C67—21no—
89C22×Q8×C67—21no—
90C23×C4×C67—21no—
91C23×C4×C67C2311yescharacteristic
92C4⋊Q8×C67—28no—
93C4⋊Q8×C67—22no—
94C4⋊Q8×C67—22no—
95C4⋊Q8×C67C2×C414yes—
96C4⋊Q8×C67C2×C414yes—
97#125×C67—28no—
98#125×C67—22no—
99#125×C67—22no—
100#125×C67C2×C414yes—
101#125×C67C2×C414yes—
102C22⋊Q8×C67—28no—
103C22⋊Q8×C67—28no—
104#117×C67—22no—
105#117×C67C2×C414yes—
106#117×C67C2×C414yes—
107#117×C67D414yes—
108C2×C4⋊C4×C67—21no—
109C2×C4⋊C4×C67—22no—
110C2×C4⋊C4×C67—21no—
111C2×C4⋊C4×C67—21no—
112C2×C4⋊C4×C67—21no—
113C2×C4⋊C4×C67—21no—
114C2×C4⋊C4×C67—21no—
115C2×C4⋊C4×C67—21no—
116C2×C4⋊C4×C67C2×C414yes—
117C2×C4⋊C4×C67D411yescharacteristic
118C2×C4⋊C4×C67D411yescharacteristic
119C2×C4⋊C4×C67C2311yescharacteristic
120C2×C4⋊C4×C67C2311yescharacteristic
121C2×C4⋊C4×C67C2311yescharacteristic
122C2×C4⋊C4×C67C2311yescharacteristic
123C2×C22⋊C4×C67—21no—
124C2×C22⋊C4×C67D411yescharacteristic
125C2×C22⋊C4×C67D411yescharacteristic
126C2×C42×C67—21no—
127C2×C42×C67C2311yescharacteristic
128C2×C42×C67C2311yescharacteristic
129C22×Dic67—21no—
130C22×Dic67—21no—
131C22×Dic67—21no—
132C22×Dic67—21no—
133C2×C67⋊C8—21no—
134C2×C67⋊C8—21no—
135C2×C67⋊C8—21no—
136C2×C67⋊C8—21no—
137C2×C67⋊C8—21no—
138C2×C67⋊C8—21no—
139C2×C67⋊C8—21no—
140C2×C67⋊C8—21no—
141C2×C67⋊C8—21no—
142C2×C67⋊C8—21no—
143C2×C67⋊C8—21no—
144C2×C67⋊C8—21no—
145C4⋊Dic67—21no—
146C4⋊Dic67—21no—
147C4⋊Dic67—21no—
148C4⋊Dic67—21no—
149C4⋊Dic67—21no—
150C4⋊Dic67—21no—
151C4⋊Dic67—21no—
152C4⋊Dic67—21no—
153C4⋊Dic67—81no—
154C4⋊Dic67—81no—
155C22⋊Dic67—81no—
156C22⋊Dic67—81no—
157C4×Dic67—81no—
158C4×Dic67—81no—
159C24×C67C2×D411yescharacteristic
160C2×Q8×C67—24no—
161C2×Q8×C67—24no—
162C2×Q8×C67—81no—
163C2×Q8×C67—81no—
164C2×Q8×C67—81no—
165C2×Q8×C67—81no—
166C22×C4×C67—21no—
167C22×C4×C67—21no—
168C22×C4×C67—21no—
169C22×C4×C67—21no—
170C22×C4×C67—21no—
171C22×C4×C67—21no—
172C22×C4×C67—21no—
173C22×C4×C67—21no—
174C22×C4×C67—24no—
175C22×C4×C67—81no—
176C22×C4×C67—81no—
177C22×C4×C67C22×C412yes—
178C22×C4×C67C2×D411yescharacteristic
179C22×C4×C67C2×D411yescharacteristic
180C22×C4×C67C2×D411yescharacteristic
181C22×C4×C67C2×D411yescharacteristic
182C22×C4×C67C2×D411yescharacteristic
183C22×C4×C67C2×D412yes—
184C22×C4×C67C2411yescharacteristic
185C4⋊C4×C67—24no—
186C4⋊C4×C67—24no—
187C4⋊C4×C67—24no—
188C4⋊C4×C67—24no—
189C4⋊C4×C67—24no—
190C4⋊C4×C67—24no—
191C4⋊C4×C67—24no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67C22⋊C414yes—
197C4⋊C4×C67C22⋊C414yes—
198C4⋊C4×C67C22×C412yes—
199C4⋊C4×C67C22×C412yes—
200C4⋊C4×C67C22×C412yes—
201C4⋊C4×C67C22×C412yes—
202C22⋊C4×C67—24no—
203C22⋊C4×C67C22⋊C414yes—
204C22⋊C4×C67C22⋊C414yes—
205C42×C67—24no—
206C42×C67C22×C412yes—
207C42×C67C22×C412yes—
208C42×C67C2×D412yes—
209C42×C67C2×D412yes—
210C23×C67—21no—
211C23×C67—21no—
212C23×C67—21no—
213C23×C67—21no—
214C23×C67—21no—
215C23×C67—21no—
216C23×C67C2×C22⋊C412yes—
217C23×C67C22×D411yescharacteristic
218Q8×C67—84no—
219Q8×C67—84no—
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—22no—
231C2×C4×C67—22no—
232C2×C4×C67—22no—
233C2×C4×C67—22no—
234C2×C4×C67—22no—
235C2×C4×C67—22no—
236C2×C4×C67—22no—
237C2×C4×C67—22no—
238C2×C4×C67—41no—
239C2×C4×C67—41no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—41no—
245C2×C4×C67—41no—
246C2×C4×C67—84no—
247C2×C4×C67C2×C22⋊C412yes—
248C2×C4×C67C2×C22⋊C412yes—
249C2×C4×C67C2×C22⋊C412yes—
250C2×C4×C67C2×C22⋊C412yes—
251C2×C4×C67C2×C22⋊C412yes—
252C2×C4×C67C23×C411yescharacteristic
253C2×C4×C67C22×D411yescharacteristic
254C2×C4×C672−1+411yescharacteristic
255C2×C4×C672−1+411yescharacteristic
256C2×C4×C672−1+411yescharacteristic
257C2×C4×C672−1+411yescharacteristic
258C2×Dic67—41no—
259C2×Dic67—41no—
260C2×Dic67—41no—
261C2×Dic67—41no—
262C2×Dic67—41no—
263C2×Dic67—41no—
264C2×Dic67—41no—
265C2×Dic67—41no—
266C2×Dic67—41no—
267C2×Dic67—41no—
268C2×Dic67—41no—
269C2×Dic67—41no—
270C67⋊C8—41no—
271C67⋊C8—41no—
272C67⋊C8—41no—
273C67⋊C8—41no—
274C22×C67—22no—
275C22×C67—22no—
276C22×C67—22no—
277C22×C67—41no—
278C22×C67—41no—
279C22×C67—41no—
280C22×C67—41no—
281C22×C67C22×C22⋊C411yescharacteristic
282C22×C67#53611yescharacteristic
283C22×C67#53611yescharacteristic
284C22×C67#57611yescharacteristic
285C22×C67#57711yescharacteristic
286C22×C67#57711yescharacteristic
287C22×C67#57811yescharacteristic
288C4×C67—22no—
289C4×C67—22no—
290C4×C67—22no—
291C4×C67—22no—
292C4×C67—22no—
293C4×C67—42no—
294C4×C67—42no—
295C4×C67—42no—
296C4×C67—42no—
297C4×C67#51911yescharacteristic
298C4×C67#51911yescharacteristic
299C4×C67#53611yescharacteristic
300C4×C67#53611yescharacteristic
301Dic67—81no—
302Dic67—81no—
303Dic67—81no—
304Dic67—81no—
305#21843—671nomaximal, Sylow
306C2×C67—42no—
307C2×C67#231011yescharacteristic
308C2×C67#289711yescharacteristic
309C2×C67#289811yescharacteristic
310C2×C67#324111yescharacteristic
311C2×C67#324211yescharacteristic
312C2×C67#324311yescharacteristic
313C2×C67#324411yescharacteristic
314C2×#544D6711yescharacteristic
315C2×#433—671no—
316#2314—671no—
317#1993—671no—
318#1992—671no—
319#1991—671no—
320#1990—671no—
321#1989—671no—
322#1989—671no—
323#1988—671no—
324#1987—671no—
325#1825—671no—
326#1825—671no—
327#1824—671no—
328#1823—671no—
329C67#2184311yescharacteristic, complemented, Sylow
330#544—24no—
331#544Dic6718yescomplemented
332C2×C4⋊Q8—21no—
333C2×C4⋊Q8C2×D6711yescharacteristic
334C2×C4⋊Q8C2×D6711yescharacteristic
335C2×#125—21no—
336C2×#125C2×D6711yescharacteristic
337C2×#125C2×D6711yescharacteristic
338C2×C22⋊Q8—21no—
339C2×C22⋊Q8—21no—
340C2×#117C2×D6711yescharacteristic
341C2×#117C2×D6711yescharacteristic
342C22×C4⋊C4C2×D6711yescharacteristic
343#433—1344no—
344C2×C4⋊C8—671no—
345C2×C4⋊C8—671no—
346C2×C22⋊C8—671no—
347C2×C22⋊C8—671no—
348C2×#97—671no—
349C2×C4×C8—671no—
350#388—1341no—
351#388—1341no—
352#387—1341no—
353#387—1341no—
354#384—671no—
355#382—671no—
356#380—671no—
357#380—671no—
358C4×C4⋊C4—671no—
359C4×C4⋊C4—671no—
360#340—671no—
361#340—671no—
362#340—671no—
363#340—671no—
364#340—671no—
365#340—671no—
366#340—671no—
367#340—671no—
368#340—671no—
369#340—671no—
370#340—671no—
371#340—671no—
372#340—671no—
373#340—671no—
374#340—671no—
375#340—671no—
376C22×Q8—21no—
377C22×Q8—21no—
378C23×C4—21no—
379C23×C4C22×D6711yescharacteristic
380C22×C8—671no—
381C22×C8—671no—
382C22×C8—671no—
383C22×C8—671no—
384C4⋊Q8—28no—
385C4⋊Q8—22no—
386C4⋊Q8—22no—
387C4⋊Q8C2×Dic6714yes—
388C4⋊Q8C2×Dic6714yes—
389#125—28no—
390#125—22no—
391#125—22no—
392#125C2×Dic6714yes—
393#125C2×Dic6714yes—
394C22⋊Q8—28no—
395C22⋊Q8—28no—
396#117—22no—
397#117C2×Dic6714yes—
398#117C2×Dic6714yes—
399#117C22⋊D6714yes—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—671no—
409C2×C4⋊C4—671no—
410C2×C4⋊C4—671no—
411C2×C4⋊C4—671no—
412C2×C4⋊C4—1341no—
413C2×C4⋊C4—1341no—
414C2×C4⋊C4C2×Dic6714yes—
415C2×C4⋊C4C22⋊D6711yescharacteristic
416C2×C4⋊C4C22⋊D6711yescharacteristic
417C2×C4⋊C4C22×D6711yescharacteristic
418C2×C4⋊C4C22×D6711yescharacteristic
419C2×C4⋊C4C22×D6711yescharacteristic
420C2×C4⋊C4C22×D6711yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1341no—
423C2×C22⋊C4—1341no—
424C2×C22⋊C4C22⋊D6711yescharacteristic
425C2×C22⋊C4C22⋊D6711yescharacteristic
426C2×C42—21no—
427C2×C42—1341no—
428C2×C42—1341no—
429C2×C42C22×D6711yescharacteristic
430C2×C42C22×D6711yescharacteristic
431C4⋊C8—1342no—
432C4⋊C8—1342no—
433C22⋊C8—1342no—
434C22⋊C8—1342no—
435#97—1342no—
436C4×C8—1342no—
437#95—1341no—
438#95—1341no—
439#95—1341no—
440#95—1341no—
441#95—1341no—
442#95—1341no—
443C24C2×C22⋊D6711yescharacteristic
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—81no—
447C2×Q8—81no—
448C2×Q8—81no—
449C2×Q8—81no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—1341no—
462C22×C4—1341no—
463C22×C4—1341no—
464C22×C4—1341no—
465C22×C4C22×Dic6712yes—
466C22×C4C2×C22⋊D6711yescharacteristic
467C22×C4C2×C22⋊D6711yescharacteristic
468C22×C4C2×C22⋊D6711yescharacteristic
469C22×C4C2×C22⋊D6711yescharacteristic
470C22×C4C2×C22⋊D6711yescharacteristic
471C22×C4C2×C22⋊D6712yes—
472C22×C4C23×D6711yescharacteristic
473C2×C8—1341no—
474C2×C8—1341no—
475C2×C8—1341no—
476C2×C8—1341no—
477C2×C8—1341no—
478C2×C8—1341no—
479C2×C8—1341no—
480C2×C8—1341no—
481C2×C8—1341no—
482C2×C8—1341no—
483C2×C8—1341no—
484C2×C8—1341no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—5361no—
497C4⋊C4—5361no—
498C4⋊C4—1341no—
499C4⋊C4—1341no—
500C4⋊C4—1341no—
501C4⋊C4—1341no—
502C4⋊C4—1341no—
503C4⋊C4—1341no—
504C4⋊C4—1341no—
505C4⋊C4—1341no—
506C4⋊C4C22⋊Dic6714yes—
507C4⋊C4C22⋊Dic6714yes—
508C4⋊C4C22×Dic6712yes—
509C4⋊C4C22×Dic6712yes—
510C4⋊C4C22×Dic6712yes—
511C4⋊C4C22×Dic6712yes—
512C22⋊C4—24no—
513C22⋊C4—5361no—
514C22⋊C4—5361no—
515C22⋊C4C22⋊Dic6714yes—
516C22⋊C4C22⋊Dic6714yes—
517C42—24no—
518C42—5361no—
519C42—5361no—
520C42C22×Dic6712yes—
521C42C22×Dic6712yes—
522C42C2×C22⋊D6712yes—
523C42C2×C22⋊D6712yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×C22⋊Dic6712yes—
531C23C22×C22⋊D6711yescharacteristic
532Q8—84no—
533Q8—84no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—2681no—
562C2×C4—2681no—
563C2×C4—2681no—
564C2×C4—2681no—
565C2×C4—2681no—
566C2×C4—2681no—
567C2×C4—2681no—
568C2×C4—2681no—
569C2×C4—2681no—
570C2×C4—2681no—
571C2×C4—2681no—
572C2×C4—2681no—
573C2×C4C2×C22⋊Dic6712yes—
574C2×C4C2×C22⋊Dic6712yes—
575C2×C4C2×C22⋊Dic6712yes—
576C2×C4C2×C22⋊Dic6712yes—
577C2×C4C2×C22⋊Dic6712yes—
578C2×C4C23×Dic6711yescharacteristic
579C2×C4C22×C22⋊D6711yescharacteristic
580C2×C4#1452409111yescharacteristic
581C2×C4#1452409111yescharacteristic
582C2×C4#1452409111yescharacteristic
583C2×C4#1452409111yescharacteristic
584C8—2681no—
585C8—2681no—
586C8—2681no—
587C8—2681no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×C22⋊Dic6711yescharacteristic
596C22#1686397011yescharacteristic
597C22#1686397011yescharacteristic
598C22#1686419911yescharacteristic
599C22#1686420411yescharacteristic
600C22#1686420711yescharacteristic
601C22#1686421111yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—5361no—
612C4—5361no—
613C4—5361no—
614C4—5361no—
615C4#1686387411yescharacteristic
616C4#1686387411yescharacteristic
617C4#1686397011yescharacteristic
618C4#1686397011yescharacteristic
619C2—42no—
620C2#2256279511yescharacteristic
621C2#2256873511yescharacteristic
622C2#2256874211yescharacteristic
623C2#2257146211yescharacteristic
624C2#2257147711yescharacteristic
625C2#2257149211yescharacteristic
626C2#2257150711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#544×C67C2
socleC23×C67C22×D4
prime radicalC24×C67C2×D4
cosocleC22×C4×C67C24
hypercenterC2×#544D67
hypocenterC67#21843
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #22561183C2semidirect
4. #22561179C2semidirect
5. #22561175C2semidirect
6. #22561171C2semidirect
7. #22561167C2semidirect
8. #22561167C2semidirect
9. #22561162C2semidirect
10. #22561158C2semidirect
50. #544×C67C4semidirect
329. C67#21843semidirect
331. #544Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168634572
#225628182
#225611832
#225611792
#225611752
#225611712
#225611672
#225611622
#225611582
#225603042
#225602992
#225602952
C2×#544×C672
#2184367