Finite Group Databasebeta

← #30825176

Subgroups — #30825176

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22569339C211yesmaximal, characteristic
2#22569331C211yesmaximal, characteristic
3#22569173C211yesmaximal, characteristic
4#22558996C211yesmaximal, characteristic, complemented
5#22558996C211yesmaximal, characteristic, complemented
6#22558989C211yesmaximal, characteristic, complemented
7#22558986C211yesmaximal, characteristic, complemented
8#22558978C211yesmaximal, characteristic, complemented
9#22558974C211yesmaximal, characteristic, complemented
10#22558973C211yesmaximal, characteristic, complemented
11#22558963C211yesmaximal, characteristic, complemented
12#22558959C211yesmaximal, characteristic, complemented
13#22558958C211yesmaximal, characteristic, complemented
14#22558954C211yesmaximal, characteristic, complemented
15C67×#1570C211yesmaximal, characteristic, complemented
16C4×#11761628C2211yescharacteristic
17#16863525—21no—
18#16863525C411yescharacteristic
19#16863525C411yescharacteristic
20#16863520—21no—
21#16863473—21no—
22#16863472—21no—
23#16863472—21no—
24#16863471C411yescharacteristic
25#16863471C411yescharacteristic
26#16863470C411yescharacteristic
27#16863470C411yescharacteristic
28#16863467—21no—
29#16863466—21no—
30#16863466C411yescharacteristic
31#16863466C411yescharacteristic
32C8×C22⋊D67—21no—
33#16863453C2211yescharacteristic
34#16863452C2211yescharacteristic
35#16863449C2211yescharacteristic
36#16863448C2211yescharacteristic
37#16863448C2211yescharacteristic
38#16863448C2211yescharacteristic
39#16863447C2211yescharacteristic
40#16863447C2211yescharacteristic
41#16863418C2211yescharacteristic
42#16863418C2211yescharacteristic
43#16863418C2211yescharacteristic
44C4⋊C8×D67C2211yescharacteristic
45C4×#11761612C2211yescharacteristic
46C4×#11761611C2211yescharacteristic
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#431×C67C2211yescharacteristic
61#423×C67C2211yescharacteristic
62C2×C4⋊C8×C67C2211yescharacteristic
63#334×C67C2211yescharacteristic
64#334×C67C2211yescharacteristic
65#334×C67C2211yescharacteristic
66#334×C67C2211yescharacteristic
67C2×#11761628Q811yescharacteristic
68C2×#11761612C2×C411yescharacteristic
69C2×#11761612C2×C411yescharacteristic
70C2×#11761611—21no—
71C2×#11761611—21no—
72C2×#11761611—21no—
73C2×#11761611C2×C411yescharacteristic
74C2×#11761611C2×C411yescharacteristic
75C2×C8×D67—21no—
76C2×C8×D67C2×C411yescharacteristic
77C2×C8×D67C2×C411yescharacteristic
78C4×Dic134C2×C411yescharacteristic
79C4×Dic134D411yescharacteristic
80C4×D268C2×C411yescharacteristic
81C4×D268D411yescharacteristic
82C4×C22⋊D67C2×C411yescharacteristic
83C4×C22⋊D67C2×C411yescharacteristic
84C4×C22⋊D67D411yescharacteristic
85C4×C22⋊D67D411yescharacteristic
86#14524002Q811yescharacteristic
87#14524000Q811yescharacteristic
88#14524000Q811yescharacteristic
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#14523954C2×C411yescharacteristic
115#14523954C2×C411yescharacteristic
116#14523953—21no—
117#14523953C2×C411yescharacteristic
118#14523953C2×C411yescharacteristic
119C4×C67⋊C8—21no—
120C4×C67⋊C8C2311yescharacteristic
121C4×C67⋊C8C2311yescharacteristic
122C8×Dic67—21no—
123C8×Dic67—21no—
124C8×Dic67—21no—
125C8×Dic67C2×C411yescharacteristic
126C8×Dic67C2×C411yescharacteristic
127C2×M16×C67—21no—
128C22×C8×C67—21no—
129C2×C42×C67C2311yescharacteristic
130C4⋊C8×C67—21no—
131C4⋊C8×C67—21no—
132C4⋊C8×C67C2×C411yescharacteristic
133C4⋊C8×C67C2×C411yescharacteristic
134C4⋊C8×C67C2311yescharacteristic
135C4⋊C8×C67C2311yescharacteristic
136C4⋊C8×C67C2311yescharacteristic
137C4⋊C8×C67C2311yescharacteristic
138C22⋊C8×C67C2×C411yescharacteristic
139C22⋊C8×C67C2×C411yescharacteristic
140C4×C8×C67C2311yescharacteristic
141C4×C8×C67C2311yescharacteristic
142#11761628—42no—
143C2×Dic134C4⋊C411yescharacteristic
144C2×D268C4⋊C411yescharacteristic
145C2×C22⋊D67C4⋊C411yescharacteristic
146C2×C22⋊D67C4⋊C411yescharacteristic
147C2×C4×D67C22×C411yescharacteristic
148C2×C4×D67C22×C411yescharacteristic
149C2×C4×D67C2×D411yescharacteristic
150C2×C4×D67C2×D411yescharacteristic
151C2×C4×D67C2×Q811yescharacteristic
152C2×C4×D67C2×Q811yescharacteristic
153#11761612—22no—
154#11761612—41no—
155#11761611—22no—
156#11761611—41no—
157#11761611—81no—
158#11761611—81no—
159#11761611—81no—
160C2×C67⋊C8—21no—
161C2×C67⋊C8—21no—
162C2×C67⋊C8—21no—
163C2×C67⋊C8—21no—
164C2×C67⋊C8C22×C411yescharacteristic
165C2×C67⋊C8C22×C411yescharacteristic
166C2×C67⋊C8C22×C411yescharacteristic
167C2×C67⋊C8C22×C411yescharacteristic
168C8×D67—22no—
169C8×D67—41no—
170C8×D67—81no—
171C4⋊Dic67C4⋊C411yescharacteristic
172C4⋊Dic67C4⋊C411yescharacteristic
173Dic67⋊C4C4⋊C411yescharacteristic
174Dic67⋊C4C4⋊C411yescharacteristic
175Dic67⋊C4C4⋊C411yescharacteristic
176Dic67⋊C4C4⋊C411yescharacteristic
177C22⋊Dic67C4⋊C411yescharacteristic
178C22⋊Dic67C4⋊C411yescharacteristic
179(C2×D67)⋊C4C4⋊C411yescharacteristic
180(C2×D67)⋊C4C4⋊C411yescharacteristic
181(C2×D67)⋊C4C4⋊C411yescharacteristic
182(C2×D67)⋊C4C4⋊C411yescharacteristic
183C4×Dic67C22×C411yescharacteristic
184C4×Dic67C22×C411yescharacteristic
185C4×Dic67C2×D411yescharacteristic
186C4×Dic67C2×D411yescharacteristic
187C4×Dic67C2×Q811yescharacteristic
188C4×Dic67C2×Q811yescharacteristic
189C22×C4×C67C22×C411yescharacteristic
190C22×C4×C67C2×D411yescharacteristic
191C22×C4×C67C2×Q811yescharacteristic
192M16×C67—81no—
193C2×C8×C67—21no—
194C2×C8×C67—21no—
195C2×C8×C67—21no—
196C2×C8×C67—21no—
197C2×C8×C67—81no—
198C2×C8×C67C22×C411yescharacteristic
199C2×C8×C67C22×C411yescharacteristic
200C2×C8×C67C22×C411yescharacteristic
201C2×C8×C67C22×C411yescharacteristic
202C42×C67C22×C411yescharacteristic
203C42×C67C2×D411yescharacteristic
204C42×C67C2×Q811yescharacteristic
205C42×C67C2411yescharacteristic
206C23×C67C2×C4⋊C411yescharacteristic
207C22×D67C2×C4⋊C411yescharacteristic
208C22×D67C2×C4⋊C411yescharacteristic
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—41no—
214C2×C4×C67C2×C4⋊C411yescharacteristic
215C2×C4×C67C2×C4⋊C411yescharacteristic
216C2×C4×C67C2×C4⋊C411yescharacteristic
217C2×C4×C67C23×C411yescharacteristic
218C2×C4×C67C22×D411yescharacteristic
219C2×C4×C67C22×Q811yescharacteristic
220C22⋊D67—22no—
221C22⋊D67—41no—
222C2×Dic67C2×C4⋊C411yescharacteristic
223C2×Dic67C2×C4⋊C411yescharacteristic
224C2×Dic67C2×C4⋊C411yescharacteristic
225C2×Dic67C2×C4⋊C411yescharacteristic
226C2×Dic67C2×C4⋊C411yescharacteristic
227C2×Dic67C2×C4⋊C411yescharacteristic
228D268—41no—
229C4×D67—21no—
230C4×D67—21no—
231C4×D67—21no—
232C4×D67—21no—
233C4×D67—21no—
234C4×D67—21no—
235C4×D67—21no—
236C4×D67—21no—
237C4×D67—22no—
238C4×D67—41no—
239Dic134—22no—
240C8×C67—21no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—41no—
245C8×C67—41no—
246C8×C67—41no—
247C8×C67—41no—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—21no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8—41no—
255C67⋊C8#13112yes—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67—21no—
259C22×C67C22×C4⋊C411yescharacteristic
260C2×D67—21no—
261C2×D67—21no—
262C2×D67—21no—
263C2×D67—21no—
264C2×D67—21no—
265C2×D67—21no—
266C4×C67—21no—
267C4×C67—21no—
268C4×C67—21no—
269C4×C67—21no—
270C4×C67C2×#13111yescharacteristic
271C4×C67#56811yescharacteristic
272C4×C67#57711yescharacteristic
273C4×C67#57711yescharacteristic
274Dic67—21no—
275Dic67—21no—
276Dic67—21no—
277Dic67—21no—
278Dic67—21no—
279Dic67#43012yes—
280#17841—671nomaximal, Sylow
281C2×C67—41no—
282C2×C67#290611yescharacteristic
283C2×C67#291211yescharacteristic
284C2×C67C2×#43011yescharacteristic
285D67—41no—
286D67—41no—
287#2991—671no—
288#2990—671no—
289#2965—671no—
290#1574—671no—
291#1574—671no—
292#1572—671no—
293#1572—671no—
294#1570—671no—
295#1570D6711yescharacteristic, complemented
296#1569—671no—
297#1569—671no—
298#1566—671no—
299#1565—671no—
300#1565—671no—
301#1564—671no—
302C67#1784111yescharacteristic, complemented, Sylow
303C4×#41—671no—
304#446—671no—
305#446—671no—
306#446—1341no—
307C8×Q8—1341no—
308#436—671no—
309#436—671no—
310#436—1341no—
311#436—1341no—
312#436—1341no—
313#435—671no—
314#435—671no—
315#435—671no—
316#435—671no—
317#435—1341no—
318#435—1341no—
319C8×D4—1341no—
320#432—671no—
321#432—671no—
322#431—671no—
323#431—671no—
324#431—671no—
325#431—671no—
326#431—671no—
327#431—671no—
328#431C2×D6711yescharacteristic
329#423—671no—
330#423—671no—
331#423—671no—
332#423C2×D6711yescharacteristic
333C2×C4⋊C8—671no—
334C2×C4⋊C8C2×D6711yescharacteristic
335C4×M16—671no—
336C4×M16—671no—
337C2×C4×C8—671no—
338#335—671no—
339#335—671no—
340#335—671no—
341#335—671no—
342#335—671no—
343#335—671no—
344#334—671no—
345#334—671no—
346#334—671no—
347#334—671no—
348#334—671no—
349#334—671no—
350#334C2×D6711yescharacteristic
351#334C2×D6711yescharacteristic
352#334C2×D6711yescharacteristic
353#334C2×D6711yescharacteristic
354C2×#41—671no—
355C2×M16—21no—
356C2×M16—671no—
357C2×M16—671no—
358C2×M16—671no—
359C2×M16—671no—
360C2×M16—1341no—
361C2×M16—1341no—
362C2×M16—1341no—
363C22×C8—21no—
364C22×C8—671no—
365C22×C8—671no—
366C22×C8—1341no—
367C4×Q8—671no—
368C4×Q8—671no—
369C4×D4—671no—
370C4×D4—671no—
371C4×D4—671no—
372C4×D4—671no—
373C4×D4—671no—
374C4×D4—671no—
375#117—671no—
376#117—671no—
377#117—671no—
378C2×C42—671no—
379C2×C42—671no—
380C2×C42C22×D6711yescharacteristic
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—671no—
384C4⋊C8—671no—
385C4⋊C8—671no—
386C4⋊C8—671no—
387C4⋊C8—671no—
388C4⋊C8—671no—
389C4⋊C8—671no—
390C4⋊C8—671no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8—1341no—
395C4⋊C8C4×D6711yescharacteristic
396C4⋊C8C4×D6711yescharacteristic
397C4⋊C8C22×D6711yescharacteristic
398C4⋊C8C22×D6711yescharacteristic
399C4⋊C8C22×D6711yescharacteristic
400C4⋊C8C22×D6711yescharacteristic
401C22⋊C8—671no—
402C22⋊C8—671no—
403C22⋊C8—671no—
404C22⋊C8—671no—
405C22⋊C8—1341no—
406C22⋊C8—1341no—
407C22⋊C8—1341no—
408C22⋊C8—1341no—
409C22⋊C8—1341no—
410C22⋊C8—1341no—
411C22⋊C8C4×D6711yescharacteristic
412C22⋊C8C4×D6711yescharacteristic
413#97—671no—
414#97—671no—
415#97—671no—
416#97—671no—
417#97—1341no—
418#97—1341no—
419C4×C8—671no—
420C4×C8—671no—
421C4×C8—671no—
422C4×C8—671no—
423C4×C8—1341no—
424C4×C8—1341no—
425C4×C8—1341no—
426C4×C8—1341no—
427C4×C8C22×D6711yescharacteristic
428C4×C8C22×D6711yescharacteristic
429#41—2682no—
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—81no—
444M16—2681no—
445M16—2681no—
446M16—5361no—
447M16—5361no—
448M16—5361no—
449M16—1342no—
450M16—1342no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—81no—
456C2×C8—2681no—
457C2×C8—5361no—
458C2×C8—671no—
459C2×C8—671no—
460C2×C8—671no—
461C2×C8—671no—
462C2×C8—1342no—
463C2×C8—1341no—
464C2×C8—1341no—
465C2×C8—1341no—
466C2×C8—1341no—
467C2×C8C2×C4×D6711yescharacteristic
468C2×C8C2×C4×D6711yescharacteristic
469C2×C8C2×C4×D6711yescharacteristic
470C2×C8C2×C4×D6711yescharacteristic
471C4⋊C4—671no—
472C4⋊C4—671no—
473C4⋊C4—671no—
474C4⋊C4—671no—
475C4⋊C4—671no—
476C4⋊C4—671no—
477C22⋊C4—671no—
478C22⋊C4—671no—
479C22⋊C4—671no—
480C22⋊C4—671no—
481C22⋊C4—671no—
482C22⋊C4—671no—
483C42—671no—
484C42—671no—
485C42—671no—
486C42—671no—
487C42—671no—
488C42—671no—
489C42C2×C4×D6711yescharacteristic
490C42D4×D6711yescharacteristic
491C42Q8×D6711yescharacteristic
492C42C23×D6711yescharacteristic
493C23—671no—
494C23—671no—
495C23C4⋊C4×D6711yescharacteristic
496Q8—1342no—
497D4—2681no—
498D4—2681no—
499D4—1342no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—2681no—
506C2×C4—671no—
507C2×C4—671no—
508C2×C4—671no—
509C2×C4—671no—
510C2×C4—671no—
511C2×C4—671no—
512C2×C4—1342no—
513C2×C4—1341no—
514C2×C4—1341no—
515C2×C4—1341no—
516C2×C4—1341no—
517C2×C4—1341no—
518C2×C4—1341no—
519C2×C4—1341no—
520C2×C4—1341no—
521C2×C4C4⋊C4×D6711yescharacteristic
522C2×C4C4⋊C4×D6711yescharacteristic
523C2×C4C4⋊C4×D6711yescharacteristic
524C2×C4C22×C4×D6711yescharacteristic
525C2×C4C2×D4×D6711yescharacteristic
526C2×C4C2×Q8×D6711yescharacteristic
527C8—21no—
528C8—21no—
529C8—21no—
530C8—21no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—2681no—
536C8—2681no—
537C8—2681no—
538C8—2681no—
539C8—672no—
540C8—1341no—
541C8—1341no—
542C8—1341no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—1341no—
547C22—1341no—
548C22—1341no—
549C22—1341no—
550C22—1341no—
551C22—1341no—
552C22C2×C4⋊C4×D6711yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—672no—
558C4—1341no—
559C4—1341no—
560C4—1341no—
561C4—1341no—
562C4—1341no—
563C4#131×D6711yescharacteristic
564C4#1686416111yescharacteristic
565C4#1686420211yescharacteristic
566C4#1686420211yescharacteristic
567C2—41no—
568C2—2681no—
569C2—2681no—
570C2#2256877611yescharacteristic
571C2#2256880011yescharacteristic
572C2#430×D6711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1570C2
socleC22×C67C22×C4⋊C4
prime radicalC2×#11761628Q8
cosocleC42×C67C24
hypercenter#1570D67
hypocenterC67#17841
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
4. #22558996C2semidirect
5. #22558996C2semidirect
6. #22558989C2semidirect
7. #22558986C2semidirect
8. #22558978C2semidirect
9. #22558974C2semidirect
10. #22558973C2semidirect
11. #22558963C2semidirect
12. #22558959C2semidirect
13. #22558958C2semidirect
14. #22558954C2semidirect
15. C67×#1570C2semidirect
295. #1570D67semidirect
302. C67#17841semidirect

Maximal subgroups

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

subgroupindex
#225693392
#225693312
#225691732
#225589962
#225589892
#225589862
#225589782
#225589742
#225589732
#225589632
#225589592
#225589582
#225589542
C67×#15702
#1784167