Finite Group Databasebeta

← #557×#64408 #34030696

Subgroups — #557×#64408 #34030696

Order 18688.

2220 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626465C216yesmaximal
2#22626460C213yesmaximal
3#22626450C213yesmaximal
4#22626446C212yesmaximal
5Q8×C4⋊#64408C211yesmaximal, characteristic
6C4⋊Q8×#64408C213yesmaximal
7#125×#64408C216yesmaximal
8C4×Q8×#64408C213yesmaximal
9C4×Q8×#64408C212yesmaximal
10C4×Q8×#64408C211yesmaximal, characteristic
11#557×D73C211yesmaximal, characteristic
12C2×Q8×#64408C2213yes—
13C2×Q8×#64408C2212yes—
14C2×Q8×#64408C2211yescharacteristic
15C2×Q8×#64408C2213yes—
16C2×Q8×#64408C2211yescharacteristic
17(C2×#64408)⋊Q8C2216yes—
18#17989538C2216yes—
19#17989538C2216yes—
20#17989538C2216yes—
21#17989538C2216yes—
22#17989536C22112yes—
23#17989536C22112yes—
24#17989533C2213yes—
25#17989531C2216yes—
26C4⋊C4×#64408C2211yescharacteristic
27C4⋊C4×#64408C2216yes—
28C4⋊C4×#64408C2213yes—
29C4⋊C4×#64408C2216yes—
30C4⋊C4×#64408C2213yes—
31C4⋊C4×#64408C2216yes—
32C4⋊C4×#64408C2216yes—
33C4⋊C4×#64408C2213yes—
34C4⋊C4×#64408C2216yes—
35C4×C4⋊#64408C2213yes—
36C4×C4⋊#64408C2216yes—
37C4×C4⋊#64408C2213yes—
38C4×C4⋊#64408C2216yes—
39C42×#64408C2216yes—
40C42×#64408C2213yes—
41#17989316C411yescharacteristic, complemented
42#17989315C413yescomplemented
43#17989314C416yescomplemented
44#17989313C412yescomplemented
45#17989312C413yescomplemented
46C4⋊Q8×D73C2213yes—
47#125×D73C2216yes—
48C4×Q8×D73C2211yescharacteristic
49C4×Q8×D73C2212yes—
50C4×Q8×D73C2213yes—
51#557×C73C411yescharacteristic, complemented
52Q8×#64408—212no—
53Q8×#64408—212no—
54Q8×#64408—28no—
55Q8×#64408—24no—
56Q8×#64408Q818yescomplemented
57C2×C4⋊#64408C2313yes—
58C2×C4⋊#64408C2316yes—
59C2×C4⋊#64408C2313yes—
60C2×C4⋊#64408C2316yes—
61C2×C4⋊#64408C2316yes—
62C2×C4⋊#64408C2316yes—
63C2×C4⋊#64408C2313yes—
64C2×C4⋊#64408C2311yescharacteristic
65C2×C4×#64408C2313yes—
66C2×C4×#64408C2316yes—
67C2×C4×#64408C2313yes—
68C2×C4×#64408C2316yes—
69C2×C4×#64408C2316yes—
70C2×C4×#64408C2312yes—
71C2×C4×#64408C2313yes—
72C2×C4×#64408C2316yes—
73C2×C4×#64408C2313yes—
74C2×C4×#64408C2313yes—
75C2×C4×#64408C2311yescharacteristic
76(C2×#64408)⋊C4C2312yes—
77(C2×#64408)⋊C4C2316yes—
78(C2×#64408)⋊C4C2316yes—
79(C2×#64408)⋊C4C2316yes—
80(C2×#64408)⋊C4C23112yes—
81(C2×#64408)⋊C4C23112yes—
82C2×Q8×D73C2313yes—
83C2×Q8×D73C2311yescharacteristic
84#14556591C2×C413yes—
85#14556590C2×C416yes—
86#14556590C2×C416yes—
87#14556589C2×C416yes—
88#14556583C2×C416yes—
89#14556582C2×C416yes—
90#14556582C2×C416yes—
91#14556581C2×C4112yes—
92#14556581C2×C4112yes—
93Q8×Dic73C2×C413yes—
94Q8×Dic73C2×C411yescharacteristic
95Q8×Dic73C2×C412yes—
96C4×Dic146C2×C413yes—
97C4×Dic146C2×C413yes—
98C4×Dic146C2×C416yes—
99C4×Dic146C2×C416yes—
100#14556556C2×C416yes—
101#14556556C2×C416yes—
102#14556556C2×C413yes—
103#14556556C2×C413yes—
104C4⋊C4×D73C2316yes—
105C4⋊C4×D73C2316yes—
106C4⋊C4×D73C2311yescharacteristic
107C4⋊C4×D73C2316yes—
108C4⋊C4×D73C2313yes—
109C42×D73C2313yes—
110C42×D73C2316yes—
111C4⋊Q8×C73C2×C413yes—
112#125×C73C2×C416yes—
113C4×Q8×C73C2×C412yes—
114C4×Q8×C73C2×C413yes—
115C4×Q8×C73C2×C411yescharacteristic
116C22×#64408C2413yes—
117C22×#64408C2416yes—
118C22×#64408C2413yes—
119C22×#64408C2412yes—
120C22×#64408C2411yescharacteristic
121C22×#64408C2411yescharacteristic
122C4⋊#64408—212no—
123C4⋊#64408—212no—
124C4⋊#64408—26no—
125C4⋊#64408—212no—
126C4⋊#64408—26no—
127C4⋊#64408—46no—
128C4⋊#64408C2×Q8112yes—
129C4⋊#64408#4114yes—
130C4×#64408—212no—
131C4×#64408—212no—
132C4×#64408—24no—
133C4×#64408—26no—
134C4×#64408—26no—
135C4×#64408—212no—
136C4×#64408—26no—
137C4×#64408—46no—
138C4×#64408—43no—
139C4×#64408C2×Q8112yes—
140C4×#64408#4114yes—
141C2×Dic146C22×C413yes—
142C2×Dic146C22×C416yes—
143C2×Dic146C22×C413yes—
144Q8×D73—26no—
145Q8×D73—26no—
146Q8×D73C2×Q814yes—
147Q8×D73#4114yes—
148C2×C4×D73C2412yes—
149C2×C4×D73C2413yes—
150C2×C4×D73C2416yes—
151C2×C4×D73C2413yes—
152C2×C4×D73C2411yescharacteristic
153C4⋊Dic73C22×C411yescharacteristic
154C4⋊Dic73C22×C416yes—
155C4⋊Dic73C22×C413yes—
156C4⋊Dic73C22×C416yes—
157C4⋊Dic73C22×C416yes—
158#11924007C22×C412yes—
159#11924007C22×C416yes—
160#11924007C22×C416yes—
161#11924007C22×C416yes—
162#11924007C22×C4112yes—
163#11924007C22×C4112yes—
164C4×Dic73C22×C413yes—
165C4×Dic73C22×C416yes—
166C4×Dic73C22×C416yes—
167C4×Dic73C22×C416yes—
168C4×Dic73C22×C413yes—
169C4×Dic73C22×C413yes—
170C2×Q8×C73C22×C413yes—
171C2×Q8×C73C22×C411yescharacteristic
172C4⋊C4×C73C22×C411yescharacteristic
173C4⋊C4×C73C22×C416yes—
174C4⋊C4×C73C22×C413yes—
175C4⋊C4×C73C22×C416yes—
176C4⋊C4×C73C22×C416yes—
177C42×C73C22×C416yes—
178C42×C73C22×C413yes—
179C22×D73C2511yescharacteristic
180C2×#64408—23no—
181C2×#64408—21no—
182C2×#64408—22no—
183C2×#64408—22no—
184C2×#64408—23no—
185C2×#64408—21no—
186C2×#64408—26no—
187C2×#64408—26no—
188C2×#64408—23no—
189C2×#64408—23no—
190C2×#64408—23no—
191C2×#64408—26no—
192C2×#64408C22×Q816yes—
193C2×#64408C22×Q812yes—
194C2×#64408C2×#4114yes—
195C2×#64408C2×#4112yes—
196C2×#64408C2×#4112yes—
197C2×#644082−1+412yes—
198Q8×C73—26no—
199Q8×C73C4×Q814yescomplemented
200C2×C4×C73C23×C412yes—
201C2×C4×C73C23×C413yes—
202C2×C4×C73C23×C416yes—
203C2×C4×C73C23×C413yes—
204C2×C4×C73C23×C411yescharacteristic
205C2×Dic73C23×C412yes—
206C2×Dic73C23×C413yes—
207C2×Dic73C23×C416yes—
208C2×Dic73C23×C413yes—
209C2×Dic73C23×C411yescharacteristic
210C4×D73—26no—
211C4×D73—26no—
212C4×D73—22no—
213C4×D73—23no—
214C4×D73—22no—
215C4×D73—23no—
216C4×D73C22×Q816yes—
217C4×D73C2×#4116yes—
218C4×D73C2×#4112yes—
219C4×D73C2×#4112yes—
220Dic146—26no—
221Dic146—212no—
222Dic146C4×Q8112yescomplemented
223C22×C73C24×C411yescharacteristic
224C2×D73C23×Q811yescharacteristic
225C2×D73C22×#4111yescharacteristic
226C2×D73C22×#4111yescharacteristic
227C2×D73C22×#4111yescharacteristic
228C2×D73C2×2−1+411yescharacteristic
229C2×D73#58511yescharacteristic
230#64408—26no—
231#64408—22no—
232#64408—24no—
233#64408—46no—
234#64408—43no—
235#64408#55714yesdirect product factor
236C4×C73—26no—
237C4×C73—22no—
238C4×C73—23no—
239C4×C73C2×C4×Q816yes—
240C4×C73C4×#4112yes—
241Dic73—26no—
242Dic73—22no—
243Dic73—23no—
244Dic73C2×C4×Q816yes—
245Dic73C4×#4112yes—
246C4×#557—731nomaximal, Sylow
247C2×C73C22×C4×Q811yescharacteristic
248C2×C73C2×C4×#4111yescharacteristic
249C2×C73C4×2−1+411yescharacteristic
250D73C2×#55711yescharacteristic
251D73Q8×#4111yescharacteristic
252D73#348011yescharacteristic
253D73#348211yescharacteristic
254C2×#557—731no—
255#2370—736no—
256#2365—733no—
257#2355—733no—
258#2351—732no—
259Q8×C4⋊C4—731no—
260C4×C4⋊Q8—733no—
261C4×#125—736no—
262C42×Q8—731no—
263C42×Q8—732no—
264C42×Q8—733no—
265C73C4×#55711yescharacteristic, complemented, Sylow
266#557—733no—
267#557—736no—
268#557—732no—
269#557—733no—
270#557—731no—
271#557#6440811yescharacteristic, direct product factor
272C2×C4⋊Q8—733no—
273C2×#125—736no—
274C2×C4×Q8—731no—
275C2×C4×Q8—731no—
276C2×C4×Q8—732no—
277C2×C4×Q8—733no—
278C2×C4×Q8—733no—
279C2×C4×Q8—732no—
280C2×C4×Q8—731no—
281C2×C4×Q8—733no—
282#391—736no—
283#389—736no—
284#389—736no—
285#389—736no—
286#389—736no—
287#387—7312no—
288#387—7312no—
289#384—733no—
290#382—736no—
291C4×C4⋊C4—736no—
292C4×C4⋊C4—736no—
293C4×C4⋊C4—733no—
294C4×C4⋊C4—736no—
295C4×C4⋊C4—733no—
296C4×C4⋊C4—736no—
297C4×C4⋊C4—733no—
298C4×C4⋊C4—733no—
299C4×C4⋊C4—736no—
300C4×C4⋊C4—736no—
301C4×C4⋊C4—736no—
302C4×C4⋊C4—733no—
303C4×C4⋊C4—731no—
304C43—736no—
305C43—733no—
306C22×Q8—731no—
307C22×Q8—733no—
308C4⋊Q8—736no—
309C4⋊Q8—736no—
310C4⋊Q8—736no—
311C4⋊Q8—733no—
312C4⋊Q8C2×#6440813yes—
313#125—7312no—
314#125—7312no—
315#125—736no—
316#125—736no—
317#125—736no—
318#125C2×#6440816yes—
319C4×Q8—738no—
320C4×Q8—732no—
321C4×Q8—733no—
322C4×Q8—731no—
323C4×Q8—736no—
324C4×Q8—736no—
325C4×Q8—733no—
326C4×Q8—733no—
327C4×Q8—736no—
328C4×Q8—733no—
329C4×Q8—736no—
330C4×Q8—733no—
331C4×Q8—14612no—
332C4×Q8—14612no—
333C4×Q8—1468no—
334C4×Q8—1464no—
335C4×Q8C2×#6440812yes—
336C4×Q8C2×#6440813yes—
337C4×Q8C2×#6440811yescharacteristic
338C2×C4⋊C4—736no—
339C2×C4⋊C4—731no—
340C2×C4⋊C4—733no—
341C2×C4⋊C4—731no—
342C2×C4⋊C4—736no—
343C2×C4⋊C4—733no—
344C2×C4⋊C4—736no—
345C2×C4⋊C4—736no—
346C2×C4⋊C4—733no—
347C2×C4⋊C4—736no—
348C2×C4⋊C4—733no—
349C2×C4⋊C4—736no—
350C2×C4⋊C4—736no—
351C2×C42—736no—
352C2×C42—733no—
353C2×C42—731no—
354C2×C42—733no—
355C2×C42—736no—
356C2×C42—733no—
357C2×C42—736no—
358C2×C42—736no—
359C2×C42—736no—
360C2×C42—733no—
361C2×C42—733no—
362C2×C42—732no—
363C2×C42—733no—
364#95—732no—
365#95—736no—
366#95—736no—
367#95—736no—
368#95—7312no—
369#95—7312no—
370C2×Q8—734no—
371C2×Q8—734no—
372C2×Q8—733no—
373C2×Q8—736no—
374C2×Q8—733no—
375C2×Q8—1466no—
376C2×Q8—1466no—
377C2×Q8C22×#6440813yes—
378C2×Q8C22×#6440811yescharacteristic
379C22×C4—731no—
380C22×C4—733no—
381C22×C4—731no—
382C22×C4—733no—
383C22×C4—732no—
384C22×C4—731no—
385C22×C4—732no—
386C22×C4—733no—
387C22×C4—736no—
388C22×C4—733no—
389C22×C4—736no—
390C4⋊C4—2926no—
391C4⋊C4—7312no—
392C4⋊C4—734no—
393C4⋊C4—732no—
394C4⋊C4—736no—
395C4⋊C4—736no—
396C4⋊C4—736no—
397C4⋊C4—7312no—
398C4⋊C4—7312no—
399C4⋊C4—731no—
400C4⋊C4—736no—
401C4⋊C4—733no—
402C4⋊C4—736no—
403C4⋊C4—736no—
404C4⋊C4—1466no—
405C4⋊C4—14612no—
406C4⋊C4—1466no—
407C4⋊C4—14612no—
408C4⋊C4—14612no—
409C4⋊C4C22×#6440811yescharacteristic
410C4⋊C4C22×#6440816yes—
411C4⋊C4C22×#6440813yes—
412C4⋊C4C22×#6440816yes—
413C4⋊C4C22×#6440816yes—
414C42—2926no—
415C42—2923no—
416C42—7312no—
417C42—734no—
418C42—736no—
419C42—736no—
420C42—736no—
421C42—733no—
422C42—733no—
423C42—733no—
424C42—1466no—
425C42—1464no—
426C42—14612no—
427C42—1466no—
428C42—14612no—
429C42—1466no—
430C42—14612no—
431C42C22×#6440816yes—
432C42C22×#6440813yes—
433C23—731no—
434Q8—26no—
435Q8—7312no—
436Q8—14612no—
437Q8—1466no—
438Q8Q8×#6440814yescomplemented
439C2×C4—732no—
440C2×C4—732no—
441C2×C4—732no—
442C2×C4—733no—
443C2×C4—731no—
444C2×C4—736no—
445C2×C4—736no—
446C2×C4—732no—
447C2×C4—732no—
448C2×C4—736no—
449C2×C4—732no—
450C2×C4—734no—
451C2×C4—732no—
452C2×C4—733no—
453C2×C4—736no—
454C2×C4—1462no—
455C2×C4—1463no—
456C2×C4—1466no—
457C2×C4—1466no—
458C2×C4—1462no—
459C2×C4—1463no—
460C2×C4—1466no—
461C2×C4—1463no—
462C2×C4—1466no—
463C2×C4—1463no—
464C2×C4—1466no—
465C2×C4—1463no—
466C2×C4—1463no—
467C2×C4—1461no—
468C2×C4—1462no—
469C2×C4—1461no—
470C2×C4—1463no—
471C2×C4—1462no—
472C2×C4C23×#6440812yes—
473C2×C4C23×#6440813yes—
474C2×C4C23×#6440816yes—
475C2×C4C23×#6440813yes—
476C2×C4C23×#6440811yescharacteristic
477C22—731no—
478C22—731no—
479C22—731no—
480C22—731no—
481C22—731no—
482C22—731no—
483C22C24×#6440811yescharacteristic
484C4—26no—
485C4—22no—
486C4—23no—
487C4—2926no—
488C4—2923no—
489C4—734no—
490C4—736no—
491C4—732no—
492C4—1462no—
493C4—1464no—
494C4—1466no—
495C4—1466no—
496C4—1462no—
497C4—1463no—
498C4C2×Q8×#6440816yes—
499C4#41×#6440812yes—
500C2—731no—
501C2—731no—
502C2—731no—
503C2—731no—
504C2C22×Q8×#6440811yescharacteristic
505C2C2×#41×#6440811yescharacteristic
506C22−1+4×#6440811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
Fitting#557×C73C4
prime radical, cosocleC22×D73C25
hypercenter#557#64408
hypocenterC73C4×#557
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
41. #17989316C4semidirect
42. #17989315C4semidirect
43. #17989314C4semidirect
44. #17989313C4semidirect
45. #17989312C4semidirect
51. #557×C73C4semidirect
56. Q8×#64408Q8semidirect
199. Q8×C73C4×Q8semidirect
222. Dic146C4×Q8semidirect
235. #64408#557direct (N × H)
265. C73C4×#557semidirect
271. #557#64408direct (N × H)
438. Q8Q8×#64408semidirect

Maximal subgroups

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

subgroupindex
#226264652
#226264602
#226264502
#226264462
Q8×C4⋊#644082
C4⋊Q8×#644082
#125×#644082
C4×Q8×#644082
#557×D732
C4×#55773