Finite Group Databasebeta

← #30873435

Subgroups — #30873435

Order 17152.

692 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22568863C211yesmaximal, characteristic
2C2×#16863432C211yesmaximal, characteristic
3#22560959C212yesmaximal, complemented
4#22560808C212yesmaximal, complemented
5#22560805C212yesmaximal, complemented
6#22560308C211yesmaximal, characteristic
7#22560306C211yesmaximal, characteristic
8#22560245C211yesmaximal, characteristic
9#22560094C212yesmaximal, complemented
10#22559988C211yesmaximal, characteristic
11C67×#2924C211yesmaximal, characteristic
12C23×C67⋊C8C2211yescharacteristic
13C2×#14524002C2211yescharacteristic
14#16863454—22no—
15#16863450—21no—
16#16863446—21no—
17C2×#14523978C2212yes—
18#16863432C411yescharacteristic, complemented
19#16863432C411yescharacteristic, complemented
20#16863432C411yescharacteristic, complemented
21#16863432C411yescharacteristic, complemented
22#16863432C2211yescharacteristic
23#16863432C2211yescharacteristic
24#16863432C2211yescharacteristic
25#16863432C2211yescharacteristic
26C2×#14523976C411yescharacteristic, complemented
27C2×#14523976C411yescharacteristic, complemented
28C2×#14523974C411yescharacteristic, complemented
29C2×#14523974C411yescharacteristic, complemented
30#16863422C2212yes—
31#16863360C2211yescharacteristic
32#16863351C2212yes—
33C2×#14523953C2212yes—
34C4×C22⋊Dic67C2211yescharacteristic
35#16863160C2212yes—
36#16863157C2212yes—
37#16863157C2212yes—
38#16863155C2212yes—
39#16863155C2212yes—
40#16863150C2212yes—
41C2×#117×C67C2211yescharacteristic
42#433×C67—22no—
43#432×C67—21no—
44#431×C67—21no—
45#424×C67C2212yes—
46#407×C67C2211yescharacteristic
47C2×C22⋊C8×C67C2211yescharacteristic
48#405×C67C2212yes—
49C2×#11761612C2312yes—
50C22×C67⋊C8—24no—
51C22×C67⋊C8C2×C411yescharacteristic
52C22×C67⋊C8C2×C411yescharacteristic
53C22×C67⋊C8C2×C411yescharacteristic
54C22×C67⋊C8C2×C411yescharacteristic
55C22×C67⋊C8C2311yescharacteristic
56C22×C67⋊C8C2311yescharacteristic
57#14524002—21no—
58#14524002—21no—
59#14524002C2×C411yescharacteristic
60#14524002C2×C411yescharacteristic
61#14524002C2311yescharacteristic
62#14524002C2311yescharacteristic
63C2×C4⋊Dic67C2×C411yescharacteristic
64C2×C4⋊Dic67C2×C411yescharacteristic
65C2×C22⋊Dic67—21no—
66C2×C22⋊Dic67—21no—
67C2×C4×Dic67—21no—
68C2×C4×Dic67C2312yes—
69#14523978—24no—
70#14523976C2×C411yescharacteristic
71#14523976C2×C411yescharacteristic
72#14523976C2×C411yescharacteristic
73#14523976C2×C411yescharacteristic
74#14523976C2×C411yescharacteristic
75#14523976C2×C411yescharacteristic
76#14523976C2×C411yescharacteristic
77#14523976C2×C411yescharacteristic
78#14523974C2×C411yescharacteristic
79#14523974C2×C411yescharacteristic
80#14523974C2×C411yescharacteristic
81#14523974C2×C411yescharacteristic
82#14523974C2×C411yescharacteristic
83#14523974C2×C411yescharacteristic
84#14523974C2×C411yescharacteristic
85#14523974C2×C411yescharacteristic
86Dic67⋊C8—22no—
87Dic67⋊C8—22no—
88C22⋊(C67⋊C8)—21no—
89C22⋊(C67⋊C8)—21no—
90#14523953—24no—
91#14523953—22no—
92C8×Dic67—22no—
93C67⋊#95—21no—
94C23×C4×C67C2311yescharacteristic
95C2×M16×C67C2312yes—
96C22×C8×C67C2312yes—
97#117×C67—21no—
98#117×C67—21no—
99#117×C67C2×C411yescharacteristic
100#117×C67C2×C411yescharacteristic
101#117×C67C2311yescharacteristic
102#117×C67C2311yescharacteristic
103C2×C4⋊C4×C67C2×C411yescharacteristic
104C2×C4⋊C4×C67C2×C411yescharacteristic
105C2×C22⋊C4×C67—21no—
106C2×C42×C67—21no—
107C4⋊C8×C67—22no—
108C4⋊C8×C67—22no—
109C22⋊C8×C67—21no—
110C22⋊C8×C67—21no—
111C22⋊C8×C67D411yescharacteristic
112C22⋊C8×C67D411yescharacteristic
113C22⋊C8×C67D412yes—
114#97×C67—22no—
115C4×C8×C67—22no—
116C22×Dic67—21no—
117C22×Dic67—21no—
118C22×Dic67—22no—
119#11761612—22no—
120#11761612—22no—
121C2×C67⋊C8—22no—
122C2×C67⋊C8—22no—
123C2×C67⋊C8—22no—
124C2×C67⋊C8—22no—
125C2×C67⋊C8—22no—
126C2×C67⋊C8—22no—
127C2×C67⋊C8C22×C411yescharacteristic
128C2×C67⋊C8C22×C411yescharacteristic
129C2×C67⋊C8C22×C411yescharacteristic
130C2×C67⋊C8C22×C411yescharacteristic
131C2×C67⋊C8#4114yes—
132C4⋊Dic67—21no—
133C4⋊Dic67—21no—
134C4⋊Dic67C22×C411yescharacteristic
135C4⋊Dic67C22×C411yescharacteristic
136C4⋊Dic67C22×C411yescharacteristic
137C4⋊Dic67C22×C411yescharacteristic
138C22⋊Dic67—21no—
139C22⋊Dic67—21no—
140C22⋊Dic67—21no—
141C22⋊Dic67—21no—
142C22⋊Dic67—42no—
143C4×Dic67—21no—
144C4×Dic67—21no—
145C4×Dic67—21no—
146C4×Dic67—21no—
147C4×Dic67—22no—
148C4×Dic67—22no—
149C24×C67#4111yescharacteristic
150C22×C4×C67—22no—
151C22×C4×C67—22no—
152C22×C4×C67—21no—
153C22×C4×C67—21no—
154C22×C4×C67C22×C411yescharacteristic
155C22×C4×C67C22×C411yescharacteristic
156C22×C4×C67C2×D411yescharacteristic
157C22×C4×C67C2×D411yescharacteristic
158C22×C4×C67#4111yescharacteristic
159C22×C4×C67C2411yescharacteristic
160M16×C67—22no—
161M16×C67—22no—
162C2×C8×C67—22no—
163C2×C8×C67—22no—
164C2×C8×C67—22no—
165C2×C8×C67C2×D412yes—
166C2×C8×C67C2×D412yes—
167C4⋊C4×C67—21no—
168C4⋊C4×C67—21no—
169C4⋊C4×C67C22×C411yescharacteristic
170C4⋊C4×C67C22×C411yescharacteristic
171C4⋊C4×C67C22×C411yescharacteristic
172C4⋊C4×C67C22×C411yescharacteristic
173C22⋊C4×C67—21no—
174C22⋊C4×C67—21no—
175C22⋊C4×C67—21no—
176C22⋊C4×C67—21no—
177C42×C67—21no—
178C42×C67—21no—
179C42×C67—21no—
180C42×C67—21no—
181C23×C67—22no—
182C23×C67—22no—
183C23×C67—41no—
184C23×C67C4×D411yescharacteristic
185C23×C67C4×D411yescharacteristic
186C23×C67C2×#4111yescharacteristic
187C2×C4×C67—22no—
188C2×C4×C67—21no—
189C2×C4×C67—21no—
190C2×C4×C67—21no—
191C2×C4×C67—21no—
192C2×C4×C67—22no—
193C2×C4×C67—22no—
194C2×C4×C67—21no—
195C2×C4×C67—21no—
196C2×C4×C67—21no—
197C2×C4×C67—21no—
198C2×C4×C67—42no—
199C2×C4×C67—81no—
200C2×C4×C67C4×D411yescharacteristic
201C2×C4×C67C4×D411yescharacteristic
202C2×C4×C67C23×C411yescharacteristic
203C2×C4×C67C22×D411yescharacteristic
204C2×C4×C67C2×#4112yes—
205C2×C4×C672−1+412yes—
206C2×Dic67—21no—
207C2×Dic67—21no—
208C2×Dic67—21no—
209C2×Dic67—21no—
210C2×Dic67—42no—
211C2×Dic67—42no—
212C2×Dic67—42no—
213C2×Dic67—42no—
214C2×Dic67—81no—
215C8×C67—22no—
216C8×C67—22no—
217C8×C67—42no—
218C67⋊C8—21no—
219C67⋊C8—21no—
220C67⋊C8—42no—
221C67⋊C8#11712yes—
222C67⋊C8#11712yes—
223C22×C67—21no—
224C22×C67—21no—
225C22×C67—21no—
226C22×C67—21no—
227C22×C67—21no—
228C22×C67—21no—
229C22×C67—21no—
230C22×C67—44no—
231C22×C67#44412yes—
232C22×C67C2×C4×D411yescharacteristic
233C22×C67#55212yes—
234C4×C67—21no—
235C4×C67—21no—
236C4×C67—41no—
237C4×C67—41no—
238C4×C67—41no—
239C4×C67—41no—
240C4×C67C2×#11711yescharacteristic
241C4×C67#51911yescharacteristic
242C4×C67#54712yes—
243Dic67—41no—
244Dic67—41no—
245Dic67—41no—
246Dic67—41no—
247Dic67—82no—
248#22607—671nomaximal, Sylow
249C2×C67—22no—
250C2×C67—22no—
251C2×C67—41no—
252C2×C67#234411yescharacteristic
253C2×C67C2×#44411yescharacteristic
254C2×C67#295711yescharacteristic
255#2924—671no—
256#2924D6711yescharacteristic
257C2×#427—671no—
258#1946—672no—
259#1922—672no—
260#1922—672no—
261#1826—671no—
262#1826—671no—
263#1810—671no—
264#1773—672no—
265#1752—671no—
266C67#2260711yescharacteristic, complemented, Sylow
267C23×C8—671no—
268C2×#117—671no—
269C2×#117C2×D6711yescharacteristic
270#433—22no—
271#433—1342no—
272#432—21no—
273#432—1341no—
274#431—21no—
275#431—1341no—
276C2×#108—672no—
277#427—671no—
278#427—671no—
279#427—671no—
280#427—671no—
281#427—671no—
282#427—671no—
283#427—671no—
284#427—671no—
285C2×#107—671no—
286C2×#107—671no—
287C2×#106—671no—
288C2×#106—671no—
289#424—672no—
290#424C2×D6712yes—
291#407—671no—
292#407C2×D6711yescharacteristic
293C2×C22⋊C8C2×D6711yescharacteristic
294#405—672no—
295#405C2×D6712yes—
296C2×#97—672no—
297C4×C22⋊C4—671no—
298#341—672no—
299#339—672no—
300#339—672no—
301#339—672no—
302#339—672no—
303#336—672no—
304C23×C4C22×D6711yescharacteristic
305C2×M16—672no—
306C2×M16C22×D6712yes—
307C22×C8—671no—
308C22×C8—671no—
309C22×C8—671no—
310C22×C8—671no—
311C22×C8—671no—
312C22×C8—671no—
313C22×C8—1344no—
314C22×C8C22×D6712yes—
315#117—21no—
316#117—21no—
317#117—671no—
318#117—671no—
319#117—671no—
320#117—671no—
321#117—1341no—
322#117—1341no—
323#117C4×D6711yescharacteristic
324#117C4×D6711yescharacteristic
325#117C22×D6711yescharacteristic
326#117C22×D6711yescharacteristic
327C2×C4⋊C4—671no—
328C2×C4⋊C4—671no—
329C2×C4⋊C4C4×D6711yescharacteristic
330C2×C4⋊C4C4×D6711yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1341no—
333C2×C22⋊C4—1341no—
334C2×C42—21no—
335C2×C42—672no—
336C2×C42—1341no—
337#108—1344no—
338#107—671no—
339#107—671no—
340#107—671no—
341#107—671no—
342#107—671no—
343#107—671no—
344#107—671no—
345#107—671no—
346#106—671no—
347#106—671no—
348#106—671no—
349#106—671no—
350#106—671no—
351#106—671no—
352#106—671no—
353#106—671no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—1342no—
357C4⋊C8—1342no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8—1341no—
361C22⋊C8—1341no—
362C22⋊C8C22⋊D6711yescharacteristic
363C22⋊C8C22⋊D6711yescharacteristic
364C22⋊C8C22⋊D6712yes—
365#97—22no—
366#97—1344no—
367#97—1342no—
368C4×C8—22no—
369C4×C8—1342no—
370#95—1341no—
371C24#1176162811yescharacteristic
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—1342no—
377C22×C4—1341no—
378C22×C4—1341no—
379C22×C4C2×C4×D6711yescharacteristic
380C22×C4C2×C4×D6711yescharacteristic
381C22×C4C2×C22⋊D6711yescharacteristic
382C22×C4C2×C22⋊D6711yescharacteristic
383C22×C4#1176162811yescharacteristic
384C22×C4C23×D6711yescharacteristic
385M16—22no—
386M16—22no—
387M16—1342no—
388M16—1342no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—671no—
393C2×C8—671no—
394C2×C8—674no—
395C2×C8—671no—
396C2×C8—671no—
397C2×C8—1342no—
398C2×C8—1342no—
399C2×C8—1342no—
400C2×C8—1342no—
401C2×C8—1342no—
402C2×C8—1342no—
403C2×C8C2×C22⋊D6712yes—
404C2×C8C2×C22⋊D6712yes—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—671no—
408C4⋊C4—671no—
409C4⋊C4—671no—
410C4⋊C4—671no—
411C4⋊C4—1341no—
412C4⋊C4—1341no—
413C4⋊C4C2×C4×D6711yescharacteristic
414C4⋊C4C2×C4×D6711yescharacteristic
415C4⋊C4C2×C4×D6711yescharacteristic
416C4⋊C4C2×C4×D6711yescharacteristic
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—2682no—
422C22⋊C4—1341no—
423C22⋊C4—1341no—
424C22⋊C4—1341no—
425C22⋊C4—1341no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—1342no—
431C42—1342no—
432C42—1341no—
433C42—1341no—
434C42—1341no—
435C42—1341no—
436C23—22no—
437C23—22no—
438C23—41no—
439C23C4×C22⋊D6711yescharacteristic
440C23C4×C22⋊D6711yescharacteristic
441C23C2×#1176162811yescharacteristic
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—81no—
455C2×C4—2682no—
456C2×C4—2682no—
457C2×C4—2682no—
458C2×C4—2682no—
459C2×C4—5361no—
460C2×C4—1341no—
461C2×C4—1341no—
462C2×C4—1341no—
463C2×C4—1341no—
464C2×C4C4×C22⋊D6711yescharacteristic
465C2×C4C4×C22⋊D6711yescharacteristic
466C2×C4C22×C4×D6711yescharacteristic
467C2×C4C22×C22⋊D6711yescharacteristic
468C2×C4#41×D6712yes—
469C2×C4#1452409012yes—
470C8—22no—
471C8—22no—
472C8—42no—
473C8—2682no—
474C8—672no—
475C8—672no—
476C8—1341no—
477C8—1341no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—44no—
486C22#1686351612yes—
487C22C2×C4×C22⋊D6711yescharacteristic
488C22#1686407912yes—
489C4—21no—
490C4—21no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—2681no—
496C4—2681no—
497C4—2681no—
498C4—2681no—
499C4—5362no—
500C4#117×D6711yescharacteristic
501C4#1686387311yescharacteristic
502C4#1686404512yes—
503C2—22no—
504C2—22no—
505C2—41no—
506C2#2256307911yescharacteristic
507C2C2×#1686351611yescharacteristic
508C2#2256912411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2924C2
socleC22×C67C2×C4×D4
prime radicalC24×C67#41
cosocleC22×C4×C67C24
hypercenter#2924D67
hypocenterC67#22607
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ #2924

Complements

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

NH ≅ G/Ntype
3. #22560959C2semidirect
4. #22560808C2semidirect
5. #22560805C2semidirect
9. #22560094C2semidirect
18. #16863432C4semidirect
19. #16863432C4semidirect
20. #16863432C4semidirect
21. #16863432C4semidirect
26. C2×#14523976C4semidirect
27. C2×#14523976C4semidirect
28. C2×#14523974C4semidirect
29. C2×#14523974C4semidirect
266. C67#22607semidirect

Maximal subgroups

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

subgroupindex
#225688632
C2×#168634322
#225609592
#225608082
#225608052
#225603082
#225603062
#225602452
#225600942
#225599882
C67×#29242
#2260767