Finite Group Databasebeta

← #197886

Subgroups — #197886

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic
3#21864C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1554C211yesmaximal, characteristic
6#9228C212yesmaximal, complemented
7#9224C212yesmaximal, complemented
8#9220C212yesmaximal, complemented
9#9216C212yesmaximal, complemented
10#8042C211yesmaximal, characteristic
11#7880C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3078—24no—
15#3078C418yescomplemented
16C2×#479—21no—
17C2×#479—21no—
18C2×#477C2211yescharacteristic
19C2×#477C2211yescharacteristic
20C2×#475C2211yescharacteristic
21C2×#475C2211yescharacteristic
22#2984—24no—
23#2984—24no—
24C2×#446—22no—
25C2×#446C2212yes—
26C2×C8×Q8C2212yes—
27C2×#432—21no—
28C2×#432C2211yescharacteristic
29C2×#431—21no—
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#428—21no—
33C2×#423C2211yescharacteristic
34C2×#419—21no—
35#2285C2211yescharacteristic
36#1990C2212yes—
37#1956C2212yes—
38#1873—21no—
39#1873—21no—
40#1824—21no—
41#1805—22no—
42#1803C2212yes—
43#1803C2212yes—
44#1787—21no—
45C4×#103—21no—
46C4×#103—21no—
47#1554—24no—
48C2×#333C2212yes—
49C2×#332C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1277C2212yes—
53#1277C2212yes—
54C23×Q8D411yescharacteristic
55C22×M16—21no—
56#533—22no—
57#533—22no—
58#533—22no—
59#533C2×C414yes—
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8C2×C414yes—
72C2×C4×Q8D414yes—
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×#117—21no—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D411yescharacteristic
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#479—28no—
84#479—28no—
85#477—22no—
86#477—22no—
87#477C2×C414yes—
88#477C2×C414yes—
89#475—22no—
90#475—22no—
91#475C2×C414yes—
92#475C2×C414yes—
93#446—216no—
94#446—24no—
95#446—24no—
96C8×Q8—24no—
97C8×Q8—24no—
98#432—28no—
99#432—22no—
100#432D414yes—
101#431—28no—
102#431—22no—
103#431—22no—
104#431—22no—
105#431—22no—
106#428—28no—
107C2×#107—21no—
108C2×#107—21no—
109C2×#106—21no—
110C2×#106—21no—
111#423—22no—
112#423C2×C414yes—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8D412yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2312yes—
121#419—28no—
122C2×#103—21no—
123C2×#103—21no—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—22no—
127C2×#103—22no—
128C2×#103—22no—
129C2×#103—22no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8D411yescharacteristic
133C2×C22⋊C8D411yescharacteristic
134C2×#97—22no—
135C2×#97—21no—
136C2×C4×C8—22no—
137C2×C4×C8—21no—
138C2×C4×C8C2312yes—
139#384C2312yes—
140#382—21no—
141#380—21no—
142#380—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#333—24no—
150#332—24no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—42no—
157C22×Q8C22⋊C414yes—
158C22×Q8C2×D411yescharacteristic
159C22×Q8C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4#4111yescharacteristic
165C2×M16—24no—
166C2×M16—44no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22×C412yes—
180C4⋊Q8C22×C412yes—
181#125—24no—
182#125—42no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—24no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8C22⋊C414yes—
202C4×Q8C22⋊C414yes—
203C4×Q8C22×C412yes—
204C4×Q8C22×C412yes—
205C4×Q8C2×D412yes—
206C4×Q8C2×D412yes—
207#117—24no—
208#117—42no—
209#117—82no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C22⋊C414yes—
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42C22×C412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42#4111yescharacteristic
246C2×C42#4112yes—
247C2×C42C2411yescharacteristic
248#107—24no—
249#107—24no—
250#106—24no—
251#106—24no—
252C4⋊C8—28no—
253C4⋊C8—28no—
254C4⋊C8—24no—
255C4⋊C8—24no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8C22×C412yes—
262C4⋊C8C22×C412yes—
263C4⋊C8C2×D414yes—
264#103—24no—
265#103—24no—
266#103—24no—
267#103—24no—
268#103—44no—
269#103—44no—
270#103—82no—
271#103—82no—
272C22⋊C8—24no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—42no—
276#97—28no—
277#97—24no—
278C4×C8—28no—
279C4×C8—22no—
280C4×C8—22no—
281C4×C8—42no—
282#95—41no—
283#95—41no—
284C24#12111yescharacteristic
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—28no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—48no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—82no—
302C2×Q8C2×C22⋊C412yes—
303C2×Q8C2×C22⋊C412yes—
304C2×Q8#13714yes—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4C2×C22⋊C412yes—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4#12012yes—
325C22×C4#12112yes—
326C22×C4#12111yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12312yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333M16—48no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—82no—
374C4⋊C4C2×C22⋊C412yes—
375C4⋊C4C2×C22⋊C412yes—
376C22⋊C4—44no—
377C22⋊C4—82no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—24no—
383C42—24no—
384C42—44no—
385C42—82no—
386C42C2×C22⋊C412yes—
387C42C4×D412yes—
388C42C4×D412yes—
389C42#12012yes—
390C42#12112yes—
391C42#12312yes—
392C42C23×C411yescharacteristic
393C42C22×D411yescharacteristic
394C42C22×D411yescharacteristic
395C42C2×#4111yescharacteristic
396C23—21no—
397C23—21no—
398C23—22no—
399C23—22no—
400C23#38612yes—
401C23C2×#12111yescharacteristic
402Q8—22no—
403Q8—24no—
404Q8—22no—
405Q8—22no—
406Q8—48no—
407Q8—42no—
408Q8—44no—
409Q8—82no—
410Q8—82no—
411Q8—82no—
412Q8#41914yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—48no—
431C2×C4—48no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4#38612yes—
444C2×C4#38612yes—
445C2×C4#38612yes—
446C2×C4C22×C22⋊C411yescharacteristic
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×C4×D411yescharacteristic
449C2×C4C2×#12011yescharacteristic
450C2×C4C2×#12111yescharacteristic
451C2×C4C2×#12311yescharacteristic
452C2×C4C2×#13712yes—
453C2×C4#57512yes—
454C2×C4#57712yes—
455C2×C4#57812yes—
456C8—42no—
457C8—42no—
458C8—42no—
459C8—42no—
460C8—44no—
461C8—82no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—42no—
466C22—42no—
467C22C2×#38611yescharacteristic
468C22#304311yescharacteristic
469C22#304411yescharacteristic
470C22#305612yes—
471C22#306712yes—
472C4—21no—
473C4—21no—
474C4—22no—
475C4—22no—
476C4—22no—
477C4—24no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—82no—
482C4—82no—
483C4—162no—
484C4C2×#41911yescharacteristic
485C4#289611yescharacteristic
486C4#294712yes—
487C4#301312yes—
488C4#305612yes—
489C2—42no—
490C2#1947212yes—
491C2#1947812yes—
492C2#2093011yescharacteristic
493C2#2097211yescharacteristic
494C2#2100311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9228C2semidirect
7. #9224C2semidirect
8. #9220C2semidirect
9. #9216C2semidirect
15. #3078C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#29842
#218642
#207342
C2×#15542
#92282
#92242
#92202
#92162
#80422
#78802