Finite Group Databasebeta

← #30855436

Subgroups — #30855436

Order 17152.

1438 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524050C211yesmaximal, characteristic
2C2×#16863439C211yesmaximal, characteristic, complemented
3C2×#16863436C211yesmaximal, characteristic, complemented
4#22562801C211yesmaximal, characteristic
5#22560240C212yesmaximal, complemented
6#22560240C212yesmaximal, complemented
7#22560236C212yesmaximal, complemented
8#22560232C212yesmaximal, complemented
9#22559925C211yesmaximal, characteristic, complemented
10C2×#16863158C211yesmaximal, characteristic, complemented
11C67×#2311C211yesmaximal, characteristic
12C2×#14524050—24no—
13C2×#14524050—24no—
14C2×#14524050C418yescomplemented
15C22×#11761612C2212yes—
16C22×#11761612C2211yescharacteristic
17C23×C67⋊C8C2211yescharacteristic
18C23×C67⋊C8C2212yes—
19C2×#14524002C2211yescharacteristic
20C22×C4⋊Dic67C2211yescharacteristic
21#16863439—24no—
22#16863436—24no—
23C2×#14523976C2212yes—
24C2×#14523976C2212yes—
25C2×#14523974C2212yes—
26C2×#14523974C2212yes—
27#16863303—21no—
28#16863299—21no—
29#16863276—22no—
30#16863256C2212yes—
31#16863256C2212yes—
32#16863158C2212yes—
33#16863158C2212yes—
34#16863158C2211yescharacteristic
35#16863158C2211yescharacteristic
36#16863158C2211yescharacteristic
37#16863158C2211yescharacteristic
38C22×#41×C67C2211yescharacteristic
39C2×#117×C67C2211yescharacteristic
40C22×C4⋊C4×C67C2211yescharacteristic
41#391×C67—21no—
42#390×C67—21no—
43#386×C67—22no—
44#380×C67C2212yes—
45#380×C67C2212yes—
46C23×Dic67—21no—
47#14524050—232no—
48C2×#11761612—24no—
49C2×#11761612—24no—
50C2×#11761612C2×C414yes—
51C2×#11761612C2×C418yescomplemented
52C2×#11761612D414yes—
53C2×#11761612D414yes—
54C22×C67⋊C8—24no—
55C22×C67⋊C8—24no—
56C22×C67⋊C8—22no—
57C22×C67⋊C8—22no—
58C22×C67⋊C8C2×C414yes—
59C22×C67⋊C8C2×C418yescomplemented
60C22×C67⋊C8C2311yescharacteristic
61C22×C67⋊C8C2311yescharacteristic
62C22×C67⋊C8C2312yes—
63#14524002—22no—
64#14524002—22no—
65C2×C4⋊Dic67—22no—
66C2×C4⋊Dic67—21no—
67C2×C4⋊Dic67—22no—
68C2×C4⋊Dic67C2312yes—
69C2×C4⋊Dic67C2311yescharacteristic
70C2×C4⋊Dic67C2311yescharacteristic
71C2×C22⋊Dic67—21no—
72C2×C22⋊Dic67—22no—
73C2×C22⋊Dic67—22no—
74C2×C4×Dic67—21no—
75#14523976—24no—
76#14523976—24no—
77#14523974—24no—
78#14523974—24no—
79C67⋊#95—22no—
80C67⋊#95—22no—
81C2×#41×C67—22no—
82C2×#41×C67—22no—
83C2×#41×C67—22no—
84C2×#41×C67—22no—
85C2×#41×C67C2×C414yes—
86C2×#41×C67Q814yes—
87C22×Q8×C67D411yescharacteristic
88C22×D4×C67D412yes—
89C22×D4×C67D411yescharacteristic
90C23×C4×C67—21no—
91C23×C4×C67C2312yes—
92C23×C4×C67C2311yescharacteristic
93#117×C67—22no—
94#117×C67—22no—
95C2×C4⋊C4×C67—22no—
96C2×C4⋊C4×C67—22no—
97C2×C4⋊C4×C67—21no—
98C2×C4⋊C4×C67C2312yes—
99C2×C4⋊C4×C67C2311yescharacteristic
100C2×C4⋊C4×C67C2311yescharacteristic
101C2×C22⋊C4×C67—21no—
102C2×C22⋊C4×C67—22no—
103C2×C22⋊C4×C67—22no—
104C2×C42×C67—21no—
105#95×C67—22no—
106#95×C67—22no—
107C22×Dic67—21no—
108C22×Dic67—21no—
109C22×Dic67—22no—
110C22×Dic67—42no—
111C22×Dic67—81no—
112C22×Dic67—81no—
113#11761612—216no—
114#11761612C22⋊C4116yescomplemented
115C2×C67⋊C8—28no—
116C2×C67⋊C8—21no—
117C2×C67⋊C8—21no—
118C2×C67⋊C8—21no—
119C2×C67⋊C8—21no—
120C2×C67⋊C8—216no—
121C2×C67⋊C8C22×C412yes—
122C2×C67⋊C8C22×C412yes—
123C2×C67⋊C8C22×C414yes—
124C2×C67⋊C8C2×D414yes—
125C2×C67⋊C8C2×D414yes—
126C4⋊Dic67—21no—
127C4⋊Dic67—22no—
128C4⋊Dic67—21no—
129C4⋊Dic67—22no—
130C4⋊Dic67—21no—
131C4⋊Dic67—21no—
132C4⋊Dic67—42no—
133C22⋊Dic67—42no—
134C22⋊Dic67—82no—
135C22⋊Dic67—82no—
136C4×Dic67—42no—
137C24×C67C2×D412yes—
138C24×C67C2×D411yescharacteristic
139#41×C67—28no—
140#41×C67—28no—
141#41×C67—28no—
142#41×C67—28no—
143C2×Q8×C67—22no—
144C2×Q8×C67—22no—
145C2×Q8×C67C4⋊C414yes—
146C2×D4×C67—24no—
147C2×D4×C67—24no—
148C2×D4×C67—22no—
149C2×D4×C67—22no—
150C2×D4×C67C4⋊C418yes—
151C2×D4×C67C4⋊C414yes—
152C22×C4×C67—22no—
153C22×C4×C67—22no—
154C22×C4×C67—22no—
155C22×C4×C67—22no—
156C22×C4×C67—21no—
157C22×C4×C67—21no—
158C22×C4×C67—22no—
159C22×C4×C67—42no—
160C22×C4×C67—81no—
161C22×C4×C67—81no—
162C22×C4×C67C22×C414yes—
163C22×C4×C67C22×C412yes—
164C22×C4×C67C2×D412yes—
165C22×C4×C67C2×D412yes—
166C22×C4×C67C2×D412yes—
167C22×C4×C67C2×D411yescharacteristic
168C22×C4×C67C2×Q814yes—
169C22×C4×C67C2×Q812yes—
170C22×C4×C67#4112yes—
171C22×C4×C67#4112yes—
172C22×C4×C67C2411yescharacteristic
173C4⋊C4×C67—22no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—22no—
177C4⋊C4×C67—21no—
178C4⋊C4×C67—21no—
179C4⋊C4×C67—42no—
180C22⋊C4×C67—42no—
181C22⋊C4×C67—82no—
182C22⋊C4×C67—82no—
183C42×C67—42no—
184C23×C67—21no—
185C23×C67—21no—
186C23×C67—21no—
187C23×C67—21no—
188C23×C67—22no—
189C23×C67—22no—
190C23×C67—22no—
191C23×C67—22no—
192C23×C67—22no—
193C23×C67—22no—
194C23×C67C2×C4⋊C414yes—
195C23×C67C2×C4⋊C412yes—
196C23×C67#12112yes—
197C23×C67#12112yes—
198C23×C67C22×D411yescharacteristic
199Q8×C67—28no—
200D4×C67—216no—
201D4×C67—28no—
202C2×C4×C67—22no—
203C2×C4×C67—22no—
204C2×C4×C67—24no—
205C2×C4×C67—24no—
206C2×C4×C67—24no—
207C2×C4×C67—24no—
208C2×C4×C67—41no—
209C2×C4×C67—41no—
210C2×C4×C67—42no—
211C2×C4×C67—41no—
212C2×C4×C67—42no—
213C2×C4×C67—41no—
214C2×C4×C67—42no—
215C2×C4×C67—41no—
216C2×C4×C67—41no—
217C2×C4×C67—84no—
218C2×C4×C67C2×C22⋊C414yes—
219C2×C4×C67C2×C4⋊C414yes—
220C2×C4×C67C2×C4⋊C412yes—
221C2×C4×C67#11714yes—
222C2×C4×C67#12112yes—
223C2×C4×C67#12112yes—
224C2×C4×C67C22⋊Q814yes—
225C2×C4×C67C22⋊Q814yes—
226C2×C4×C67C23×C411yescharacteristic
227C2×C4×C67C22×D411yescharacteristic
228C2×C4×C67C22×D411yescharacteristic
229C2×C4×C67C22×Q811yescharacteristic
230C2×C4×C67C2×#4111yescharacteristic
231C2×C4×C67C2×#4111yescharacteristic
232C2×Dic67—41no—
233C2×Dic67—41no—
234C2×Dic67—42no—
235C2×Dic67—41no—
236C2×Dic67—42no—
237C2×Dic67—41no—
238C2×Dic67—42no—
239C2×Dic67—41no—
240C2×Dic67—41no—
241C2×Dic67—84no—
242C67⋊C8—22no—
243C67⋊C8—22no—
244C67⋊C8C2×C22⋊C418yes—
245C22×C67—22no—
246C22×C67—22no—
247C22×C67—22no—
248C22×C67—22no—
249C22×C67—22no—
250C22×C67—22no—
251C22×C67—24no—
252C22×C67—24no—
253C22×C67—24no—
254C22×C67—42no—
255C22×C67—42no—
256C22×C67—42no—
257C22×C67—42no—
258C22×C67#38014yes—
259C22×C67C22×C4⋊C411yescharacteristic
260C22×C67C2×#12111yescharacteristic
261C22×C67C2×#12111yescharacteristic
262C22×C67#57611yescharacteristic
263C22×C67#57711yescharacteristic
264C22×C67#57711yescharacteristic
265C22×C67#57811yescharacteristic
266C4×C67—24no—
267C4×C67—81no—
268C4×C67—82no—
269C4×C67—81no—
270C4×C67#38014yes—
271C4×C67C22×C22⋊C411yescharacteristic
272C4×C67C2×#11711yescharacteristic
273C4×C67C2×C22⋊Q811yescharacteristic
274C4×C67C2×C22⋊Q811yescharacteristic
275Dic67—81no—
276Dic67—82no—
277Dic67—81no—
278#20664—671nomaximal, Sylow
279C2×C67—24no—
280C2×C67—44no—
281C2×C67C2×#38011yescharacteristic
282C2×C67#291211yescharacteristic
283C2×C67#291311yescharacteristic
284C2×C67#306211yescharacteristic
285C2×C67#306211yescharacteristic
286C2×C67#306311yescharacteristic
287C2×C67#306311yescharacteristic
288C22×#131—671no—
289C2×#428—671no—
290C2×#428—671no—
291#2311—671no—
292#2311D6711yescharacteristic
293#1809—672no—
294#1809—672no—
295#1808—672no—
296#1807—672no—
297#1735—671no—
298C2×#340—671no—
299C67#2066411yescharacteristic, complemented, Sylow
300C22×#41C2×D6711yescharacteristic
301C2×#131—678no—
302C2×#131—1344no—
303C2×#131—1344no—
304C22×M16—671no—
305C22×M16—672no—
306C23×C8—672no—
307C23×C8—671no—
308C2×#117—671no—
309C2×#117C2×D6711yescharacteristic
310C22×C4⋊C4—671no—
311C22×C4⋊C4C2×D6711yescharacteristic
312#428—1344no—
313#428—1344no—
314C2×#107—672no—
315C2×#107—672no—
316C2×#106—672no—
317C2×#106—672no—
318#391—21no—
319#391—1341no—
320#390—21no—
321#390—1341no—
322#386—22no—
323#386—1342no—
324#380—672no—
325#380—672no—
326#380C2×D6712yes—
327#380C2×D6712yes—
328#340—671no—
329#340—672no—
330#340—671no—
331#340—672no—
332#340—671no—
333#340—671no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41Dic13414yes—
339C2×#41C4×D6714yes—
340C22×Q8C22⋊D6711yescharacteristic
341C22×D4C22⋊D6712yes—
342C22×D4C22⋊D6711yescharacteristic
343C23×C4—21no—
344C23×C4—1341no—
345C23×C4C22×D6712yes—
346C23×C4C22×D6711yescharacteristic
347#131—13432no—
348C2×M16—674no—
349C2×M16—674no—
350C2×M16—674no—
351C2×M16—678no—
352C2×M16—1344no—
353C2×M16—1344no—
354C22×C8—672no—
355C22×C8—674no—
356C22×C8—678no—
357C22×C8—671no—
358C22×C8—671no—
359C22×C8—1344no—
360C22×C8—1344no—
361C22×C8—1342no—
362C22×C8—1342no—
363#117—22no—
364#117—22no—
365#117—1342no—
366#117—1342no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—672no—
371C2×C4⋊C4—671no—
372C2×C4⋊C4—671no—
373C2×C4⋊C4—1341no—
374C2×C4⋊C4—1342no—
375C2×C4⋊C4—1342no—
376C2×C4⋊C4C22×D6712yes—
377C2×C4⋊C4C22×D6711yescharacteristic
378C2×C4⋊C4C22×D6711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1341no—
383C2×C22⋊C4—1342no—
384C2×C22⋊C4—1342no—
385C2×C42—21no—
386C2×C42—1341no—
387#107—1344no—
388#107—1344no—
389#106—1344no—
390#106—1344no—
391#95—22no—
392#95—22no—
393#95—1342no—
394#95—1342no—
395C24C2×C22⋊D6712yes—
396C24C2×C22⋊D6711yescharacteristic
397#41—28no—
398#41—28no—
399#41—28no—
400#41—28no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8Dic67⋊C414yes—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4Dic67⋊C418yes—
409C2×D4Dic67⋊C414yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—2682no—
421C22×C4—5361no—
422C22×C4—5361no—
423C22×C4—1341no—
424C22×C4—1341no—
425C22×C4—1342no—
426C22×C4C2×C4×D6714yes—
427C22×C4C2×C4×D6712yes—
428C22×C4D4×D6712yes—
429C22×C4D4×D6712yes—
430C22×C4C2×C22⋊D6712yes—
431C22×C4C2×C22⋊D6711yescharacteristic
432C22×C4C2×Dic13414yes—
433C22×C4C2×Dic13412yes—
434C22×C4#1176162712yes—
435C22×C4#1176162712yes—
436C22×C4C23×D6711yescharacteristic
437M16—6716no—
438M16—13416no—
439C2×C8—674no—
440C2×C8—674no—
441C2×C8—674no—
442C2×C8—672no—
443C2×C8—672no—
444C2×C8—13416no—
445C2×C8—1348no—
446C2×C8—1341no—
447C2×C8—1341no—
448C2×C8—1341no—
449C2×C8—1341no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—2682no—
458C4⋊C4—1341no—
459C4⋊C4—1342no—
460C4⋊C4—1341no—
461C4⋊C4—1342no—
462C4⋊C4—1341no—
463C4⋊C4—1341no—
464C22⋊C4—42no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—2682no—
468C22⋊C4—5362no—
469C22⋊C4—5362no—
470C42—42no—
471C42—2682no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23C2×Dic67⋊C414yes—
483C23C2×Dic67⋊C412yes—
484C23#1452401512yes—
485C23#1452401512yes—
486C23C22×C22⋊D6711yescharacteristic
487Q8—28no—
488D4—216no—
489D4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—84no—
506C2×C4—2681no—
507C2×C4—2681no—
508C2×C4—2682no—
509C2×C4—2681no—
510C2×C4—2682no—
511C2×C4—2681no—
512C2×C4—2682no—
513C2×C4—2681no—
514C2×C4—2681no—
515C2×C4—5364no—
516C2×C4C22⋊C4×D6714yes—
517C2×C4C2×Dic67⋊C414yes—
518C2×C4C2×Dic67⋊C412yes—
519C2×C4#1452399914yes—
520C2×C4#1452401512yes—
521C2×C4#1452401512yes—
522C2×C4#1452402014yes—
523C2×C4#1452402014yes—
524C2×C4C22×C4×D6711yescharacteristic
525C2×C4C2×D4×D6711yescharacteristic
526C2×C4C2×D4×D6711yescharacteristic
527C2×C4C22×Dic13411yescharacteristic
528C2×C4C2×#1176162711yescharacteristic
529C2×C4C2×#1176162711yescharacteristic
530C8—678no—
531C8—1342no—
532C8—1342no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22#1686325414yes—
547C22C22×Dic67⋊C411yescharacteristic
548C22C2×#1452401511yescharacteristic
549C22C2×#1452401511yescharacteristic
550C22#1686419911yescharacteristic
551C22#1686420411yescharacteristic
552C22#1686420711yescharacteristic
553C22#1686421111yescharacteristic
554C4—24no—
555C4—82no—
556C4—81no—
557C4—81no—
558C4—5361no—
559C4—5362no—
560C4—5361no—
561C4#1686325414yes—
562C4C2×C22⋊C4×D6711yescharacteristic
563C4C2×#1452399911yescharacteristic
564C4C2×#1452402011yescharacteristic
565C4C2×#1452402011yescharacteristic
566C2—24no—
567C2—44no—
568C2C2×#1686325411yescharacteristic
569C2#2256880211yescharacteristic
570C2#2256880711yescharacteristic
571C2#2256988511yescharacteristic
572C2#2256989111yescharacteristic
573C2#2256989611yescharacteristic
574C2#2256990211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2311C2
socleC23×C67C22×D4
prime radicalC22×#41×C67C22
cosocleC22×C4×C67C24
hypercenter#2311D67
hypocenterC67#20664
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#16863439C2semidirect
3. C2×#16863436C2semidirect
5. #22560240C2semidirect
6. #22560240C2semidirect
7. #22560236C2semidirect
8. #22560232C2semidirect
9. #22559925C2semidirect
10. C2×#16863158C2semidirect
14. C2×#14524050C4semidirect
51. C2×#11761612C2×C4semidirect
59. C22×C67⋊C8C2×C4semidirect
114. #11761612C22⋊C4semidirect
299. C67#20664semidirect

Maximal subgroups

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

subgroupindex
C22×#145240502
C2×#168634392
C2×#168634362
#225628012
#225602402
#225602362
#225602322
#225599252
C2×#168631582
C67×#23112
#2066467