Finite Group Databasebeta

← #30825930

Subgroups — #30825930

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570280C211yesmaximal, characteristic
2#22569320C211yesmaximal, characteristic
3#22569306C211yesmaximal, characteristic, complemented
4#22559077C211yesmaximal, characteristic, complemented
5#22559063C211yesmaximal, characteristic, complemented
6#22559056C211yesmaximal, characteristic, complemented
7#22559042C211yesmaximal, characteristic, complemented
8#22559025C211yesmaximal, characteristic, complemented
9#22559018C211yesmaximal, characteristic, complemented
10#22559013C211yesmaximal, characteristic, complemented
11#22559012C211yesmaximal, characteristic, complemented
12#22558995C211yesmaximal, characteristic, complemented
13#22558613C211yesmaximal, characteristic, complemented
14#22558509C211yesmaximal, characteristic, complemented
15C67×#1488C211yesmaximal, characteristic
16#16863953C2211yescharacteristic
17#16863737C411yescharacteristic
18#16863737C411yescharacteristic
19#16863681C411yescharacteristic
20#16863676C411yescharacteristic
21#16863606C411yescharacteristic
22#16863601C411yescharacteristic
23#16863558C411yescharacteristic
24#16863558C411yescharacteristic
25#16863524—21no—
26#16863524—21no—
27#16863523C2211yescharacteristic
28#16863523C2211yescharacteristic
29Q8×C67⋊C8C2211yescharacteristic
30Q8×C67⋊C8C2211yescharacteristic
31#16863520C2211yescharacteristic
32#16863520C2211yescharacteristic
33C4×#11761618C2211yescharacteristic
34C4×#11761618C2211yescharacteristic
35C4×#11761616C2211yescharacteristic
36C4×#11761616C2211yescharacteristic
37D4×C67⋊C8—21no—
38D4×C67⋊C8—21no—
39D4×C67⋊C8C2211yescharacteristic
40D4×C67⋊C8C2211yescharacteristic
41#16863454—21no—
42#16863454—21no—
43#16863449—21no—
44#16863449C2211yescharacteristic
45#16863449C2211yescharacteristic
46C2×C4⋊(C67⋊C8)—21no—
47C2×C4⋊(C67⋊C8)C2211yescharacteristic
48#16863148C2211yescharacteristic
49#16863148C2211yescharacteristic
50#16863146C2211yescharacteristic
51#16863146C2211yescharacteristic
52#16863128C2211yescharacteristic
53#16863128C2211yescharacteristic
54#16863128C2211yescharacteristic
55#16863128C2211yescharacteristic
56#16863118C2211yescharacteristic
57#16863118C2211yescharacteristic
58#16863117C2211yescharacteristic
59C8×C67⋊C8C2211yescharacteristic
60C4×#41×C67C2211yescharacteristic
61#432×C67C2211yescharacteristic
62#431×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66D4⋊C8×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761616C2×C411yescharacteristic
70C2×#11761616C2×C411yescharacteristic
71C22×C67⋊C8—21no—
72C22×C67⋊C8—21no—
73C22×C67⋊C8—21no—
74#14524043C2×C411yescharacteristic
75#14524043C2×C411yescharacteristic
76#14524035—21no—
77#14524022—21no—
78#14524022C2×C411yescharacteristic
79#14524022C2×C411yescharacteristic
80C4×Dic134—21no—
81C4×Dic134C2311yescharacteristic
82C4×Dic134C2311yescharacteristic
83#14524002—21no—
84#14523976C2×C411yescharacteristic
85#14523976C2×C411yescharacteristic
86#14523974C2×C411yescharacteristic
87#14523974C2×C411yescharacteristic
88C4⋊(C67⋊C8)—21no—
89C4⋊(C67⋊C8)—21no—
90C4⋊(C67⋊C8)—21no—
91C4⋊(C67⋊C8)C2×C411yescharacteristic
92C4⋊(C67⋊C8)C2×C411yescharacteristic
93C4⋊(C67⋊C8)D411yescharacteristic
94C4⋊(C67⋊C8)D411yescharacteristic
95C4⋊(C67⋊C8)C2311yescharacteristic
96C4⋊(C67⋊C8)C2311yescharacteristic
97Dic67⋊C8—21no—
98Dic67⋊C8—21no—
99Dic67⋊C8—21no—
100Dic67⋊C8—21no—
101#14523967C2×C411yescharacteristic
102#14523967C2×C411yescharacteristic
103#14523966C2×C411yescharacteristic
104#14523966C2×C411yescharacteristic
105#14523966C2×C411yescharacteristic
106#14523966C2×C411yescharacteristic
107#14523964C2×C411yescharacteristic
108#14523964C2×C411yescharacteristic
109C22⋊(C67⋊C8)—21no—
110C22⋊(C67⋊C8)—21no—
111C22⋊(C67⋊C8)C2×C411yescharacteristic
112C22⋊(C67⋊C8)C2×C411yescharacteristic
113C22⋊(C67⋊C8)D411yescharacteristic
114C22⋊(C67⋊C8)D411yescharacteristic
115#14523953—21no—
116#14523953—21no—
117C4×C67⋊C8—21no—
118C4×C67⋊C8—21no—
119C4×C67⋊C8C2311yescharacteristic
120C4×C67⋊C8C2311yescharacteristic
121C4×C67⋊C8C2311yescharacteristic
122C4×C67⋊C8C2311yescharacteristic
123C8×Dic67—21no—
124C8×Dic67—21no—
125C2×#41×C67C2×C411yescharacteristic
126C4×Q8×C67C2×C411yescharacteristic
127C4×Q8×C67C2311yescharacteristic
128C4×D4×C67—21no—
129C4×D4×C67—21no—
130C4×D4×C67C2×C411yescharacteristic
131C4×D4×C67C2311yescharacteristic
132#117×C67—21no—
133#117×C67C2×C411yescharacteristic
134C2×C42×C67—21no—
135C2×C42×C67C2311yescharacteristic
136C4⋊C8×C67—21no—
137C4⋊C8×C67C2311yescharacteristic
138C4⋊C8×C67C2311yescharacteristic
139C22⋊C8×C67—21no—
140C22⋊C8×C67—21no—
141#97×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134—21no—
145C2×Dic134C22×C411yescharacteristic
146C2×Dic134C22×C411yescharacteristic
147#11761618—22no—
148#11761618—22no—
149#11761616—22no—
150#11761616—22no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—21no—
153C2×C67⋊C8—81no—
154C2×C67⋊C8—81no—
155C2×C67⋊C8—81no—
156C2×C67⋊C8C22×C411yescharacteristic
157C2×C67⋊C8C22×C411yescharacteristic
158C2×C67⋊C8C22×C411yescharacteristic
159C2×C67⋊C8C22×C411yescharacteristic
160C2×C67⋊C8C2×D411yescharacteristic
161C2×C67⋊C8C2×D411yescharacteristic
162C2×C67⋊C8C2×D411yescharacteristic
163C2×C67⋊C8C2×D411yescharacteristic
164C4⋊Dic67—21no—
165C4⋊Dic67—21no—
166C4⋊Dic67—21no—
167C4⋊Dic67C22×C411yescharacteristic
168C4⋊Dic67C22×C411yescharacteristic
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171Dic67⋊C4—21no—
172Dic67⋊C4—21no—
173C22⋊Dic67—21no—
174C22⋊Dic67—21no—
175C4×Dic67—21no—
176C4×Dic67—21no—
177#41×C67—22no—
178#41×C67—22no—
179C2×Q8×C67C22×C411yescharacteristic
180C2×D4×C67—21no—
181C2×D4×C67C22×C411yescharacteristic
182C22×C4×C67—21no—
183C22×C4×C67—21no—
184C22×C4×C67—21no—
185C22×C4×C67C22×C411yescharacteristic
186C22×C4×C67C2×D411yescharacteristic
187C22×C4×C67#4111yescharacteristic
188C2×C8×C67—21no—
189C2×C8×C67—21no—
190C2×C8×C67—21no—
191C2×C8×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67C22×C411yescharacteristic
195C4⋊C4×C67C22×C411yescharacteristic
196C22⋊C4×C67—21no—
197C22⋊C4×C67—21no—
198C22⋊C4×C67—21no—
199C42×C67—21no—
200C42×C67—21no—
201C42×C67—21no—
202C42×C67C22×C411yescharacteristic
203C42×C67C2×D411yescharacteristic
204C42×C67#4111yescharacteristic
205C42×C67C2411yescharacteristic
206C23×C67—21no—
207C23×C67C4×D411yescharacteristic
208Q8×C67—21no—
209Q8×C67#13112yes—
210D4×C67—24no—
211D4×C67—21no—
212D4×C67#13112yes—
213C2×C4×C67—24no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—41no—
224C2×C4×C67—81no—
225C2×C4×C67C4×D411yescharacteristic
226C2×C4×C67C4×D411yescharacteristic
227C2×C4×C67C4×D411yescharacteristic
228C2×C4×C67C23×C411yescharacteristic
229C2×C4×C67C22×D411yescharacteristic
230C2×C4×C67C2×#4111yescharacteristic
231C2×Dic67—21no—
232C2×Dic67—21no—
233C2×Dic67—21no—
234C2×Dic67—21no—
235Dic134—21no—
236Dic134—21no—
237Dic134—21no—
238Dic134—21no—
239Dic134—81no—
240C8×C67—41no—
241C8×C67—41no—
242C8×C67—41no—
243C8×C67—41no—
244C67⋊C8—21no—
245C67⋊C8—21no—
246C67⋊C8—21no—
247C67⋊C8—21no—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—21no—
251C67⋊C8—21no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C22×C67—21no—
255C22×C67—22no—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67C2×C4×D411yescharacteristic
261C4×C67—21no—
262C4×C67—22no—
263C4×C67—21no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67C2×#13111yescharacteristic
269C4×C67C2×#13111yescharacteristic
270C4×C67#57711yescharacteristic
271C4×C67#57811yescharacteristic
272Dic67—41no—
273Dic67—41no—
274Dic67—41no—
275Dic67—41no—
276#17921—671nomaximal, Sylow
277C2×C67—41no—
278C2×C67—42no—
279C2×C67#292911yescharacteristic
280C2×C67#294711yescharacteristic
281C2×C67#295711yescharacteristic
282#3107—671no—
283#2988—671no—
284#2985—671no—
285#1587—671no—
286#1585—671no—
287#1584—671no—
288#1582—671no—
289#1579—671no—
290SD16⋊C8—671no—
291C8×Q16—671no—
292C8×SD16—671no—
293#1573—671no—
294#1488D6711yescharacteristic
295#1481—671no—
296#1454—671no—
297C67#1792111yescharacteristic, complemented, Sylow
298#533—671no—
299C4×#41C2×D6711yescharacteristic
300#485—671no—
301#485—671no—
302#475—671no—
303#474—671no—
304#462—671no—
305#461—671no—
306#452—671no—
307#452—671no—
308#446—671no—
309#446—671no—
310#446—1341no—
311#446—1341no—
312C8×Q8—671no—
313C8×Q8—671no—
314C8×Q8—671no—
315C8×Q8—671no—
316C4×Q16—671no—
317C4×Q16—671no—
318C4×SD16—671no—
319C4×SD16—671no—
320C8×D4—671no—
321C8×D4—671no—
322C8×D4—1341no—
323C8×D4—1341no—
324#433—1341no—
325#433—1341no—
326#432C2×D6711yescharacteristic
327#431—671no—
328#431—671no—
329#431—1341no—
330#431C2×D6711yescharacteristic
331C2×C4⋊C8—671no—
332C2×C4⋊C8—1341no—
333#335—671no—
334#335—671no—
335#334—671no—
336#334—671no—
337Q8⋊C8—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8C2×D6711yescharacteristic
342Q8⋊C8C2×D6711yescharacteristic
343D4⋊C8C2×D6711yescharacteristic
344D4⋊C8C2×D6711yescharacteristic
345#322—671no—
346#322—671no—
347#322—671no—
348C82—671no—
349C2×#41C4×D6711yescharacteristic
350C2×Q16—671no—
351C2×Q16—671no—
352C2×SD16—671no—
353C2×SD16—671no—
354C22×C8—1341no—
355C22×C8—1341no—
356C22×C8—1341no—
357C4⋊Q8—671no—
358C4⋊Q8—671no—
359#125—1341no—
360C22⋊Q8—671no—
361C22⋊Q8—671no—
362C22⋊Q8—1341no—
363C4×Q8—671no—
364C4×Q8—671no—
365C4×Q8—1341no—
366C4×Q8C4×D6711yescharacteristic
367C4×Q8C22×D6711yescharacteristic
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C4×D6711yescharacteristic
371C4×D4C22×D6711yescharacteristic
372#117—21no—
373#117—1341no—
374#117C4×D6711yescharacteristic
375C2×C42—21no—
376C2×C42C22×D6711yescharacteristic
377#107—671no—
378#107—671no—
379#106—671no—
380#106—671no—
381C4⋊C8—21no—
382C4⋊C8—671no—
383C4⋊C8—671no—
384C4⋊C8—671no—
385C4⋊C8—671no—
386C4⋊C8—671no—
387C4⋊C8—671no—
388C4⋊C8—1341no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8—1341no—
395C4⋊C8C22×D6711yescharacteristic
396C4⋊C8C22×D6711yescharacteristic
397#103—671no—
398#103—671no—
399#103—671no—
400#103—671no—
401#103—671no—
402#103—671no—
403#102—671no—
404#102—671no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—671no—
408C22⋊C8—671no—
409C22⋊C8—671no—
410C22⋊C8—671no—
411C22⋊C8—1341no—
412C22⋊C8—1341no—
413#97—21no—
414#97—1341no—
415#97—1341no—
416C4×C8—671no—
417C4×C8—671no—
418C4×C8—671no—
419C4×C8—671no—
420C4×C8—1341no—
421C4×C8—1341no—
422C4×C8—1341no—
423C4×C8—1341no—
424C4×C8C22×D6711yescharacteristic
425C4×C8C22×D6711yescharacteristic
426#41—22no—
427#41—22no—
428C2×Q8—671no—
429C2×Q8—671no—
430C2×Q8—1341no—
431C2×Q8C2×C4×D6711yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D6711yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C2×C4×D6711yescharacteristic
438C22×C4D4×D6711yescharacteristic
439C22×C4#1176162711yescharacteristic
440Q16—1342no—
441Q16—1342no—
442SD16—1342no—
443SD16—1342no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—5361no—
449C2×C8—5361no—
450C2×C8—5361no—
451C2×C8—671no—
452C2×C8—671no—
453C2×C8—671no—
454C2×C8—671no—
455C2×C8—671no—
456C2×C8—671no—
457C2×C8—671no—
458C2×C8—671no—
459C2×C8—1341no—
460C2×C8—1341no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—671no—
464C4⋊C4—671no—
465C4⋊C4—1341no—
466C4⋊C4—1341no—
467C4⋊C4—1341no—
468C4⋊C4—1341no—
469C4⋊C4—1341no—
470C4⋊C4—1341no—
471C4⋊C4—1341no—
472C4⋊C4C2×C4×D6711yescharacteristic
473C4⋊C4C2×C4×D6711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—1341no—
478C22⋊C4—1341no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—1341no—
483C42—1341no—
484C42C2×C4×D6711yescharacteristic
485C42D4×D6711yescharacteristic
486C42#1176162711yescharacteristic
487C42C23×D6711yescharacteristic
488C23—21no—
489C23(C22⋊D67)⋊C411yescharacteristic
490Q8—21no—
491Q8—5361no—
492Q8—1341no—
493Q8—1341no—
494Q8—1341no—
495Q8—1341no—
496Q8#1452405112yes—
497D4—24no—
498D4—21no—
499D4#1452405112yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—1341no—
513C2×C4—1341no—
514C2×C4—1341no—
515C2×C4—1341no—
516C2×C4(C22⋊D67)⋊C411yescharacteristic
517C2×C4(C22⋊D67)⋊C411yescharacteristic
518C2×C4(C22⋊D67)⋊C411yescharacteristic
519C2×C4C22×C4×D6711yescharacteristic
520C2×C4C2×D4×D6711yescharacteristic
521C2×C4C2×#1176162711yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—2681no—
527C8—2681no—
528C8—1341no—
529C8—1341no—
530C8—1341no—
531C8—1341no—
532C8—1341no—
533C8—1341no—
534C8—1341no—
535C8—1341no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22C2×(C22⋊D67)⋊C411yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—2681no—
551C4—2681no—
552C4—2681no—
553C4—2681no—
554C4C2×#1452405111yescharacteristic
555C4C2×#1452405111yescharacteristic
556C4#1686420311yescharacteristic
557C4#1686421011yescharacteristic
558C2—41no—
559C2—42no—
560C2#2256889911yescharacteristic
561C2#2256905511yescharacteristic
562C2#2256912211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1488C2
socleC22×C67C2×C4×D4
prime radicalC2×#41×C67C2×C4
cosocleC42×C67C24
hypercenter#1488D67
hypocenterC67#17921
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1488

Complements

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

NH ≅ G/Ntype
3. #22569306C2semidirect
4. #22559077C2semidirect
5. #22559063C2semidirect
6. #22559056C2semidirect
7. #22559042C2semidirect
8. #22559025C2semidirect
9. #22559018C2semidirect
10. #22559013C2semidirect
11. #22559012C2semidirect
12. #22558995C2semidirect
13. #22558613C2semidirect
14. #22558509C2semidirect
297. C67#17921semidirect

Maximal subgroups

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

subgroupindex
#225702802
#225693202
#225693062
#225590772
#225590632
#225590562
#225590422
#225590252
#225590182
#225590132
#225590122
#225589952
#225586132
#225585092
C67×#14882
#1792167