Finite Group Databasebeta

← #30863034

Subgroups — #30863034

Order 17152.

1048 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863714C211yesmaximal, characteristic
2C4×#14524017C211yesmaximal, characteristic
3#22560829C212yesmaximal, complemented
4#22560794C212yesmaximal, complemented
5#22560526C211yesmaximal, characteristic
6#22560397C211yesmaximal, characteristic, complemented
7#22560299C211yesmaximal, characteristic, complemented
8#22560107C212yesmaximal, complemented
9#22560037C212yesmaximal, complemented
10#22559971C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C67C211yesmaximal, characteristic, complemented
12C2×#14524017C2211yescharacteristic
13C2×C4×D268C2211yescharacteristic
14C2×C4×C22⋊D67—21no—
15#16863714C418yescomplemented
16C2×#14523976C2212yes—
17C2×#14523974C2212yes—
18C2×#14523963C2212yes—
19C2×#14523963C2212yes—
20C2×C22⋊(C67⋊C8)C2211yescharacteristic
21C2×C22⋊(C67⋊C8)C2211yescharacteristic
22C2×#14523954C2212yes—
23#16863298C2211yescharacteristic
24#16863294C2211yescharacteristic
25#16863287—21no—
26#16863275—21no—
27#16863256C2211yescharacteristic
28C4×C4⋊Dic67C2211yescharacteristic
29C4×(C2×D67)⋊C4—21no—
30#16863158C2212yes—
31#16863158C2212yes—
32#16863158C2212yes—
33#16863158C2212yes—
34#16863151C2212yes—
35C22×C4⋊C4×C67C2211yescharacteristic
36C22×C4⋊C4×C67C2211yescharacteristic
37C22×C42×C67—21no—
38C22×C42×C67C2211yescharacteristic
39C4×C4⋊C4×C67—21no—
40C4×C4⋊C4×C67—21no—
41C4×C4⋊C4×C67C2212yes—
42C4×C4⋊C4×C67C2212yes—
43C2×#95×C67—21no—
44C22×D268C2311yescharacteristic
45C22×C22⋊D67—21no—
46C22×C4×D67—21no—
47C22×C67⋊C8C2312yes—
48C22×C67⋊C8C2312yes—
49#14524017—22no—
50#14524017C2×C414yes—
51C4×D268—24no—
52C4×C22⋊D67—44no—
53C2×C4⋊Dic67C2311yescharacteristic
54C2×C4⋊Dic67C2311yescharacteristic
55C2×C4⋊Dic67C2311yescharacteristic
56C2×Dic67⋊C4—21no—
57C2×C22⋊Dic67—21no—
58C2×(C2×D67)⋊C4—21no—
59C2×(C2×D67)⋊C4—21no—
60C2×(C2×D67)⋊C4—21no—
61C2×(C2×D67)⋊C4—21no—
62C2×C4×Dic67—21no—
63#14523976—24no—
64#14523974C2×C418yescomplemented
65#14523963—24no—
66#14523963C2×C418yescomplemented
67C22⋊(C67⋊C8)—22no—
68C22⋊(C67⋊C8)C2×C414yescomplemented
69#14523954—24no—
70C67⋊#95—21no—
71C67⋊#95—21no—
72C67⋊#95—21no—
73C23×C4×C67—21no—
74C23×C4×C67—21no—
75C23×C4×C67D411yescharacteristic
76C23×C4×C67D411yescharacteristic
77C23×C4×C67C2311yescharacteristic
78C2×C4⋊C4×C67—21no—
79C2×C4⋊C4×C67—21no—
80C2×C4⋊C4×C67—22no—
81C2×C4⋊C4×C67—24no—
82C2×C4⋊C4×C67C2×C414yescomplemented
83C2×C4⋊C4×C67C2312yes—
84C2×C4⋊C4×C67C2312yes—
85C2×C42×C67—24no—
86C2×C42×C67—22no—
87C2×C42×C67—22no—
88C2×C42×C67—44no—
89C2×C42×C67D411yescharacteristic
90C2×C42×C67D411yescharacteristic
91C2×C42×C67C2311yescharacteristic
92C2×C42×C67C2311yescharacteristic
93#95×C67—22no—
94#95×C67—22no—
95C23×D67—21no—
96C2×D268—22no—
97C2×D268—22no—
98C2×D268—21no—
99C2×D268C22×C412yes—
100C2×C22⋊D67—24no—
101C2×C22⋊D67—42no—
102C2×C22⋊D67—42no—
103C22×Dic67—21no—
104C22×Dic67—21no—
105C22×Dic67—21no—
106C2×C4×D67—42no—
107C2×C4×D67—42no—
108C2×C4×D67—42no—
109C2×C67⋊C8—22no—
110C2×C67⋊C8—22no—
111C2×C67⋊C8—22no—
112C2×C67⋊C8—22no—
113C2×C67⋊C8—22no—
114C2×C67⋊C8C22×C414yes—
115C4⋊Dic67—21no—
116C4⋊Dic67—22no—
117C4⋊Dic67—22no—
118C4⋊Dic67C22×C412yes—
119Dic67⋊C4—42no—
120C22⋊Dic67—42no—
121(C2×D67)⋊C4—24no—
122(C2×D67)⋊C4—42no—
123(C2×D67)⋊C4—42no—
124(C2×D67)⋊C4—42no—
125C4×Dic67—42no—
126C24×C67C2×D411yescharacteristic
127C22×C4×C67—22no—
128C22×C4×C67—22no—
129C22×C4×C67—22no—
130C22×C4×C67—22no—
131C22×C4×C67—22no—
132C22×C4×C67—24no—
133C22×C4×C67—24no—
134C22×C4×C67—22no—
135C22×C4×C67—22no—
136C22×C4×C67—22no—
137C22×C4×C67—22no—
138C22×C4×C67—41no—
139C22×C4×C67—42no—
140C22×C4×C67—42no—
141C22×C4×C67—42no—
142C22×C4×C67—42no—
143C22×C4×C67C22×C412yes—
144C22×C4×C67C2×D411yescharacteristic
145C22×C4×C67C2×D411yescharacteristic
146C22×C4×C67C2×D411yescharacteristic
147C22×C4×C67C2×D411yescharacteristic
148C22×C4×C67C2×D411yescharacteristic
149C22×C4×C67#4112yes—
150C22×C4×C67C2411yescharacteristic
151C4⋊C4×C67—24no—
152C4⋊C4×C67—24no—
153C4⋊C4×C67—22no—
154C4⋊C4×C67—22no—
155C4⋊C4×C67C22×C414yes—
156C4⋊C4×C67#4114yes—
157C42×C67—22no—
158C42×C67—22no—
159C42×C67—22no—
160C42×C67—44no—
161C42×C67—44no—
162C42×C67#4114yes—
163C23×C67—21no—
164C23×C67—21no—
165C23×C67—21no—
166C23×C67—21no—
167C23×C67C4×D412yes—
168C23×C67AGL(1,ℤ/8)12yes—
169C23×C67#13712yes—
170C23×C67C22×D411yescharacteristic
171C22×D67—22no—
172C22×D67—41no—
173C22×D67—41no—
174C22×D67—41no—
175C22×D67—41no—
176C22×D67—41no—
177C22×D67—41no—
178C2×C4×C67—28no—
179C2×C4×C67—21no—
180C2×C4×C67—21no—
181C2×C4×C67—21no—
182C2×C4×C67—21no—
183C2×C4×C67—21no—
184C2×C4×C67—21no—
185C2×C4×C67—21no—
186C2×C4×C67—21no—
187C2×C4×C67—24no—
188C2×C4×C67—24no—
189C2×C4×C67—24no—
190C2×C4×C67—21no—
191C2×C4×C67—24no—
192C2×C4×C67—21no—
193C2×C4×C67—21no—
194C2×C4×C67—44no—
195C2×C4×C67—48no—
196C2×C4×C67—48no—
197C2×C4×C67—42no—
198C2×C4×C67—42no—
199C2×C4×C67—42no—
200C2×C4×C67—42no—
201C2×C4×C67C4×D412yes—
202C2×C4×C67#12312yes—
203C2×C4×C67#12312yes—
204C2×C4×C67AGL(1,ℤ/8)12yes—
205C2×C4×C67C23×C411yescharacteristic
206C2×C4×C67C2×#4112yes—
207C2×C4×C67C2×#4112yes—
208C2×C4×C67C2×#4111yescharacteristic
209C22⋊D67—48no—
210C2×Dic67—22no—
211C2×Dic67—41no—
212C2×Dic67—41no—
213C2×Dic67—41no—
214C2×Dic67—41no—
215C2×Dic67—41no—
216C2×Dic67—41no—
217C2×Dic67—41no—
218C2×Dic67—41no—
219D268—24no—
220D268—42no—
221C4×D67—84no—
222C67⋊C8—24no—
223C67⋊C8—42no—
224C22×C67—22no—
225C22×C67—22no—
226C22×C67—22no—
227C22×C67—22no—
228C22×C67—22no—
229C22×C67—22no—
230C22×C67#44014yes—
231C22×C67C2×C4×D411yescharacteristic
232C22×C67C2×#12311yescharacteristic
233C22×C67#52511yescharacteristic
234C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
235C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
236C22×C67C2×#13711yescharacteristic
237C22×C67#57511yescharacteristic
238C2×D67—42no—
239C2×D67—42no—
240C2×D67—42no—
241C2×D67—81no—
242C2×D67—81no—
243C2×D67—81no—
244C2×D67—81no—
245C4×C67—22no—
246C4×C67—22no—
247C4×C67—22no—
248C4×C67—22no—
249C4×C67—41no—
250C4×C67—44no—
251C4×C67—44no—
252C4×C67C4×#4112yes—
253C4×C67#52912yes—
254Dic67—42no—
255Dic67—81no—
256Dic67—81no—
257#21580—671nomaximal, Sylow
258C2×C67—24no—
259C2×C67C4×#12311yescharacteristic
260C2×C67C2×#44011yescharacteristic
261C2×C67C4×AGL(1,ℤ/8)11yescharacteristic
262C2×C67#309211yescharacteristic
263C2×C67#309411yescharacteristic
264C2×C67#310911yescharacteristic
265C2×C67#311711yescharacteristic
266D67—82no—
267C2×#481—671no—
268C4×#121—671no—
269C2×C4×C4⋊C4D6711yescharacteristic, complemented
270#1925—672no—
271#1920—672no—
272#1874—671no—
273#1852—671no—
274#1824—671no—
275#1776—672no—
276#1762—672no—
277#1749—671no—
278C67#2158011yescharacteristic, complemented, Sylow
279C2×#121—671no—
280C2×C4×D4—671no—
281C2×C4×D4—1341no—
282C22×C4⋊C4C2×D6711yescharacteristic
283C22×C4⋊C4C2×D6711yescharacteristic
284C22×C42—21no—
285C22×C42C2×D6711yescharacteristic
286#481—678no—
287C2×#107—672no—
288C2×#106—672no—
289C2×#102—672no—
290C2×#102—672no—
291C2×C22⋊C8—671no—
292C2×C22⋊C8—671no—
293C2×#97—672no—
294#390—671no—
295#389—671no—
296#388—1341no—
297#386—1341no—
298#380—671no—
299C4×C4⋊C4—21no—
300C4×C4⋊C4—21no—
301C4×C4⋊C4—671no—
302C4×C4⋊C4C2×D6712yes—
303C4×C4⋊C4C2×D6712yes—
304C4×C22⋊C4—1341no—
305C2×#95—21no—
306#340—672no—
307#340—672no—
308#340—672no—
309#340—672no—
310#336—672no—
311C22×D4—671no—
312C22×D4—1341no—
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—1341no—
316C23×C4C22⋊D6711yescharacteristic
317C23×C4C22⋊D6711yescharacteristic
318C23×C4C22×D6711yescharacteristic
319C22×C8—672no—
320C22×C8—672no—
321#121—674no—
322#121—1342no—
323C4×D4—2684no—
324C4×D4—1344no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—671no—
330C2×C4⋊C4—671no—
331C2×C4⋊C4—671no—
332C2×C4⋊C4—1341no—
333C2×C4⋊C4C4×D6714yescomplemented
334C2×C4⋊C4C22×D6712yes—
335C2×C4⋊C4C22×D6712yes—
336C2×C22⋊C4—1341no—
337C2×C22⋊C4—1341no—
338C2×C22⋊C4—1341no—
339C2×C22⋊C4—1341no—
340C2×C22⋊C4—1341no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42—1341no—
346C2×C42C22⋊D6711yescharacteristic
347C2×C42C22⋊D6711yescharacteristic
348C2×C42C22×D6711yescharacteristic
349C2×C42C22×D6711yescharacteristic
350#107—1344no—
351#106—678no—
352#102—678no—
353#102—1344no—
354C22⋊C8—674no—
355C22⋊C8—1342no—
356#97—1344no—
357#95—22no—
358#95—22no—
359#95—1341no—
360#95—1341no—
361#95—1341no—
362C24—1341no—
363C24C2×C22⋊D6711yescharacteristic
364C2×D4—2682no—
365C2×D4—2682no—
366C2×D4—672no—
367C2×D4—1344no—
368C2×D4—1341no—
369C2×D4—1342no—
370C2×D4—1342no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—2682no—
388C22×C4—2682no—
389C22×C4—2682no—
390C22×C4—1341no—
391C22×C4—1341no—
392C22×C4—1341no—
393C22×C4C2×C4×D6712yes—
394C22×C4C2×C22⋊D6711yescharacteristic
395C22×C4C2×C22⋊D6711yescharacteristic
396C22×C4C2×C22⋊D6711yescharacteristic
397C22×C4C2×C22⋊D6711yescharacteristic
398C22×C4C2×C22⋊D6711yescharacteristic
399C22×C4#1176162812yes—
400C22×C4C23×D6711yescharacteristic
401C2×C8—674no—
402C2×C8—1342no—
403C2×C8—1342no—
404C2×C8—1342no—
405C2×C8—1342no—
406C2×C8—1342no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—2682no—
412C4⋊C4—672no—
413C4⋊C4—1342no—
414C4⋊C4—1342no—
415C4⋊C4—1341no—
416C4⋊C4C2×C4×D6714yes—
417C4⋊C4#1176162814yes—
418C22⋊C4—2682no—
419C22⋊C4—2682no—
420C22⋊C4—2682no—
421C22⋊C4—2682no—
422C22⋊C4—1344no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—44no—
427C42—44no—
428C42—2682no—
429C42#1176162814yes—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—2681no—
435C23—2681no—
436C23—2681no—
437C23—2681no—
438C23—2681no—
439C23—2681no—
440C23—1342no—
441C23C4×C22⋊D6712yes—
442C23#1452406812yes—
443C23#1452407412yes—
444C23C22×C22⋊D6711yescharacteristic
445D4—2688no—
446D4—2682no—
447D4—1344no—
448C2×C4—28no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—2681no—
472C2×C4—2681no—
473C2×C4—2681no—
474C2×C4—2681no—
475C2×C4—2681no—
476C2×C4—2681no—
477C2×C4—2681no—
478C2×C4—2681no—
479C2×C4—5364no—
480C2×C4—1342no—
481C2×C4C4×C22⋊D6712yes—
482C2×C4#1452402612yes—
483C2×C4#1452402612yes—
484C2×C4#1452406912yes—
485C2×C4C22×C4×D6711yescharacteristic
486C2×C4C2×#1176162812yes—
487C2×C4C2×#1176162812yes—
488C2×C4C2×#1176162811yescharacteristic
489C8—2682no—
490C8—1344no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—2682no—
498C22—2682no—
499C22—2682no—
500C22—5361no—
501C22—5361no—
502C22—5361no—
503C22—5361no—
504C22#1686349414yes—
505C22C2×C4×C22⋊D6711yescharacteristic
506C22C2×#1452402611yescharacteristic
507C22#1686390711yescharacteristic
508C22C2×#1452406811yescharacteristic
509C22C2×#1452406911yescharacteristic
510C22C2×#1452407411yescharacteristic
511C22#1686419411yescharacteristic
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—44no—
517C4—44no—
518C4—41no—
519C4—2682no—
520C4—5361no—
521C4—5361no—
522C4C4×#1176162812yes—
523C4#1686393012yes—
524C2—24no—
525C2—5362no—
526C2C4×#1452402611yescharacteristic
527C2C2×#1686349411yescharacteristic
528C2C4×#1452406911yescharacteristic
529C2#2257013211yescharacteristic
530C2#2257014911yescharacteristic
531C2#2257030011yescharacteristic
532C2#2257038011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×C4×C4⋊C4×C67C2
socleC23×C67C22×D4
prime radicalC2×#14524017C22
cosocleC22×C4×C67C24
hypercenterC2×C4×C4⋊C4D67
hypocenterC67#21580
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22560829C2semidirect
4. #22560794C2semidirect
6. #22560397C2semidirect
7. #22560299C2semidirect
8. #22560107C2semidirect
9. #22560037C2semidirect
10. #22559971C2semidirect
11. C2×C4×C4⋊C4×C67C2semidirect
15. #16863714C4semidirect
64. #14523974C2×C4semidirect
66. #14523963C2×C4semidirect
68. C22⋊(C67⋊C8)C2×C4semidirect
82. C2×C4⋊C4×C67C2×C4semidirect
269. C2×C4×C4⋊C4D67semidirect
278. C67#21580semidirect
333. C2×C4⋊C4C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168637142
C4×#145240172
#225608292
#225607942
#225605262
#225603972
#225602992
#225601072
#225600372
#225599712
C2×C4×C4⋊C4×C672
#2158067