Finite Group Databasebeta

← #189979

Subgroups — #189979

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2#21851C212yesmaximal
3#8729C212yesmaximal, complemented
4#8728C211yesmaximal, characteristic, complemented
5#8728C211yesmaximal, characteristic, complemented
6#8725C211yesmaximal, characteristic, complemented
7#8724C212yesmaximal, complemented
8#8723C211yesmaximal, characteristic, complemented
9#8044C212yesmaximal, complemented
10#8040C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#520C412yes—
13C2×C4×#41C2211yescharacteristic
14C2×#432C2212yes—
15C22×C4⋊C8C2212yes—
16Q8×C4⋊C4—22no—
17D4×C4⋊C4—22no—
18D4×C4⋊C4—22no—
19D4×C4⋊C4—22no—
20#2317C412yes—
21#2317C412yes—
22#2317C412yes—
23#2285C2211yescharacteristic
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#1985—22no—
30#1955—22no—
31#1869—22no—
32#1869—22no—
33#1797C2212yes—
34#1797C2212yes—
35#1797C2212yes—
36#1797C2212yes—
37#1296C2212yes—
38#1296C2211yescharacteristic
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47C22×#41C2×C411yescharacteristic
48C23×C8—22no—
49#520—48no—
50C4×#41—22no—
51C4×#41—24no—
52C4×#41—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8D412yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4D412yes—
59C2×C4×D4D412yes—
60C2×C4×D4D412yes—
61C2×#117C2×C412yes—
62C2×#117C2×C412yes—
63C2×#117C2×C412yes—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#432—24no—
77#432—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×#97—22no—
91#389—24no—
92#389—24no—
93#389—24no—
94#385—24no—
95#385—24no—
96#385—24no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384C2×C412yes—
101#384C2×C412yes—
102#384C2×C412yes—
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#382C2×C412yes—
108#382C2×C412yes—
109#382C2×C412yes—
110#380—22no—
111#380—22no—
112#380—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120C2×#41—24no—
121C2×#41—24no—
122C2×#41—22no—
123C2×#41—22no—
124C22×Q8C22⋊C411yescharacteristic
125C22×D4C22⋊C411yescharacteristic
126C22×D4C22⋊C411yescharacteristic
127C22×D4C22⋊C411yescharacteristic
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—44no—
142C22×C8—82no—
143C22×C8—82no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—48no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—48no—
154C4×D4—48no—
155C4×D4—48no—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—44no—
163#117—44no—
164#117—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42#4112yes—
213C2×C42#4114yes—
214C2×C42#4112yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—44no—
225C22⋊C8—44no—
226C22⋊C8—82no—
227C22⋊C8—82no—
228C22⋊C8—82no—
229C22⋊C8—82no—
230#97—44no—
231#95—24no—
232#95—24no—
233#95—24no—
234C24C2×C22⋊C411yescharacteristic
235C24C2×C22⋊C411yescharacteristic
236C24C2×C22⋊C411yescharacteristic
237#41—216no—
238#41—216no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4#10014yes—
250C2×D4#10114yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—84no—
285C22×C4—84no—
286C22×C4—84no—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4#11712yes—
294C22×C4#11712yes—
295C22×C4#12314yes—
296C22×C4#12314yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D412yes—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—88no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—84no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—44no—
348C42—44no—
349C42—44no—
350C42#11712yes—
351C42#11712yes—
352C42#12314yes—
353C42#12314yes—
354C42C2×#4111yescharacteristic
355C42C2×#4112yes—
356C42C2×#4111yescharacteristic
357C23—22no—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—21no—
366C23—21no—
367C23#38112yes—
368C23C2×#10012yes—
369C23C2×#10012yes—
370C23C2×#10112yes—
371C23C2×#10112yes—
372C23C22×C22⋊C411yescharacteristic
373Q8—28no—
374D4—28no—
375D4—28no—
376D4—28no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—164no—
420C2×C4—164no—
421C2×C4—164no—
422C2×C4#38114yes—
423C2×C4#38112yes—
424C2×C4C2×#10012yes—
425C2×C4C2×#10112yes—
426C2×C4C2×#11711yescharacteristic
427C2×C4C2×#11711yescharacteristic
428C2×C4C2×#12312yes—
429C2×C4C2×#12312yes—
430C2×C4#57612yes—
431C2×C4#57712yes—
432C2×C4#57712yes—
433C2×C4#57812yes—
434C8—82no—
435C8—82no—
436C8—82no—
437C8—82no—
438C22—22no—
439C22—24no—
440C22—22no—
441C22—24no—
442C22—22no—
443C22—24no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—42no—
449C22—41no—
450C22—42no—
451C22—42no—
452C22#182114yes—
453C22C2×#38111yescharacteristic
454C22C22×#10011yescharacteristic
455C22C22×#10111yescharacteristic
456C22#289712yes—
457C22#289812yes—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—24no—
464C4—42no—
465C4—42no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4—82no—
470C4#309612yes—
471C4#309712yes—
472C4#309812yes—
473C4#309912yes—
474C2—24no—
475C2—42no—
476C2—42no—
477C2#1777011yescharacteristic
478C2#1777412yes—
479C2#1777911yescharacteristic
480C2C2×#182111yescharacteristic
481C2#2070312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8729C2semidirect
4. #8728C2semidirect
5. #8728C2semidirect
6. #8725C2semidirect
7. #8724C2semidirect
8. #8723C2semidirect
9. #8044C2semidirect
10. #8040C2semidirect
11. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#341832
#218512
#87292
#87282
#87252
#87242
#87232
#80442
#80402
C2×#12962