Finite Group Databasebeta

← #30866238

Subgroups — #30866238

Order 17152.

754 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863423C211yesmaximal, characteristic
2#22562818C211yesmaximal, characteristic
3#22560977C211yesmaximal, characteristic, complemented
4#22560977C211yesmaximal, characteristic, complemented
5#22560975C211yesmaximal, characteristic, complemented
6#22560975C211yesmaximal, characteristic, complemented
7#22560972C211yesmaximal, characteristic, complemented
8#22560970C211yesmaximal, characteristic, complemented
9#22560967C211yesmaximal, characteristic, complemented
10#22560965C211yesmaximal, characteristic, complemented
11#22560240C211yesmaximal, characteristic
12#22560240C211yesmaximal, characteristic
13#22560236C211yesmaximal, characteristic
14#22560232C211yesmaximal, characteristic
15C67×#2314C211yesmaximal, characteristic
16C22×#11761612C2211yescharacteristic
17C23×C67⋊C8C2211yescharacteristic
18C2×#14523976C2211yescharacteristic
19C2×#14523976C2211yescharacteristic
20C2×#14523976C2211yescharacteristic
21C2×#14523976C2211yescharacteristic
22C2×#14523974C2211yescharacteristic
23C2×#14523974C2211yescharacteristic
24C2×#14523974C2211yescharacteristic
25C2×#14523974C2211yescharacteristic
26#16863423C418yescomplemented
27C2×C4⋊(C67⋊C8)C2211yescharacteristic
28C2×C4⋊(C67⋊C8)C2211yescharacteristic
29C2×C4⋊(C67⋊C8)C2211yescharacteristic
30C2×C4⋊(C67⋊C8)C2211yescharacteristic
31#16863289—21no—
32#16863289—21no—
33#16863282—21no—
34#16863282—21no—
35#16863266C2211yescharacteristic
36#16863262C2211yescharacteristic
37#16863256C2211yescharacteristic
38#16863256C2211yescharacteristic
39C4×C4⋊Dic67C2211yescharacteristic
40C4×C4⋊Dic67C2211yescharacteristic
41#16863158C2211yescharacteristic
42#16863158C2211yescharacteristic
43#16863158C2211yescharacteristic
44#16863158C2211yescharacteristic
45#16863158C2211yescharacteristic
46#16863158C2211yescharacteristic
47#16863158C2211yescharacteristic
48#16863158C2211yescharacteristic
49C2×#117×C67C2211yescharacteristic
50#388×C67—21no—
51#388×C67—21no—
52#387×C67—21no—
53#387×C67—21no—
54#384×C67C2211yescharacteristic
55#382×C67C2211yescharacteristic
56#380×C67C2211yescharacteristic
57#380×C67C2211yescharacteristic
58C4×C4⋊C4×C67C2211yescharacteristic
59C4×C4⋊C4×C67C2211yescharacteristic
60C2×#11761612—41no—
61C2×#11761612—41no—
62C2×#11761612C2×C414yes—
63C22×C67⋊C8—41no—
64C22×C67⋊C8—41no—
65C22×C67⋊C8C2×C414yes—
66C22×C67⋊C8C2311yescharacteristic
67C22×C67⋊C8C2311yescharacteristic
68C22×C67⋊C8C2311yescharacteristic
69C22×C67⋊C8C2311yescharacteristic
70C2×C4⋊Dic67—21no—
71C2×C4⋊Dic67—21no—
72C2×C4⋊Dic67C2311yescharacteristic
73C2×C4⋊Dic67C2311yescharacteristic
74C2×C4⋊Dic67C2311yescharacteristic
75C2×C4⋊Dic67C2311yescharacteristic
76C2×C22⋊Dic67—21no—
77C2×C22⋊Dic67—21no—
78C2×C4×Dic67—21no—
79C2×C4×Dic67—21no—
80#14523976—41no—
81#14523976—41no—
82#14523976—41no—
83#14523976—41no—
84#14523974—41no—
85#14523974—41no—
86#14523974—41no—
87#14523974—41no—
88C4⋊(C67⋊C8)C2×C414yes—
89C4⋊(C67⋊C8)C2×C414yes—
90C4⋊(C67⋊C8)C2×C414yes—
91C4⋊(C67⋊C8)C2×C414yes—
92C67⋊#95—21no—
93C67⋊#95—21no—
94C67⋊#95—21no—
95C67⋊#95—21no—
96C67⋊#95—21no—
97C67⋊#95—21no—
98C23×C4×C67C2311yescharacteristic
99#117×C67C2×C414yes—
100#117×C67Q814yes—
101C2×C4⋊C4×C67—21no—
102C2×C4⋊C4×C67—21no—
103C2×C4⋊C4×C67D411yescharacteristic
104C2×C4⋊C4×C67D411yescharacteristic
105C2×C4⋊C4×C67C2311yescharacteristic
106C2×C4⋊C4×C67C2311yescharacteristic
107C2×C4⋊C4×C67C2311yescharacteristic
108C2×C4⋊C4×C67C2311yescharacteristic
109C2×C22⋊C4×C67—21no—
110C2×C22⋊C4×C67—21no—
111C2×C22⋊C4×C67D411yescharacteristic
112C2×C22⋊C4×C67D411yescharacteristic
113C2×C42×C67—21no—
114C2×C42×C67—21no—
115C2×C42×C67C2311yescharacteristic
116C2×C42×C67C2311yescharacteristic
117#95×C67—21no—
118#95×C67—21no—
119#95×C67—21no—
120#95×C67—21no—
121#95×C67—21no—
122#95×C67—21no—
123C22×Dic67—21no—
124C22×Dic67—21no—
125C22×Dic67—21no—
126C22×Dic67—21no—
127#11761612—44no—
128C2×C67⋊C8—21no—
129C2×C67⋊C8—21no—
130C2×C67⋊C8—21no—
131C2×C67⋊C8—21no—
132C2×C67⋊C8—21no—
133C2×C67⋊C8—21no—
134C2×C67⋊C8—21no—
135C2×C67⋊C8—21no—
136C2×C67⋊C8—44no—
137C2×C67⋊C8C22×C412yes—
138C2×C67⋊C8C22×C412yes—
139C2×C67⋊C8C22×C412yes—
140C2×C67⋊C8C22×C412yes—
141C4⋊Dic67—21no—
142C4⋊Dic67—21no—
143C4⋊Dic67—21no—
144C4⋊Dic67—21no—
145C4⋊Dic67—21no—
146C4⋊Dic67—21no—
147C4⋊Dic67—21no—
148C4⋊Dic67—21no—
149C4⋊Dic67—81no—
150C4⋊Dic67—81no—
151C22⋊Dic67—81no—
152C22⋊Dic67—81no—
153C4×Dic67—81no—
154C4×Dic67—81no—
155C24×C67C2×D411yescharacteristic
156C22×C4×C67—21no—
157C22×C4×C67—21no—
158C22×C4×C67—21no—
159C22×C4×C67—21no—
160C22×C4×C67—21no—
161C22×C4×C67—21no—
162C22×C4×C67—21no—
163C22×C4×C67—21no—
164C22×C4×C67C22×C412yes—
165C22×C4×C67C2×D411yescharacteristic
166C22×C4×C67C2×D411yescharacteristic
167C22×C4×C67C2×D411yescharacteristic
168C22×C4×C67C2×D411yescharacteristic
169C22×C4×C67C2×D411yescharacteristic
170C22×C4×C67C2×Q812yes—
171C22×C4×C67C2411yescharacteristic
172C4⋊C4×C67—21no—
173C4⋊C4×C67—21no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—21no—
177C4⋊C4×C67—21no—
178C4⋊C4×C67—21no—
179C4⋊C4×C67—21no—
180C4⋊C4×C67—81no—
181C4⋊C4×C67—81no—
182C4⋊C4×C67C4⋊C414yes—
183C4⋊C4×C67C4⋊C414yes—
184C22⋊C4×C67—81no—
185C22⋊C4×C67—81no—
186C22⋊C4×C67C4⋊C414yes—
187C22⋊C4×C67C4⋊C414yes—
188C42×C67—81no—
189C42×C67—81no—
190C42×C67C22×C412yes—
191C42×C67C22×C412yes—
192C42×C67C2×Q812yes—
193C42×C67C2×Q812yes—
194C23×C67—21no—
195C23×C67—21no—
196C23×C67—21no—
197C23×C67—21no—
198C23×C67—21no—
199C23×C67—21no—
200C23×C67C2×C4⋊C412yes—
201C23×C67C22×D411yescharacteristic
202C2×C4×C67—21no—
203C2×C4×C67—21no—
204C2×C4×C67—21no—
205C2×C4×C67—21no—
206C2×C4×C67—21no—
207C2×C4×C67—21no—
208C2×C4×C67—21no—
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—22no—
213C2×C4×C67—22no—
214C2×C4×C67—22no—
215C2×C4×C67—22no—
216C2×C4×C67—41no—
217C2×C4×C67—41no—
218C2×C4×C67—41no—
219C2×C4×C67—41no—
220C2×C4×C67—41no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—41no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67—41no—
227C2×C4×C67—41no—
228C2×C4×C67C2×C4⋊C412yes—
229C2×C4×C67C2×C4⋊C412yes—
230C2×C4×C67C2×C4⋊C412yes—
231C2×C4×C67C2×C4⋊C412yes—
232C2×C4×C67C2×C4⋊C412yes—
233C2×C4×C67C23×C411yescharacteristic
234C2×C4×C67C22×Q811yescharacteristic
235C2×C4×C67H(5,𝔽2)11yescharacteristic
236C2×C4×C67H(5,𝔽2)11yescharacteristic
237C2×C4×C672−1+411yescharacteristic
238C2×C4×C672−1+411yescharacteristic
239C2×Dic67—41no—
240C2×Dic67—41no—
241C2×Dic67—41no—
242C2×Dic67—41no—
243C2×Dic67—41no—
244C2×Dic67—41no—
245C2×Dic67—41no—
246C2×Dic67—41no—
247C2×Dic67—41no—
248C2×Dic67—41no—
249C2×Dic67—41no—
250C2×Dic67—41no—
251C67⋊C8—22no—
252C67⋊C8—22no—
253C67⋊C8—22no—
254C67⋊C8—22no—
255C22×C67—22no—
256C22×C67—22no—
257C22×C67—22no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67—41no—
262C22×C67C22×C4⋊C411yescharacteristic
263C22×C67#53711yescharacteristic
264C22×C67#53711yescharacteristic
265C22×C67#57611yescharacteristic
266C22×C67#57711yescharacteristic
267C22×C67#57711yescharacteristic
268C22×C67#57811yescharacteristic
269C4×C67—22no—
270C4×C67—22no—
271C4×C67—22no—
272C4×C67—22no—
273C4×C67—22no—
274C4×C67—81no—
275C4×C67—81no—
276C4×C67—81no—
277C4×C67—81no—
278C4×C67#51811yescharacteristic
279C4×C67#51911yescharacteristic
280C4×C67#54411yescharacteristic
281C4×C67#54411yescharacteristic
282Dic67—81no—
283Dic67—81no—
284Dic67—81no—
285Dic67—81no—
286#21966—671nomaximal, Sylow
287C2×C67—42no—
288C2×C67#231611yescharacteristic
289C2×C67#291211yescharacteristic
290C2×C67#291311yescharacteristic
291C2×C67#325711yescharacteristic
292C2×C67#325711yescharacteristic
293C2×C67#325811yescharacteristic
294C2×C67#325811yescharacteristic
295C2×#424—671no—
296#2314—671no—
297#2314D6711yescharacteristic
298#1949—671no—
299#1949—671no—
300#1949—671no—
301#1949—671no—
302#1948—671no—
303#1948—671no—
304#1947—671no—
305#1947—671no—
306#1809—671no—
307#1809—671no—
308#1808—671no—
309#1807—671no—
310C67#2196611yescharacteristic, complemented, Sylow
311C22×M16—671no—
312C23×C8—671no—
313C2×#117C2×D6711yescharacteristic
314C2×#107—671no—
315C2×#107—671no—
316C2×#107—671no—
317C2×#107—671no—
318C2×#106—671no—
319C2×#106—671no—
320C2×#106—671no—
321C2×#106—671no—
322#424—678no—
323C2×C4⋊C8—671no—
324C2×C4⋊C8—671no—
325C2×C4⋊C8—671no—
326C2×C4⋊C8—671no—
327#388—21no—
328#388—21no—
329#388—1341no—
330#388—1341no—
331#387—21no—
332#387—21no—
333#387—1341no—
334#387—1341no—
335#384—671no—
336#384C2×D6711yescharacteristic
337#382—671no—
338#382C2×D6711yescharacteristic
339#380—671no—
340#380—671no—
341#380C2×D6711yescharacteristic
342#380C2×D6711yescharacteristic
343C4×C4⋊C4—671no—
344C4×C4⋊C4—671no—
345C4×C4⋊C4C2×D6711yescharacteristic
346C4×C4⋊C4C2×D6711yescharacteristic
347#340—671no—
348#340—671no—
349#340—671no—
350#340—671no—
351#340—671no—
352#340—671no—
353#340—671no—
354#340—671no—
355C23×C4C22×D6711yescharacteristic
356C2×M16—2681no—
357C2×M16—2681no—
358C2×M16—674no—
359C22×C8—2681no—
360C22×C8—2681no—
361C22×C8—674no—
362C22×C8—671no—
363C22×C8—671no—
364C22×C8—671no—
365C22×C8—671no—
366#117Dic13414yes—
367#117C4×D6714yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—671no—
371C2×C4⋊C4—671no—
372C2×C4⋊C4—671no—
373C2×C4⋊C4—671no—
374C2×C4⋊C4—1341no—
375C2×C4⋊C4—1341no—
376C2×C4⋊C4C22⋊D6711yescharacteristic
377C2×C4⋊C4C22⋊D6711yescharacteristic
378C2×C4⋊C4C22×D6711yescharacteristic
379C2×C4⋊C4C22×D6711yescharacteristic
380C2×C4⋊C4C22×D6711yescharacteristic
381C2×C4⋊C4C22×D6711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1341no—
385C2×C22⋊C4—1341no—
386C2×C22⋊C4C22⋊D6711yescharacteristic
387C2×C22⋊C4C22⋊D6711yescharacteristic
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—1341no—
391C2×C42—1341no—
392C2×C42C22×D6711yescharacteristic
393C2×C42C22×D6711yescharacteristic
394#107—2681no—
395#107—2681no—
396#107—2681no—
397#107—2681no—
398#106—2681no—
399#106—2681no—
400#106—2681no—
401#106—2681no—
402C4⋊C8—674no—
403C4⋊C8—674no—
404C4⋊C8—674no—
405C4⋊C8—674no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—1341no—
413#95—1341no—
414#95—1341no—
415#95—1341no—
416#95—1341no—
417#95—1341no—
418C24C2×C22⋊D6711yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—1341no—
428C22×C4—1341no—
429C22×C4—1341no—
430C22×C4—1341no—
431C22×C4C2×C4×D6712yes—
432C22×C4C2×C22⋊D6711yescharacteristic
433C22×C4C2×C22⋊D6711yescharacteristic
434C22×C4C2×C22⋊D6711yescharacteristic
435C22×C4C2×C22⋊D6711yescharacteristic
436C22×C4C2×C22⋊D6711yescharacteristic
437C22×C4C2×Dic13412yes—
438C22×C4C23×D6711yescharacteristic
439M16—2684no—
440C2×C8—2684no—
441C2×C8—672no—
442C2×C8—672no—
443C2×C8—672no—
444C2×C8—672no—
445C2×C8—1341no—
446C2×C8—1341no—
447C2×C8—1341no—
448C2×C8—1341no—
449C2×C8—1341no—
450C2×C8—1341no—
451C2×C8—1341no—
452C2×C8—1341no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—5361no—
464C4⋊C4—5361no—
465C4⋊C4—1341no—
466C4⋊C4—1341no—
467C4⋊C4—1341no—
468C4⋊C4—1341no—
469C4⋊C4—1341no—
470C4⋊C4—1341no—
471C4⋊C4—1341no—
472C4⋊C4—1341no—
473C4⋊C4Dic67⋊C414yes—
474C4⋊C4Dic67⋊C414yes—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—5361no—
478C22⋊C4—5361no—
479C22⋊C4Dic67⋊C414yes—
480C22⋊C4Dic67⋊C414yes—
481C42—81no—
482C42—81no—
483C42—5361no—
484C42—5361no—
485C42C2×C4×D6712yes—
486C42C2×C4×D6712yes—
487C42C2×Dic13412yes—
488C42C2×Dic13412yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C2×Dic67⋊C412yes—
496C23C22×C22⋊D6711yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—2681no—
524C2×C4—2681no—
525C2×C4—2681no—
526C2×C4—2681no—
527C2×C4—2681no—
528C2×C4—2681no—
529C2×C4—2681no—
530C2×C4—2681no—
531C2×C4—2681no—
532C2×C4—2681no—
533C2×C4—2681no—
534C2×C4—2681no—
535C2×C4C2×Dic67⋊C412yes—
536C2×C4C2×Dic67⋊C412yes—
537C2×C4C2×Dic67⋊C412yes—
538C2×C4C2×Dic67⋊C412yes—
539C2×C4C2×Dic67⋊C412yes—
540C2×C4C22×C4×D6711yescharacteristic
541C2×C4C22×Dic13411yescharacteristic
542C2×C4#1452408911yescharacteristic
543C2×C4#1452408911yescharacteristic
544C2×C4#1452409011yescharacteristic
545C2×C4#1452409011yescharacteristic
546C8—1342no—
547C8—1342no—
548C8—1342no—
549C8—1342no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C22×Dic67⋊C411yescharacteristic
558C22#1686397411yescharacteristic
559C22#1686397411yescharacteristic
560C22#1686419911yescharacteristic
561C22#1686420411yescharacteristic
562C22#1686420711yescharacteristic
563C22#1686421111yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—5361no—
574C4—5361no—
575C4—5361no—
576C4—5361no—
577C4#1686387011yescharacteristic
578C4#1686387311yescharacteristic
579C4#1686402411yescharacteristic
580C4#1686402411yescharacteristic
581C2—42no—
582C2#2256282511yescharacteristic
583C2#2256880211yescharacteristic
584C2#2256880711yescharacteristic
585C2#2257161411yescharacteristic
586C2#2257162011yescharacteristic
587C2#2257162911yescharacteristic
588C2#2257163511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2314C2
socleC23×C67C22×D4
prime radicalC24×C67C2×D4
cosocleC22×C4×C67C24
hypercenter#2314D67
hypocenterC67#21966
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
3. #22560977C2semidirect
4. #22560977C2semidirect
5. #22560975C2semidirect
6. #22560975C2semidirect
7. #22560972C2semidirect
8. #22560970C2semidirect
9. #22560967C2semidirect
10. #22560965C2semidirect
26. #16863423C4semidirect
310. C67#21966semidirect

Maximal subgroups

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

subgroupindex
C2×#168634232
#225628182
#225609772
#225609752
#225609722
#225609702
#225609672
#225609652
#225602402
#225602362
#225602322
C67×#23142
#2196667