Finite Group Databasebeta

← #30827028

Subgroups — #30827028

Order 17152.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570291C211yesmaximal, characteristic
2#22569371C211yesmaximal, characteristic
3#22569296C211yesmaximal, characteristic, complemented
4#22559107C211yesmaximal, characteristic, complemented
5#22559101C211yesmaximal, characteristic, complemented
6#22559089C211yesmaximal, characteristic, complemented
7#22559082C211yesmaximal, characteristic, complemented
8#22559082C211yesmaximal, characteristic, complemented
9#22559068C211yesmaximal, characteristic, complemented
10C8×Dic268C211yesmaximal, characteristic, complemented
11C8×#11761617C211yesmaximal, characteristic, complemented
12#22558994C211yesmaximal, characteristic, complemented
13#22558670C211yesmaximal, characteristic
14C2×#16863128C211yesmaximal, characteristic, complemented
15C67×#1450C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17C2×C4×Dic134C2211yescharacteristic
18#16863727C411yescharacteristic
19#16863710C411yescharacteristic
20#16863704C411yescharacteristic
21#16863704C411yescharacteristic
22#16863619C411yescharacteristic
23#16863619C411yescharacteristic
24#16863554C411yescharacteristic
25#16863537C411yescharacteristic
26#16863526C2211yescharacteristic
27#16863526C2211yescharacteristic
28#16863526C2211yescharacteristic
29#16863525—21no—
30#16863525—21no—
31C8×Dic134C2211yescharacteristic
32C8×Dic134C2211yescharacteristic
33C8×Dic134C2211yescharacteristic
34C4×Dic268C2211yescharacteristic
35C4×Dic268C2211yescharacteristic
36C4×#11761617C2211yescharacteristic
37C4×#11761617C2211yescharacteristic
38#16863475C2211yescharacteristic
39#16863464—21no—
40C8×D268C2211yescharacteristic
41(C22⋊D67)⋊C8—21no—
42#16863456—21no—
43#16863453C2211yescharacteristic
44#16863450—21no—
45#16863446—21no—
46#16863421—21no—
47C2×C4⋊(C67⋊C8)C2211yescharacteristic
48#16863149C2211yescharacteristic
49#16863149C2211yescharacteristic
50#16863147C2211yescharacteristic
51#16863147C2211yescharacteristic
52#16863128C2211yescharacteristic
53#16863128C2211yescharacteristic
54#16863128C2211yescharacteristic
55#16863128C2211yescharacteristic
56#16863128C2211yescharacteristic
57#16863128C2211yescharacteristic
58#16863125C2211yescharacteristic
59#16863125C2211yescharacteristic
60#431×C67C2211yescharacteristic
61#431×C67C2211yescharacteristic
62C2×C4×C8×C67C2211yescharacteristic
63#322×C67C2211yescharacteristic
64#322×C67C2211yescharacteristic
65C82×C67C2211yescharacteristic
66C82×C67C2211yescharacteristic
67C22×Dic134C2×C411yescharacteristic
68C2×Dic268C2×C411yescharacteristic
69C2×Dic268C2×C411yescharacteristic
70C2×#11761617C2×C411yescharacteristic
71C2×#11761617C2×C411yescharacteristic
72C2×#11761611—21no—
73C22×C67⋊C8—21no—
74C2×C8×D67—21no—
75#14524037—21no—
76#14524035C2×C411yescharacteristic
77#14524031C2×C411yescharacteristic
78#14524026—21no—
79#14524022C2×C411yescharacteristic
80#14524017C2×C411yescharacteristic
81C4×Dic134—21no—
82C4×Dic134—21no—
83C4×Dic134C2×C411yescharacteristic
84C4×Dic134C2×C411yescharacteristic
85C4×Dic134C2311yescharacteristic
86C4×Dic134C2311yescharacteristic
87C4×Dic134C2311yescharacteristic
88C4×D268C2311yescharacteristic
89C4×C22⋊D67—21no—
90#14524000—21no—
91C2×C4⋊Dic67C2×C411yescharacteristic
92C2×Dic67⋊C4—21no—
93C2×C4×Dic67—21no—
94#14523977C2×C411yescharacteristic
95#14523977C2×C411yescharacteristic
96#14523975C2×C411yescharacteristic
97#14523975C2×C411yescharacteristic
98C4⋊(C67⋊C8)—21no—
99C4⋊(C67⋊C8)C2311yescharacteristic
100C4⋊(C67⋊C8)C2311yescharacteristic
101C4⋊(C67⋊C8)C2311yescharacteristic
102C4⋊(C67⋊C8)C2311yescharacteristic
103Dic67⋊C8—21no—
104Dic67⋊C8—21no—
105Dic67⋊C8—21no—
106Dic67⋊C8—21no—
107Dic67⋊C8—21no—
108Dic67⋊C8—21no—
109#14523968C2×C411yescharacteristic
110#14523968C2×C411yescharacteristic
111#14523968C2×C411yescharacteristic
112#14523968C2×C411yescharacteristic
113#14523968C2×C411yescharacteristic
114#14523968C2×C411yescharacteristic
115#14523965C2×C411yescharacteristic
116#14523965C2×C411yescharacteristic
117C22⋊(C67⋊C8)—21no—
118(C2×D67)⋊C8—21no—
119(C2×D67)⋊C8—21no—
120#14523954—21no—
121#14523953—21no—
122#14523953—21no—
123#14523953—21no—
124C8×Dic67—21no—
125C8×Dic67—21no—
126C8×Dic67—21no—
127C22×C8×C67—21no—
128C2×C42×C67C2311yescharacteristic
129C4⋊C8×C67C2×C411yescharacteristic
130C4⋊C8×C67C2×C411yescharacteristic
131C4⋊C8×C67D411yescharacteristic
132C4⋊C8×C67D411yescharacteristic
133C22⋊C8×C67C2×C411yescharacteristic
134C22⋊C8×C67C2×C411yescharacteristic
135C22⋊C8×C67D411yescharacteristic
136C22⋊C8×C67D411yescharacteristic
137C4×C8×C67—21no—
138C4×C8×C67C2311yescharacteristic
139C4×C8×C67C2311yescharacteristic
140C4×C8×C67C2311yescharacteristic
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134—22no—
145C2×Dic134—21no—
146C2×Dic134—41no—
147C2×Dic134C22×C411yescharacteristic
148C2×Dic134C22×C411yescharacteristic
149C2×Dic134C22×C411yescharacteristic
150C2×D268C22×C411yescharacteristic
151C2×C22⋊D67—21no—
152C22×Dic67—21no—
153C22×Dic67—21no—
154C2×C4×D67—21no—
155Dic268—22no—
156Dic268—41no—
157#11761617—22no—
158#11761617—41no—
159#11761611—81no—
160C2×C67⋊C8—21no—
161C2×C67⋊C8—21no—
162C2×C67⋊C8—21no—
163C2×C67⋊C8—21no—
164C2×C67⋊C8—81no—
165C8×D67—81no—
166C4⋊Dic67—21no—
167C4⋊Dic67C22×C411yescharacteristic
168C4⋊Dic67C22×C411yescharacteristic
169C4⋊Dic67C22×C411yescharacteristic
170C4⋊Dic67C22×C411yescharacteristic
171Dic67⋊C4—21no—
172Dic67⋊C4—21no—
173Dic67⋊C4—21no—
174Dic67⋊C4—21no—
175Dic67⋊C4—21no—
176Dic67⋊C4—21no—
177Dic67⋊C4—21no—
178Dic67⋊C4—21no—
179C22⋊Dic67—21no—
180(C2×D67)⋊C4—21no—
181(C2×D67)⋊C4—21no—
182(C2×D67)⋊C4—21no—
183(C2×D67)⋊C4—21no—
184C4×Dic67—21no—
185C4×Dic67—21no—
186C4×Dic67—21no—
187C4×Dic67—21no—
188C4×Dic67—21no—
189C22×C4×C67C22×C411yescharacteristic
190C22×C4×C67C2×D411yescharacteristic
191C22×C4×C67#4111yescharacteristic
192C2×C8×C67—24no—
193C2×C8×C67—21no—
194C2×C8×C67—21no—
195C2×C8×C67C22×C411yescharacteristic
196C2×C8×C67C22×C411yescharacteristic
197C2×C8×C67C22×C411yescharacteristic
198C2×C8×C67C22×C411yescharacteristic
199C2×C8×C67C2×D411yescharacteristic
200C2×C8×C67C2×D411yescharacteristic
201C2×C8×C67C2×D411yescharacteristic
202C2×C8×C67C2×D411yescharacteristic
203C42×C67C22×C411yescharacteristic
204C42×C67C2×D411yescharacteristic
205C42×C67#4111yescharacteristic
206C42×C67C2411yescharacteristic
207C23×C67C4×D411yescharacteristic
208C22×D67—21no—
209C2×C4×C67—22no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67C4×D411yescharacteristic
215C2×C4×C67C4×D411yescharacteristic
216C2×C4×C67C4×D411yescharacteristic
217C2×C4×C67C23×C411yescharacteristic
218C2×C4×C67C22×D411yescharacteristic
219C2×C4×C67C2×#4111yescharacteristic
220C22⋊D67—42no—
221C2×Dic67—21no—
222C2×Dic67—21no—
223C2×Dic67—21no—
224C2×Dic67—21no—
225C2×Dic67—21no—
226C2×Dic67—21no—
227C2×Dic67—21no—
228C2×Dic67—42no—
229C2×Dic67—42no—
230D268—21no—
231D268—21no—
232C4×D67—41no—
233C4×D67—41no—
234Dic134—21no—
235Dic134—21no—
236Dic134—21no—
237Dic134—21no—
238Dic134—21no—
239Dic134—42no—
240Dic134#13112yes—
241C8×C67—22no—
242C8×C67—22no—
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—21no—
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—21no—
249C8×C67—21no—
250C8×C67—21no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8—41no—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67#44312yes—
258C22×C67C2×C4×D411yescharacteristic
259C2×D67—41no—
260C2×D67—41no—
261C2×D67—41no—
262C4×C67—21no—
263C4×C67—21no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67C2×#13111yescharacteristic
267C4×C67#56811yescharacteristic
268C4×C67#57711yescharacteristic
269C4×C67#57811yescharacteristic
270Dic67—22no—
271Dic67—41no—
272Dic67—41no—
273Dic67—41no—
274Dic67—41no—
275Dic67—41no—
276Dic67—41no—
277#18017—671nomaximal, Sylow
278C2×C67—22no—
279C2×C67#293311yescharacteristic
280C2×C67#294711yescharacteristic
281C2×C67C2×#44311yescharacteristic
282D67—81no—
283#3108—671no—
284#2994—671no—
285#2984—671no—
286#1591—671no—
287#1590—671no—
288#1588—671no—
289#1587—671no—
290#1587—671no—
291#1585—671no—
292C8×Q16—671no—
293C8×SD16—671no—
294#1573—671no—
295#1494—671no—
296C2×Q8⋊C8—671no—
297#1450D6711yescharacteristic, complemented
298C67#1801711yescharacteristic, complemented, Sylow
299#529—671no—
300C2×C4×Q8—671no—
301#483—671no—
302#480—671no—
303#479—671no—
304#479—671no—
305#464—671no—
306#464—671no—
307#451—671no—
308#448—671no—
309#446—671no—
310#446—671no—
311#446—671no—
312#446—1341no—
313#446—1341no—
314C8×Q8—671no—
315C8×Q8—671no—
316C8×Q8—671no—
317C4×Q16—671no—
318C4×Q16—671no—
319C4×SD16—671no—
320C4×SD16—671no—
321#436—671no—
322#435—1341no—
323C8×D4—671no—
324C8×D4—1341no—
325#433—1341no—
326#432—671no—
327#432—1341no—
328#431—1341no—
329#431C2×D6711yescharacteristic
330#431C2×D6711yescharacteristic
331#424—1341no—
332C2×C4⋊C8—671no—
333C2×C4×C8C2×D6711yescharacteristic
334#335—671no—
335#335—671no—
336#334—671no—
337#334—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343Q8⋊C8—671no—
344D4⋊C8—671no—
345D4⋊C8—671no—
346#322C2×D6711yescharacteristic
347#322C2×D6711yescharacteristic
348C82C2×D6711yescharacteristic
349C82C2×D6711yescharacteristic
350C22×Q8—671no—
351C2×Q16—671no—
352C2×Q16—671no—
353C2×SD16—671no—
354C2×SD16—671no—
355C2×M16—1341no—
356C22×C8—21no—
357C22×C8—1341no—
358C22×C8—1341no—
359#126—1341no—
360#125—671no—
361#124—671no—
362#123—1341no—
363C22⋊Q8—671no—
364#121—671no—
365C4×Q8—671no—
366C4×Q8—671no—
367C4×Q8—671no—
368C4×Q8—671no—
369C4×Q8—671no—
370C4×Q8—1341no—
371C4×Q8—1341no—
372C4×D4—671no—
373C4×D4—1341no—
374#117—1341no—
375C2×C4⋊C4—671no—
376C2×C4⋊C4—1341no—
377C2×C42—1341no—
378C2×C42C22×D6711yescharacteristic
379#107—671no—
380#107—671no—
381#106—671no—
382#106—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⋊C8C4×D6711yescharacteristic
395C4⋊C8C4×D6711yescharacteristic
396C4⋊C8D26811yescharacteristic
397C4⋊C8D26811yescharacteristic
398#103—671no—
399#103—671no—
400#103—671no—
401#103—671no—
402#103—671no—
403#103—671no—
404#102—671no—
405#102—671no—
406C22⋊C8—1341no—
407C22⋊C8—1341no—
408C22⋊C8—1341no—
409C22⋊C8C4×D6711yescharacteristic
410C22⋊C8C4×D6711yescharacteristic
411C22⋊C8D26811yescharacteristic
412C22⋊C8D26811yescharacteristic
413#97—1341no—
414#97—1341no—
415#97—1341no—
416#97—1341no—
417C4×C8—21no—
418C4×C8—1341no—
419C4×C8—1341no—
420C4×C8—1341no—
421C4×C8C22×D6711yescharacteristic
422C4×C8C22×D6711yescharacteristic
423C4×C8C22×D6711yescharacteristic
424C4×C8C22×D6711yescharacteristic
425C4×C8C22×D6711yescharacteristic
426C4×C8C22×D6711yescharacteristic
427C2×Q8—2681no—
428C2×Q8—671no—
429C2×Q8—671no—
430C2×Q8—671no—
431C2×Q8—1342no—
432C2×Q8—1341no—
433C2×D4—671no—
434C2×D4—1341no—
435C22×C4—1341no—
436C22×C4—1341no—
437C22×C4—1341no—
438C22×C4C2×C4×D6711yescharacteristic
439C22×C4C2×D26811yescharacteristic
440C22×C4#1176162811yescharacteristic
441Q16—2681no—
442Q16—1342no—
443SD16—2681no—
444SD16—1342no—
445M16—5361no—
446C2×C8—24no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—5361no—
450C2×C8—5361no—
451C2×C8—1341no—
452C2×C8—1341no—
453C2×C8—1341no—
454C2×C8—1341no—
455C2×C8C2×C4×D6711yescharacteristic
456C2×C8C2×C4×D6711yescharacteristic
457C2×C8C2×C4×D6711yescharacteristic
458C2×C8C2×C4×D6711yescharacteristic
459C2×C8C2×D26811yescharacteristic
460C2×C8C2×D26811yescharacteristic
461C2×C8C2×D26811yescharacteristic
462C2×C8C2×D26811yescharacteristic
463C4⋊C4—671no—
464C4⋊C4—671no—
465C4⋊C4—671no—
466C4⋊C4—671no—
467C4⋊C4—1341no—
468C4⋊C4—1341no—
469C4⋊C4—1341no—
470C4⋊C4—1341no—
471C4⋊C4—1341no—
472C4⋊C4—1341no—
473C4⋊C4—1341no—
474C4⋊C4—1341no—
475C4⋊C4—1341no—
476C22⋊C4—1341no—
477C22⋊C4—1341no—
478C22⋊C4—1341no—
479C22⋊C4—1341no—
480C22⋊C4—1341no—
481C42—1341no—
482C42—1341no—
483C42—1341no—
484C42—1341no—
485C42—1341no—
486C42C2×C4×D6711yescharacteristic
487C42C2×D26811yescharacteristic
488C42#1176162811yescharacteristic
489C42C23×D6711yescharacteristic
490C23—1341no—
491C23C4×D26811yescharacteristic
492Q8—2682no—
493Q8—672no—
494Q8—1341no—
495Q8—1341no—
496Q8—1341no—
497Q8—1341no—
498Q8—1341no—
499D4—2682no—
500D4—1341no—
501D4—1341no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—2682no—
508C2×C4—2682no—
509C2×C4—2681no—
510C2×C4—2681no—
511C2×C4—1341no—
512C2×C4—1341no—
513C2×C4—1341no—
514C2×C4—1341no—
515C2×C4—1341no—
516C2×C4—1341no—
517C2×C4—1341no—
518C2×C4C4×D26811yescharacteristic
519C2×C4C4×D26811yescharacteristic
520C2×C4C4×D26811yescharacteristic
521C2×C4C22×C4×D6711yescharacteristic
522C2×C4C22×D26811yescharacteristic
523C2×C4C2×#1176162811yescharacteristic
524C8—22no—
525C8—22no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—2681no—
535C8—2681no—
536C8—2681no—
537C8—2681no—
538C22—21no—
539C22—21no—
540C22—2681no—
541C22—2681no—
542C22—2681no—
543C22#1686351412yes—
544C22C2×C4×D26811yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—2681no—
550C4—2681no—
551C4—2681no—
552C4—2681no—
553C4—2681no—
554C4—2681no—
555C4—1342no—
556C4#131×D6711yescharacteristic
557C4#1686416111yescharacteristic
558C4#1686420811yescharacteristic
559C4#1686421311yescharacteristic
560C2—22no—
561C2—5361no—
562C2#2256893711yescharacteristic
563C2#2256905411yescharacteristic
564C2C2×#1686351411yescharacteristic
subgroupG/H
centerC2×C4C22×D268
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1450C2
socleC22×C67C2×C4×D4
prime radical#14524017C2×C4
cosocleC42×C67C24
hypercenter#1450D67
hypocenterC67#18017
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450

Complements

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

NH ≅ G/Ntype
3. #22569296C2semidirect
4. #22559107C2semidirect
5. #22559101C2semidirect
6. #22559089C2semidirect
7. #22559082C2semidirect
8. #22559082C2semidirect
9. #22559068C2semidirect
10. C8×Dic268C2semidirect
11. C8×#11761617C2semidirect
12. #22558994C2semidirect
14. C2×#16863128C2semidirect
15. C67×#1450C2semidirect
297. #1450D67semidirect
298. C67#18017semidirect

Maximal subgroups

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

subgroupindex
#225702912
#225693712
#225692962
#225591072
#225591012
#225590892
#225590822
#225590682
C8×Dic2682
C8×#117616172
#225589942
#225586702
C2×#168631282
C67×#14502
#1801767