Finite Group Databasebeta

← #198396

Subgroups — #198396

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4#21332C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic
6#9258C218yesmaximal
7#7981C211yesmaximal, characteristic
8#7924C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3209—22no—
12#3209—22no—
13#3209—22no—
14#3209—22no—
15C2×#494C412yescomplemented
16C2×#487C412yescomplemented
17C2×#462C414yescomplemented
18C2×#451C2214yes—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2369—22no—
22#2365—22no—
23#2345—22no—
24Q8×C22⋊C4C2214yes—
25#2331—22no—
26#2316—21no—
27C4×C4⋊Q8—22no—
28#2285C2211yescharacteristic
29#2285C2211yescharacteristic
30#2274—21no—
31C4×#117C2211yescharacteristic
32#1996—22no—
33#1996C2214yes—
34#1981—22no—
35#1956—22no—
36#1787C2214yes—
37#1740—21no—
38#1740—21no—
39#1520C2214yes—
40#1518—22no—
41#1503—22no—
42#1289C2214yes—
43#1282C2214yes—
44C23×Q8C2312yes—
45C22×Q16C2×C418yes—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8D412yes—
66C2×C4⋊Q8D412yes—
67C2×C4⋊Q8D412yes—
68C2×#125—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8C2312yes—
71C2×C4×Q8—24no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2×C412yes—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C42C2311yescharacteristic
82#494—24no—
83#494—24no—
84#487—24no—
85#487—24no—
86#462—28no—
87#462—28no—
88#451—28no—
89#451—28no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2×C412yes—
100C2×#103—28no—
101C2×#103C2×C418yes—
102C2×C22⋊C8C2312yes—
103C2×C22⋊C8C2312yes—
104C2×C4×C8—21no—
105C2×C4×C8C2×C412yes—
106#391—24no—
107#391—28no—
108#389—24no—
109#389—24no—
110#389—24no—
111#387—28no—
112#384—21no—
113#384—21no—
114#382—21no—
115#382D412yes—
116#380—21no—
117#380—24no—
118#380—21no—
119#380C2312yes—
120#380C2312yes—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D412yes—
130C4×C22⋊C4—24no—
131C4×C22⋊C4D412yes—
132C4×C22⋊C4C2312yes—
133#376—21no—
134C43—21no—
135#342—24no—
136#342—24no—
137#328—28no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—84no—
143C22×Q8C22×C412yes—
144C22×Q8C22×C414yes—
145C22×Q8C2×D414yes—
146C22×Q8C2×D414yes—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C2×Q16—216no—
152C2×Q16—88no—
153C22×C8—22no—
154C22×C8C22×C414yes—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8C22⋊C414yes—
176C4⋊Q8C22⋊C414yes—
177C4⋊Q8C22⋊C414yes—
178C4⋊Q8C22⋊C414yes—
179#125—41no—
180#125—41no—
181#125—41no—
182#125—41no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—88no—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211#117Q1614yescomplemented
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4C22×C412yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4C2×D412yes—
243C2×C22⋊C4C2×D412yes—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—28no—
251C2×C42—41no—
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—41no—
267C4⋊C8—41no—
268#103—216no—
269#103—88no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—41no—
277C4×C8—41no—
278#95—28no—
279#95—28no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—41no—
285#95C2×D414yes—
286C24C22×D411yescharacteristic
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—44no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—82no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8C2×C22⋊C418yes—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—84no—
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4C2×Q1612yes—
328C22×C4#13512yes—
329C22×C4#13712yes—
330C22×C4#13712yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337Q16—816no—
338C2×C8—24no—
339C2×C8—44no—
340C2×C8—44no—
341C2×C8—42no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—82no—
369C4⋊C4—88no—
370C4⋊C4—82no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4C2×Q1614yes—
381C42—24no—
382C42—44no—
383C42—44no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—44no—
388C42—88no—
389C42—82no—
390C42C2×C22⋊C411yescharacteristic
391C42C2×C22⋊C411yescharacteristic
392C42C2×C22⋊C411yescharacteristic
393C42C2×C22⋊C411yescharacteristic
394C42C4×D411yescharacteristic
395C42C4×D411yescharacteristic
396C42C4×D411yescharacteristic
397C42C4×D411yescharacteristic
398C42#12111yescharacteristic
399C42#12111yescharacteristic
400C42#12111yescharacteristic
401C42#12111yescharacteristic
402C42#12311yescharacteristic
403C42#12311yescharacteristic
404C42#12311yescharacteristic
405C42#12311yescharacteristic
406C23—41no—
407C23—41no—
408C23C2×C4×D411yescharacteristic
409C23C2×#12011yescharacteristic
410C23C2×#12111yescharacteristic
411C23C22×Q1611yescharacteristic
412C23C2×#13511yescharacteristic
413C23C2×#13711yescharacteristic
414C23C2×#13711yescharacteristic
415Q8—216no—
416Q8—48no—
417Q8—44no—
418Q8—84no—
419Q8—84no—
420Q8—82no—
421Q8—82no—
422Q8—82no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4#38611yescharacteristic
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4#38611yescharacteristic
462C2×C4#38611yescharacteristic
463C2×C4C4×Q1612yes—
464C2×C4C4×Q1612yes—
465C2×C4#44112yes—
466C2×C4#44112yes—
467C2×C4#45112yes—
468C2×C4#45112yes—
469C2×C4#45212yes—
470C2×C4#45212yes—
471C2×C4#46212yes—
472C2×C4#46212yes—
473C2×C4#46412yes—
474C2×C4#46412yes—
475C2×C4C22×C22⋊C411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C2×#12111yescharacteristic
478C2×C4C2×#12311yescharacteristic
479C8—44no—
480C8—82no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22C2×#38611yescharacteristic
492C22C2×C4×Q1611yescharacteristic
493C22C2×#44111yescharacteristic
494C22C2×#45111yescharacteristic
495C22C2×#45211yescharacteristic
496C22C2×#46211yescharacteristic
497C22C2×#46411yescharacteristic
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—44no—
505C4—48no—
506C4—82no—
507C4—84no—
508C4—82no—
509C4—82no—
510C4—162no—
511C4#187812yes—
512C4#187812yes—
513C4#187812yes—
514C4#187812yes—
515C4#190512yes—
516C4#288312yes—
517C2—21no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—21no—
522C2—21no—
523C2C2×#187811yescharacteristic
524C2#2098511yescharacteristic
525C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C2×#494C4semidirect
16. C2×#487C4semidirect
17. C2×#462C4semidirect
211. #117Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#342162
#218042
#213322
#175752
#92582
#79812
#79242