Finite Group Databasebeta

← #30821900

Subgroups — #30821900

Order 17152.

1450 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863360C212yesmaximal, complemented
2#22558557C218yesmaximal, complemented
3#22558539C212yesmaximal, complemented
4#22558536C212yesmaximal, complemented
5C22×#123×C67C211yesmaximal, characteristic
6C22×#11761612—22no—
7#16863360—22no—
8#16863360—22no—
9#16863360C2214yes—
10#16863360C2214yescomplemented
11C2×C22⋊(C67⋊C8)—22no—
12C2×C22⋊(C67⋊C8)C2214yescomplemented
13#16863122C2218yescomplemented
14#16863120C2218yescomplemented
15C23×D4×C67C411yescharacteristic
16C24×C4×C67C2211yescharacteristic
17C2×#123×C67—22no—
18C2×#123×C67—22no—
19C2×#123×C67—22no—
20C2×#123×C67—22no—
21C2×#123×C67C414yes—
22C2×#123×C67C2214yes—
23C22×C4⋊C4×C67C411yescharacteristic
24C22×C4⋊C4×C67C2211yescharacteristic
25C22×C22⋊C4×C67C411yescharacteristic
26C22×C22⋊C4×C67C411yescharacteristic
27C22×C22⋊C4×C67C2211yescharacteristic
28C2×#11761612—24no—
29C2×#11761612—24no—
30C2×#11761612—48no—
31C22×C67⋊C8—24no—
32C22⋊(C67⋊C8)—24no—
33C22⋊(C67⋊C8)—24no—
34C22⋊(C67⋊C8)—24no—
35C22⋊(C67⋊C8)C2318yes—
36C25×C67C2×C411yescharacteristic
37C25×C67C2×C411yescharacteristic
38C22×D4×C67—22no—
39C22×D4×C67—21no—
40C22×D4×C67—21no—
41C22×D4×C67—44no—
42C22×D4×C67C2×C414yes—
43C23×C4×C67—44no—
44C23×C4×C67C2×C411yescharacteristic
45C23×C4×C67C2×C411yescharacteristic
46C23×C4×C67C2×C411yescharacteristic
47C23×C4×C67C2×C411yescharacteristic
48C23×C4×C67C2×C412yes—
49C23×C4×C67D412yes—
50C23×C4×C67D412yes—
51C23×C4×C67D412yes—
52C23×C4×C67D412yes—
53C23×C4×C67C2311yescharacteristic
54C23×C4×C67C2312yes—
55#123×C67—24no—
56#123×C67—24no—
57#123×C67—24no—
58#123×C67—24no—
59#123×C67—24no—
60#123×C67—24no—
61#123×C67—24no—
62#123×C67—24no—
63C2×C4⋊C4×C67—22no—
64C2×C4⋊C4×C67—21no—
65C2×C4⋊C4×C67—21no—
66C2×C4⋊C4×C67—22no—
67C2×C4⋊C4×C67—22no—
68C2×C4⋊C4×C67C2×C414yes—
69C2×C4⋊C4×C67C2×C412yes—
70C2×C4⋊C4×C67C2312yes—
71C2×C22⋊C4×C67—22no—
72C2×C22⋊C4×C67—21no—
73C2×C22⋊C4×C67—21no—
74C2×C22⋊C4×C67—22no—
75C2×C22⋊C4×C67—21no—
76C2×C22⋊C4×C67—21no—
77C2×C22⋊C4×C67—22no—
78C2×C22⋊C4×C67—22no—
79C2×C22⋊C4×C67C2×C414yes—
80C2×C22⋊C4×C67C2×C414yes—
81C2×C22⋊C4×C67C2×C412yes—
82C2×C22⋊C4×C67C2312yes—
83#11761612—48no—
84#11761612—48no—
85C2×C67⋊C8—28no—
86C2×C67⋊C8—48no—
87C24×C67—24no—
88C24×C67—21no—
89C24×C67—21no—
90C24×C67—42no—
91C24×C67—42no—
92C24×C67—42no—
93C24×C67—42no—
94C24×C67—42no—
95C24×C67C22⋊C412yes—
96C24×C67C22⋊C411yescharacteristic
97C24×C67C22⋊C411yescharacteristic
98C24×C67C22×C411yescharacteristic
99C24×C67C22×C412yes—
100C24×C67C22×C412yes—
101C2×D4×C67—24no—
102C2×D4×C67—24no—
103C2×D4×C67—48no—
104C2×D4×C67—44no—
105C2×D4×C67—44no—
106C2×D4×C67—48no—
107C22×C4×C67—28no—
108C22×C4×C67—21no—
109C22×C4×C67—21no—
110C22×C4×C67—21no—
111C22×C4×C67—21no—
112C22×C4×C67—21no—
113C22×C4×C67—21no—
114C22×C4×C67—42no—
115C22×C4×C67—44no—
116C22×C4×C67—44no—
117C22×C4×C67—44no—
118C22×C4×C67—44no—
119C22×C4×C67—44no—
120C22×C4×C67—42no—
121C22×C4×C67—42no—
122C22×C4×C67—42no—
123C22×C4×C67—42no—
124C22×C4×C67C22⋊C412yes—
125C22×C4×C67C22⋊C412yes—
126C22×C4×C67C22⋊C412yes—
127C22×C4×C67C22⋊C411yescharacteristic
128C22×C4×C67C22⋊C411yescharacteristic
129C22×C4×C67C22⋊C412yes—
130C22×C4×C67C22⋊C412yes—
131C22×C4×C67C22×C412yes—
132C22×C4×C67C22×C412yes—
133C22×C4×C67C22×C412yes—
134C22×C4×C67C22×C412yes—
135C22×C4×C67C22×C411yescharacteristic
136C22×C4×C67C2×D412yes—
137C22×C4×C67C2×D412yes—
138C22×C4×C67C2×D412yes—
139C22×C4×C67C2×D412yes—
140C22×C4×C67C2×D412yes—
141C22×C4×C67C2×D412yes—
142C22×C4×C67C2411yescharacteristic
143C4⋊C4×C67—24no—
144C4⋊C4×C67—24no—
145C4⋊C4×C67—24no—
146C4⋊C4×C67—24no—
147C22⋊C4×C67—24no—
148C22⋊C4×C67—24no—
149C22⋊C4×C67—24no—
150C22⋊C4×C67—24no—
151C22⋊C4×C67—24no—
152C22⋊C4×C67—24no—
153C23×C67—22no—
154C23×C67—22no—
155C23×C67—22no—
156C23×C67—24no—
157C23×C67—24no—
158C23×C67—22no—
159C23×C67—24no—
160C23×C67—24no—
161C23×C67—22no—
162C23×C67—24no—
163C23×C67—41no—
164C23×C67—42no—
165C23×C67—41no—
166C23×C67—41no—
167C23×C67—41no—
168C23×C67—42no—
169C23×C67—416no—
170C23×C67—42no—
171C23×C67—42no—
172C23×C67—42no—
173C23×C67—42no—
174C23×C67—42no—
175C23×C67—42no—
176C23×C67—42no—
177C23×C67—42no—
178C23×C67—88no—
179C23×C67#9914yes—
180C23×C67C2×C22⋊C412yes—
181C23×C67C2×C22⋊C412yes—
182C23×C67C2×C22⋊C411yescharacteristic
183C23×C67C2×C22⋊C411yescharacteristic
184C23×C67C23×C411yescharacteristic
185D4×C67—48no—
186D4×C67—48no—
187C2×C4×C67—24no—
188C2×C4×C67—24no—
189C2×C4×C67—24no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—24no—
193C2×C4×C67—22no—
194C2×C4×C67—22no—
195C2×C4×C67—22no—
196C2×C4×C67—24no—
197C2×C4×C67—41no—
198C2×C4×C67—41no—
199C2×C4×C67—42no—
200C2×C4×C67—42no—
201C2×C4×C67—42no—
202C2×C4×C67—41no—
203C2×C4×C67—42no—
204C2×C4×C67—42no—
205C2×C4×C67—42no—
206C2×C4×C67—41no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—42no—
211C2×C4×C67—42no—
212C2×C4×C67—42no—
213C2×C4×C67—42no—
214C2×C4×C67—42no—
215C2×C4×C67C2×C22⋊C412yes—
216C2×C4×C67C2×C22⋊C412yes—
217C2×C4×C67C2×C22⋊C412yes—
218C2×C4×C67C22×D412yes—
219C67⋊C8—48no—
220C22×C67—28no—
221C22×C67—21no—
222C22×C67—21no—
223C22×C67—22no—
224C22×C67—21no—
225C22×C67—21no—
226C22×C67—21no—
227C22×C67—21no—
228C22×C67—22no—
229C22×C67—22no—
230C22×C67—22no—
231C22×C67—22no—
232C22×C67—44no—
233C22×C67—48no—
234C22×C67—48no—
235C22×C67—44no—
236C22×C67—42no—
237C22×C67—42no—
238C22×C67—42no—
239C22×C67—84no—
240C22×C67—84no—
241C22×C67—84no—
242C22×C67C2×#9912yes—
243C22×C67C2×#9914yes—
244C22×C67#41312yes—
245C22×C67C22×C22⋊C411yescharacteristic
246C22×C67#56812yes—
247C4×C67—22no—
248C4×C67—22no—
249C4×C67—22no—
250C4×C67—22no—
251C4×C67—42no—
252C4×C67—42no—
253C4×C67—42no—
254#17392—671nomaximal, Sylow
255C2×C67—22no—
256C2×C67—22no—
257C2×C67—22no—
258C2×C67—24no—
259C2×C67—42no—
260C2×C67—82no—
261C2×C67#287911yescharacteristic
262C2×C67C22×#9911yescharacteristic
263C2×C67C2×#41311yescharacteristic
264C22×#123D6711yescharacteristic
265C2×#407—672no—
266#1468—678no—
267#1463—672no—
268#1462—672no—
269C67#1739211yescharacteristic, complemented, Sylow
270C23×D4Dic6711yescharacteristic
271C24×C4C2×D6711yescharacteristic
272C22×M16—1342no—
273C2×#123—22no—
274C2×#123—22no—
275C2×#123—22no—
276C2×#123—22no—
277C2×#123Dic6714yes—
278C2×#123C2×D6714yes—
279C22×C4⋊C4Dic6711yescharacteristic
280C22×C4⋊C4C2×D6711yescharacteristic
281C22×C22⋊C4Dic6711yescharacteristic
282C22×C22⋊C4Dic6711yescharacteristic
283C22×C22⋊C4C2×D6711yescharacteristic
284#407—674no—
285#407—674no—
286#407—1342no—
287#407—1342no—
288C2×C22⋊C8—674no—
289C2×C22⋊C8—1342no—
290#324—678no—
291#323—678no—
292C25C2×Dic6711yescharacteristic
293C25C2×Dic6711yescharacteristic
294C22×D4—22no—
295C22×D4—21no—
296C22×D4—21no—
297C22×D4—44no—
298C22×D4C2×Dic6714yes—
299C23×C4—44no—
300C23×C4C2×Dic6711yescharacteristic
301C23×C4C2×Dic6711yescharacteristic
302C23×C4C2×Dic6711yescharacteristic
303C23×C4C2×Dic6711yescharacteristic
304C23×C4C2×Dic6712yes—
305C23×C4C22⋊D6712yes—
306C23×C4C22⋊D6712yes—
307C23×C4C22⋊D6712yes—
308C23×C4C22⋊D6712yes—
309C23×C4C22×D6711yescharacteristic
310C23×C4C22×D6712yes—
311C2×M16—2688no—
312C2×M16—1344no—
313C2×M16—1344no—
314C22×C8—1344no—
315#123—24no—
316#123—24no—
317#123—24no—
318#123—24no—
319#123—24no—
320#123—24no—
321#123—24no—
322#123—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4C2×Dic6714yes—
329C2×C4⋊C4C2×Dic6712yes—
330C2×C4⋊C4C22×D6712yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×Dic6714yes—
340C2×C22⋊C4C2×Dic6714yes—
341C2×C22⋊C4C2×Dic6712yes—
342C2×C22⋊C4C22×D6712yes—
343C22⋊C8—678no—
344C22⋊C8—1344no—
345C22⋊C8—1344no—
346C22⋊C8—1344no—
347C24—24no—
348C24—21no—
349C24—21no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24C22⋊Dic6712yes—
356C24C22⋊Dic6711yescharacteristic
357C24C22⋊Dic6711yescharacteristic
358C24C22×Dic6711yescharacteristic
359C24C22×Dic6712yes—
360C24C22×Dic6712yes—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—48no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—48no—
367C22×C4—28no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4C22⋊Dic6712yes—
385C22×C4C22⋊Dic6712yes—
386C22×C4C22⋊Dic6712yes—
387C22×C4C22⋊Dic6711yescharacteristic
388C22×C4C22⋊Dic6711yescharacteristic
389C22×C4C22⋊Dic6712yes—
390C22×C4C22⋊Dic6712yes—
391C22×C4C22×Dic6712yes—
392C22×C4C22×Dic6712yes—
393C22×C4C22×Dic6712yes—
394C22×C4C22×Dic6712yes—
395C22×C4C22×Dic6711yescharacteristic
396C22×C4C2×C22⋊D6712yes—
397C22×C4C2×C22⋊D6712yes—
398C22×C4C2×C22⋊D6712yes—
399C22×C4C2×C22⋊D6712yes—
400C22×C4C2×C22⋊D6712yes—
401C22×C4C2×C22⋊D6712yes—
402C22×C4C23×D6711yescharacteristic
403M16—2688no—
404M16—2688no—
405C2×C8—2688no—
406C2×C8—1348no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—22no—
426C23—24no—
427C23—41no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—416no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—88no—
443C23#1452395814yes—
444C23C2×C22⋊Dic6712yes—
445C23C2×C22⋊Dic6712yes—
446C23C2×C22⋊Dic6711yescharacteristic
447C23C2×C22⋊Dic6711yescharacteristic
448C23C23×Dic6711yescharacteristic
449D4—48no—
450D4—48no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4C2×C22⋊Dic6712yes—
480C2×C4C2×C22⋊Dic6712yes—
481C2×C4C2×C22⋊Dic6712yes—
482C2×C4C22×C22⋊D6712yes—
483C8—2688no—
484C22—28no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—44no—
497C22—48no—
498C22—48no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—84no—
504C22—84no—
505C22—84no—
506C22C2×#1452395812yes—
507C22C2×#1452395814yes—
508C22#1686338012yes—
509C22C22×C22⋊Dic6711yescharacteristic
510C22#1686416312yes—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C2—22no—
519C2—22no—
520C2—22no—
521C2—24no—
522C2—42no—
523C2—82no—
524C2#2256865411yescharacteristic
525C2C22×#1452395811yescharacteristic
526C2C2×#1686338011yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic67
commutatorC23×C67C23×C4
FrattiniC22×C4C23×D67
FittingC22×#123×C67C2
socleC22×C67C22×C22⋊C4
prime radicalC23×D4×C67C4
cosocleC22×C4×C67C24
hypercenterC22×#123D67
hypocenterC67#17392
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16863360C2semidirect
2. #22558557C2semidirect
3. #22558539C2semidirect
4. #22558536C2semidirect
10. #16863360C22semidirect
12. C2×C22⋊(C67⋊C8)C22semidirect
13. #16863122C22semidirect
14. #16863120C22semidirect
269. C67#17392semidirect

Maximal subgroups

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

subgroupindex
C2×#168633602
#225585572
#225585392
#225585362
C22×#123×C672
#1739267