Finite Group Databasebeta

← #238184

Subgroups — #238184

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#21259C211yesmaximal, characteristic
5#10356C214yesmaximal, complemented
6#10350C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7897C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#419—21no—
15C2×#419—21no—
16C22×C4×C8C2211yescharacteristic
17#2868—21no—
18#2859—21no—
19#2660—22no—
20#2552—21no—
21#2552C2212yes—
22#2551—21no—
23#2551C2212yes—
24#2440—22no—
25#2440—22no—
26#2317—21no—
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1990—22no—
31#1990—21no—
32#1990—21no—
33#1990C2214yes—
34#1988C2214yes—
35#1981—21no—
36#1981—21no—
37#1981C2212yes—
38#1981C2212yes—
39#1823—22no—
40C2×#340—22no—
41#1296C2214yes—
42#1279C2212yes—
43#1279C2214yes—
44#1279C2212yes—
45C23×Q8—21no—
46C22×M16—21no—
47C23×C8—21no—
48#544—44no—
49#544—44no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8Q814yescomplemented
61C2×C4⋊Q8C2312yes—
62C2×#125—22no—
63C2×#125—22no—
64C2×C22⋊Q8—24no—
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C42C2311yescharacteristic
74#431—22no—
75#431—22no—
76#423—22no—
77#423—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81#419—44no—
82#419—44no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×C22⋊C8—21no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2312yes—
93#398—24no—
94#398—24no—
95#395—24no—
96#395—24no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#385—24no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384—21no—
108#384—21no—
109#384D411yescharacteristic
110#384D411yescharacteristic
111#384D412yes—
112#384C2312yes—
113#382—22no—
114#382—21no—
115#380—21no—
116#380D412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2312yes—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125#376—21no—
126C43—21no—
127#340—24no—
128#340—24no—
129#340—28no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—22no—
133C22×Q8—41no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C23×C4—22no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—42no—
145C2×M16—42no—
146C2×M16—42no—
147C22×C8—24no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—42no—
152C22×C8—42no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—48no—
160C4⋊Q8—44no—
161C4⋊Q8—44no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—84no—
166C4⋊Q8—84no—
167C4⋊Q8C2×Q814yes—
168#125—48no—
169#125—44no—
170#125—44no—
171C22⋊Q8—416no—
172#117—22no—
173#117—22no—
174#117—42no—
175#117—42no—
176#117D814yes—
177#117SD1614yes—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D414yes—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×Q812yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—42no—
234#103—44no—
235#103—44no—
236#103—44no—
237#103—44no—
238C22⋊C8—42no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—42no—
242C4×C8—42no—
243#95—24no—
244#95—24no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—22no—
251C24#12011yescharacteristic
252C2×Q8—24no—
253C2×Q8—48no—
254C2×Q8—42no—
255C2×Q8—44no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—48no—
261C2×Q8—48no—
262C2×Q8—84no—
263C2×Q8—84no—
264C2×Q8—82no—
265C2×Q8—82no—
266C2×Q8—84no—
267C2×Q8—84no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4C22⋊Q812yes—
290C22×C4C22⋊Q812yes—
291C22×C4C22⋊Q812yes—
292C22×C4#12412yes—
293C22×C4#12412yes—
294C22×C4#12412yes—
295C22×C4C2×D812yes—
296C22×C4C2×SD1612yes—
297C22×C4AGL(1,ℤ/8)12yes—
298C22×C4#13712yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302M16—84no—
303C2×C8—24no—
304C2×C8—42no—
305C2×C8—48no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—44no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—82no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4C2×D812yes—
342C4⋊C4C2×D812yes—
343C4⋊C4C2×SD1612yes—
344C4⋊C4C2×SD1612yes—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—44no—
348C42—24no—
349C42—24no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42C22⋊Q812yes—
357C42C22⋊Q812yes—
358C42C22⋊Q812yes—
359C42#12412yes—
360C42#12412yes—
361C42#12412yes—
362C42C22×Q811yescharacteristic
363C42C2×#4111yescharacteristic
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23#39312yes—
371C23#44712yes—
372C23#44812yes—
373C23C2×#12011yescharacteristic
374Q8—44no—
375Q8—44no—
376Q8—82no—
377Q8—82no—
378Q8—88no—
379Q8—168no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—88no—
413C2×C4—164no—
414C2×C4#39312yes—
415C2×C4#39312yes—
416C2×C4#39312yes—
417C2×C4#44712yes—
418C2×C4#44812yes—
419C2×C4#45512yes—
420C2×C4#45612yes—
421C2×C4#47414yes—
422C2×C4#47614yes—
423C2×C4#48614yes—
424C2×C4#48914yes—
425C2×C4C2×C22⋊Q811yescharacteristic
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×#12411yescharacteristic
429C2×C4C2×#12411yescharacteristic
430C2×C4C2×#12411yescharacteristic
431C2×C4C22×D811yescharacteristic
432C2×C4C22×SD1611yescharacteristic
433C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57712yes—
436C2×C4#57812yes—
437C8—44no—
438C8—82no—
439C8—82no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22#202314yes—
447C22C2×#39311yescharacteristic
448C22C2×#44711yescharacteristic
449C22C2×#44811yescharacteristic
450C22#300911yescharacteristic
451C22#301111yescharacteristic
452C22C2×#45511yescharacteristic
453C22C2×#45611yescharacteristic
454C4—22no—
455C4—21no—
456C4—21no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—84no—
462C4—84no—
463C4—82no—
464C4—82no—
465C4#202314yes—
466C4#202814yes—
467C4C2×#47411yescharacteristic
468C4C2×#47611yescharacteristic
469C4#308212yes—
470C4C2×#48611yescharacteristic
471C4C2×#48911yescharacteristic
472C4#313712yes—
473C2—24no—
474C2#1863011yescharacteristic
475C2#1863211yescharacteristic
476C2#1865011yescharacteristic
477C2#1865211yescharacteristic
478C2C2×#202311yescharacteristic
479C2#2345711yescharacteristic
480C2C2×#202811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
5. #10356C2semidirect
6. #10350C2semidirect
60. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#218332
C2×#19902
#212592
#103562
#103502
#80432
#78972
#78902