Finite Group Databasebeta

← #30825060

Subgroups — #30825060

Order 17152.

610 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22569165C211yesmaximal, characteristic
2#22568876C211yesmaximal, characteristic
3#22568870C211yesmaximal, characteristic
4#22558989C211yesmaximal, characteristic, complemented
5#22558986C211yesmaximal, characteristic, complemented
6#22558986C211yesmaximal, characteristic, complemented
7#22558982C211yesmaximal, characteristic, complemented
8#22558982C211yesmaximal, characteristic, complemented
9#22558971C211yesmaximal, characteristic, complemented
10#22558970C211yesmaximal, characteristic, complemented
11#22558963C211yesmaximal, characteristic, complemented
12#22558956C211yesmaximal, characteristic, complemented
13#22558955C211yesmaximal, characteristic, complemented
14#334×D67C211yesmaximal, characteristic, complemented
15C67×#1572C211yesmaximal, characteristic, complemented
16C4×#11761628C2211yescharacteristic
17#16863520C411yescharacteristic
18#16863520C411yescharacteristic
19#16863461C411yescharacteristic
20#16863461C411yescharacteristic
21(C22⋊D67)⋊C8C411yescharacteristic
22(C22⋊D67)⋊C8C411yescharacteristic
23(C22⋊D67)⋊C8C411yescharacteristic
24(C22⋊D67)⋊C8C411yescharacteristic
25#16863458—21no—
26#16863455—21no—
27#16863454—21no—
28#16863454—21no—
29#16863453C2211yescharacteristic
30#16863452C2211yescharacteristic
31#16863452C2211yescharacteristic
32#16863449C2211yescharacteristic
33#16863448C2211yescharacteristic
34#16863448C2211yescharacteristic
35#16863447C2211yescharacteristic
36#16863447C2211yescharacteristic
37#16863418C2211yescharacteristic
38#16863418C2211yescharacteristic
39C4⋊C8×D67C2211yescharacteristic
40C4⋊C8×D67C2211yescharacteristic
41#16863361—21no—
42#16863361—21no—
43#16863353—21no—
44#16863350—21no—
45C2×C4×C67⋊C8C2211yescharacteristic
46C4×C8×D67C2211yescharacteristic
47C4×C8×D67C2211yescharacteristic
48#16863149C2211yescharacteristic
49#16863149C2211yescharacteristic
50#16863148C2211yescharacteristic
51#16863148C2211yescharacteristic
52#16863148C2211yescharacteristic
53#16863148C2211yescharacteristic
54#16863147C2211yescharacteristic
55#16863147C2211yescharacteristic
56#16863146C2211yescharacteristic
57#16863146C2211yescharacteristic
58#16863146C2211yescharacteristic
59#16863146C2211yescharacteristic
60#432×C67C2211yescharacteristic
61#431×C67C2211yescharacteristic
62#431×C67C2211yescharacteristic
63#335×C67C2211yescharacteristic
64#335×C67C2211yescharacteristic
65#334×C67C2211yescharacteristic
66#334×C67C2211yescharacteristic
67C2×#11761628D411yescharacteristic
68C2×#11761611—21no—
69C2×#11761611—21no—
70C22×C67⋊C8C2×C411yescharacteristic
71C22×C67⋊C8C2×C411yescharacteristic
72C2×C8×D67—21no—
73C2×C8×D67—21no—
74C2×C8×D67C2×C411yescharacteristic
75C2×C8×D67C2×C411yescharacteristic
76C2×C8×D67C2×C411yescharacteristic
77C2×C8×D67C2×C411yescharacteristic
78C4×Dic134C2×C411yescharacteristic
79C4×Dic134Q811yescharacteristic
80C4×D268C2×C411yescharacteristic
81C4×D268Q811yescharacteristic
82C4×C22⋊D67C2×C411yescharacteristic
83C4×C22⋊D67C2×C411yescharacteristic
84C4×C22⋊D67Q811yescharacteristic
85C4×C22⋊D67Q811yescharacteristic
86#14524002D411yescharacteristic
87#14524000D411yescharacteristic
88#14524000D411yescharacteristic
89C42×D67C2311yescharacteristic
90C42×D67C2311yescharacteristic
91C4⋊(C67⋊C8)C2311yescharacteristic
92C4⋊(C67⋊C8)C2311yescharacteristic
93C4⋊(C67⋊C8)C2311yescharacteristic
94C4⋊(C67⋊C8)C2311yescharacteristic
95Dic67⋊C8—21no—
96Dic67⋊C8—21no—
97Dic67⋊C8—21no—
98Dic67⋊C8—21no—
99Dic67⋊C8C2×C411yescharacteristic
100Dic67⋊C8C2×C411yescharacteristic
101Dic67⋊C8C2×C411yescharacteristic
102Dic67⋊C8C2×C411yescharacteristic
103C22⋊(C67⋊C8)—21no—
104C22⋊(C67⋊C8)—21no—
105(C2×D67)⋊C8—21no—
106(C2×D67)⋊C8—21no—
107(C2×D67)⋊C8—21no—
108(C2×D67)⋊C8—21no—
109(C2×D67)⋊C8C2×C411yescharacteristic
110(C2×D67)⋊C8C2×C411yescharacteristic
111(C2×D67)⋊C8C2×C411yescharacteristic
112(C2×D67)⋊C8C2×C411yescharacteristic
113#14523954—21no—
114#14523953—21no—
115#14523953—21no—
116C4×C67⋊C8—21no—
117C4×C67⋊C8C2×C411yescharacteristic
118C4×C67⋊C8C2×C411yescharacteristic
119C4×C67⋊C8C2311yescharacteristic
120C4×C67⋊C8C2311yescharacteristic
121C8×Dic67—21no—
122C8×Dic67—21no—
123C8×Dic67C2×C411yescharacteristic
124C8×Dic67C2×C411yescharacteristic
125C8×Dic67C2×C411yescharacteristic
126C8×Dic67C2×C411yescharacteristic
127C2×C42×C67C2311yescharacteristic
128C4⋊C8×C67C2×C411yescharacteristic
129C4⋊C8×C67C2×C411yescharacteristic
130C4⋊C8×C67C2311yescharacteristic
131C4⋊C8×C67C2311yescharacteristic
132C4⋊C8×C67C2311yescharacteristic
133C4⋊C8×C67C2311yescharacteristic
134C22⋊C8×C67—21no—
135C22⋊C8×C67—21no—
136C22⋊C8×C67C2×C411yescharacteristic
137C22⋊C8×C67C2×C411yescharacteristic
138#97×C67—21no—
139C4×C8×C67—21no—
140C4×C8×C67C2311yescharacteristic
141C4×C8×C67C2311yescharacteristic
142#11761628—22no—
143#11761628—22no—
144C2×Dic134C4⋊C411yescharacteristic
145C2×D268C4⋊C411yescharacteristic
146C2×C22⋊D67C4⋊C411yescharacteristic
147C2×C22⋊D67C4⋊C411yescharacteristic
148C2×C4×D67C22×C411yescharacteristic
149C2×C4×D67C22×C411yescharacteristic
150C2×C4×D67C2×D411yescharacteristic
151C2×C4×D67C2×D411yescharacteristic
152C2×C4×D67C2×Q811yescharacteristic
153C2×C4×D67C2×Q811yescharacteristic
154#11761611—42no—
155#11761611—42no—
156C2×C67⋊C8—22no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8—21no—
160C2×C67⋊C8—21no—
161C2×C67⋊C8—41no—
162C2×C67⋊C8C22×C411yescharacteristic
163C2×C67⋊C8C22×C411yescharacteristic
164C2×C67⋊C8C22×C411yescharacteristic
165C2×C67⋊C8C22×C411yescharacteristic
166C8×D67—22no—
167C8×D67—22no—
168C8×D67—41no—
169C8×D67—41no—
170C8×D67—42no—
171C8×D67—42no—
172C4⋊Dic67C4⋊C411yescharacteristic
173C4⋊Dic67C4⋊C411yescharacteristic
174Dic67⋊C4C4⋊C411yescharacteristic
175Dic67⋊C4C4⋊C411yescharacteristic
176Dic67⋊C4C4⋊C411yescharacteristic
177Dic67⋊C4C4⋊C411yescharacteristic
178C22⋊Dic67C4⋊C411yescharacteristic
179C22⋊Dic67C4⋊C411yescharacteristic
180(C2×D67)⋊C4C4⋊C411yescharacteristic
181(C2×D67)⋊C4C4⋊C411yescharacteristic
182(C2×D67)⋊C4C4⋊C411yescharacteristic
183(C2×D67)⋊C4C4⋊C411yescharacteristic
184C4×Dic67C22×C411yescharacteristic
185C4×Dic67C22×C411yescharacteristic
186C4×Dic67C2×D411yescharacteristic
187C4×Dic67C2×D411yescharacteristic
188C4×Dic67C2×Q811yescharacteristic
189C4×Dic67C2×Q811yescharacteristic
190C22×C4×C67C22×C411yescharacteristic
191C22×C4×C67C2×D411yescharacteristic
192C22×C4×C67C2×Q811yescharacteristic
193C2×C8×C67—21no—
194C2×C8×C67—21no—
195C2×C8×C67—21no—
196C2×C8×C67—21no—
197C2×C8×C67C22×C411yescharacteristic
198C2×C8×C67C22×C411yescharacteristic
199C2×C8×C67C22×C411yescharacteristic
200C2×C8×C67C22×C411yescharacteristic
201C42×C67C22×C411yescharacteristic
202C42×C67C2×D411yescharacteristic
203C42×C67C2×Q811yescharacteristic
204C42×C67C2411yescharacteristic
205C23×C67C2×C4⋊C411yescharacteristic
206C22×D67C2×C4⋊C411yescharacteristic
207C22×D67C2×C4⋊C411yescharacteristic
208C2×C4×C67—21no—
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—41no—
213C2×C4×C67C2×C4⋊C411yescharacteristic
214C2×C4×C67C2×C4⋊C411yescharacteristic
215C2×C4×C67C2×C4⋊C411yescharacteristic
216C2×C4×C67C23×C411yescharacteristic
217C2×C4×C67C22×D411yescharacteristic
218C2×C4×C67C22×Q811yescharacteristic
219C22⋊D67—22no—
220C22⋊D67—22no—
221C2×Dic67C2×C4⋊C411yescharacteristic
222C2×Dic67C2×C4⋊C411yescharacteristic
223C2×Dic67C2×C4⋊C411yescharacteristic
224C2×Dic67C2×C4⋊C411yescharacteristic
225C2×Dic67C2×C4⋊C411yescharacteristic
226C2×Dic67C2×C4⋊C411yescharacteristic
227D268—22no—
228C4×D67—21no—
229C4×D67—21no—
230C4×D67—21no—
231C4×D67—21no—
232C4×D67—21no—
233C4×D67—21no—
234C4×D67—41no—
235C4×D67—41no—
236C4×D67#13512yes—
237C4×D67#13512yes—
238Dic134—22no—
239C8×C67—21no—
240C8×C67—21no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—41no—
244C8×C67—41no—
245C8×C67—41no—
246C8×C67—41no—
247C67⋊C8—21no—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—41no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8#13112yes—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67C22×C4⋊C411yescharacteristic
259C2×D67—21no—
260C2×D67—21no—
261C2×D67—21no—
262C2×D67—21no—
263C2×D67—21no—
264C2×D67#42712yes—
265C4×C67—21no—
266C4×C67—21no—
267C4×C67—21no—
268C4×C67—21no—
269C4×C67C2×#13111yescharacteristic
270C4×C67#56811yescharacteristic
271C4×C67C2×#13511yescharacteristic
272C4×C67C2×#13511yescharacteristic
273Dic67—21no—
274Dic67—21no—
275Dic67—21no—
276Dic67—21no—
277Dic67—21no—
278Dic67#42712yes—
279#17827—671nomaximal, Sylow
280C2×C67—41no—
281C2×C67#290611yescharacteristic
282C2×C67C2×#42711yescharacteristic
283C2×C67#291611yescharacteristic
284D67—22no—
285D67—41no—
286C8×#41—671no—
287#2925—671no—
288#2925—671no—
289#1572—671no—
290#1572—671no—
291#1572—671no—
292#1572D6711yescharacteristic, complemented
293#1571—671no—
294#1571—671no—
295#1568—671no—
296#1568—671no—
297#1566—671no—
298#1564—671no—
299#1564—671no—
300C2×#334—671no—
301C67#1782711yescharacteristic, complemented, Sylow
302C4×#41—671no—
303C8×Q8—671no—
304C8×Q8—671no—
305C8×D4—671no—
306C8×D4—671no—
307C8×D4—671no—
308C8×D4—671no—
309C8×D4—671no—
310C8×D4—671no—
311#433—1341no—
312#433—1341no—
313#433—1341no—
314#433—1341no—
315#432—671no—
316#432—671no—
317#432—671no—
318#432C2×D6711yescharacteristic
319#431—671no—
320#431—671no—
321#431—671no—
322#431—671no—
323#431—671no—
324#431C2×D6711yescharacteristic
325#431C2×D6711yescharacteristic
326#423—671no—
327#423—671no—
328C2×C4⋊C8—671no—
329C2×C4⋊C8—671no—
330#408—1341no—
331#408—1341no—
332#405—1341no—
333#405—1341no—
334C2×C4×C8—671no—
335C2×C4×C8—671no—
336C2×C4×C8—671no—
337#335—671no—
338#335—671no—
339#335—671no—
340#335—671no—
341#335—671no—
342#335—671no—
343#335C2×D6711yescharacteristic
344#335C2×D6711yescharacteristic
345#334—671no—
346#334—671no—
347#334—671no—
348#334—671no—
349#334—671no—
350#334—671no—
351#334C2×D6711yescharacteristic
352#334C2×D6711yescharacteristic
353C2×#41—671no—
354C2×M16—1341no—
355C2×M16—1341no—
356C22×C8—671no—
357C22×C8—671no—
358C22×C8—671no—
359C22×C8—671no—
360C22×C8—671no—
361C22×C8—671no—
362C22×C8—1341no—
363C22×C8—1341no—
364C4×Q8—671no—
365C4×Q8—671no—
366C4×D4—671no—
367C4×D4—671no—
368C4×D4—671no—
369C4×D4—671no—
370C4×D4—671no—
371C4×D4—671no—
372#117—671no—
373#117—671no—
374#117—671no—
375C2×C42—671no—
376C2×C42—671no—
377C2×C42C22×D6711yescharacteristic
378C4⋊C8—671no—
379C4⋊C8—671no—
380C4⋊C8—671no—
381C4⋊C8—671no—
382C4⋊C8—671no—
383C4⋊C8—671no—
384C4⋊C8—671no—
385C4⋊C8—671no—
386C4⋊C8—1341no—
387C4⋊C8—1341no—
388C4⋊C8—1341no—
389C4⋊C8—1341no—
390C4⋊C8C4×D6711yescharacteristic
391C4⋊C8C4×D6711yescharacteristic
392C4⋊C8C22×D6711yescharacteristic
393C4⋊C8C22×D6711yescharacteristic
394C4⋊C8C22×D6711yescharacteristic
395C4⋊C8C22×D6711yescharacteristic
396C22⋊C8—21no—
397C22⋊C8—21no—
398C22⋊C8—671no—
399C22⋊C8—671no—
400C22⋊C8—671no—
401C22⋊C8—671no—
402C22⋊C8—1341no—
403C22⋊C8—1341no—
404C22⋊C8—1341no—
405C22⋊C8—1341no—
406C22⋊C8—1341no—
407C22⋊C8—1341no—
408C22⋊C8C4×D6711yescharacteristic
409C22⋊C8C4×D6711yescharacteristic
410#97—21no—
411#97—1341no—
412#97—1341no—
413#97—1341no—
414C4×C8—21no—
415C4×C8—671no—
416C4×C8—671no—
417C4×C8—671no—
418C4×C8—671no—
419C4×C8—671no—
420C4×C8—671no—
421C4×C8—671no—
422C4×C8—671no—
423C4×C8—1341no—
424C4×C8—1341no—
425C4×C8—1341no—
426C4×C8C22×D6711yescharacteristic
427C4×C8C22×D6711yescharacteristic
428#41—1342no—
429#41—1342no—
430C2×Q8—671no—
431C2×D4—671no—
432C2×D4—671no—
433C2×D4—671no—
434C22×C4—671no—
435C22×C4—671no—
436C22×C4—671no—
437C22×C4—671no—
438C22×C4—671no—
439C22×C4—671no—
440C22×C4C2×C4×D6711yescharacteristic
441C22×C4D4×D6711yescharacteristic
442C22×C4Q8×D6711yescharacteristic
443M16—2682no—
444M16—2682no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—2681no—
450C2×C8—2681no—
451C2×C8—2682no—
452C2×C8—2682no—
453C2×C8—2681no—
454C2×C8—671no—
455C2×C8—671no—
456C2×C8—671no—
457C2×C8—671no—
458C2×C8—1342no—
459C2×C8—1342no—
460C2×C8—1341no—
461C2×C8—1341no—
462C2×C8—1341no—
463C2×C8—1341no—
464C2×C8—1342no—
465C2×C8C2×C4×D6711yescharacteristic
466C2×C8C2×C4×D6711yescharacteristic
467C2×C8C2×C4×D6711yescharacteristic
468C2×C8C2×C4×D6711yescharacteristic
469C4⋊C4—671no—
470C4⋊C4—671no—
471C4⋊C4—671no—
472C4⋊C4—671no—
473C4⋊C4—671no—
474C4⋊C4—671no—
475C22⋊C4—671no—
476C22⋊C4—671no—
477C22⋊C4—671no—
478C22⋊C4—671no—
479C22⋊C4—671no—
480C22⋊C4—671no—
481C42—671no—
482C42—671no—
483C42—671no—
484C42—671no—
485C42—671no—
486C42—671no—
487C42C2×C4×D6711yescharacteristic
488C42D4×D6711yescharacteristic
489C42Q8×D6711yescharacteristic
490C42C23×D6711yescharacteristic
491C23—671no—
492C23—671no—
493C23C4⋊C4×D6711yescharacteristic
494Q8—1342no—
495D4—1342no—
496D4—1342no—
497D4—1342no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—2681no—
504C2×C4—2681no—
505C2×C4—671no—
506C2×C4—672no—
507C2×C4—672no—
508C2×C4—671no—
509C2×C4—671no—
510C2×C4—671no—
511C2×C4—671no—
512C2×C4—671no—
513C2×C4—1341no—
514C2×C4—1341no—
515C2×C4—1341no—
516C2×C4—1341no—
517C2×C4—1341no—
518C2×C4—1341no—
519C2×C4C4⋊C4×D6711yescharacteristic
520C2×C4C4⋊C4×D6711yescharacteristic
521C2×C4C4⋊C4×D6711yescharacteristic
522C2×C4C22×C4×D6711yescharacteristic
523C2×C4C2×D4×D6711yescharacteristic
524C2×C4C2×Q8×D6711yescharacteristic
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—2681no—
534C8—2681no—
535C8—2681no—
536C8—2681no—
537C8—672no—
538C8—1341no—
539C8—1341no—
540C8—1341no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—672no—
545C22—1341no—
546C22—1341no—
547C22—1341no—
548C22—1341no—
549C22—1341no—
550C22C2×C4⋊C4×D6711yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—672no—
556C4—1341no—
557C4—1341no—
558C4—1341no—
559C4—1341no—
560C4—1341no—
561C4#131×D6711yescharacteristic
562C4#1686416111yescharacteristic
563C4#135×D6711yescharacteristic
564C4#135×D6711yescharacteristic
565C2—41no—
566C2—2681no—
567C2—1342no—
568C2#2256877611yescharacteristic
569C2#427×D6711yescharacteristic
570C2#2256881911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1572C2
socleC22×C67C22×C4⋊C4
prime radicalC2×#11761628D4
cosocleC42×C67C24
hypercenter#1572D67
hypocenterC67#17827
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
4. #22558989C2semidirect
5. #22558986C2semidirect
6. #22558986C2semidirect
7. #22558982C2semidirect
8. #22558982C2semidirect
9. #22558971C2semidirect
10. #22558970C2semidirect
11. #22558963C2semidirect
12. #22558956C2semidirect
13. #22558955C2semidirect
14. #334×D67C2semidirect
15. C67×#1572C2semidirect
292. #1572D67semidirect
301. C67#17827semidirect

Maximal subgroups

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

subgroupindex
#225691652
#225688762
#225688702
#225589892
#225589862
#225589822
#225589712
#225589702
#225589632
#225589562
#225589552
#334×D672
C67×#15722
#1782767