Finite Group Databasebeta

← #238181

Subgroups — #238181

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4C2×#1956C211yesmaximal, characteristic
5#10355C214yesmaximal, complemented
6#10349C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7897C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×#103—22no—
15C22×C4×C8C2211yescharacteristic
16#2868—21no—
17#2859—21no—
18#2660—22no—
19#2552—21no—
20#2552C2212yes—
21#2551—21no—
22#2551C2212yes—
23#2440—22no—
24#2440—22no—
25#2317—21no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1990—21no—
30#1990—21no—
31#1988—21no—
32#1988—21no—
33#1987C2212yes—
34#1987C2212yes—
35#1987C2214yes—
36#1956—22no—
37#1956C2214yes—
38#1824—22no—
39#1736—21no—
40#1736—21no—
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—
81C2×#103—28no—
82C2×#103—48no—
83C2×C22⋊C8—21no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#398—24no—
90#398—24no—
91#395—24no—
92#395—24no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#385—24no—
100#384—22no—
101#384—22no—
102#384—22no—
103#384—21no—
104#384—21no—
105#384D411yescharacteristic
106#384D411yescharacteristic
107#384D412yes—
108#384C2312yes—
109#382—22no—
110#382—21no—
111#380—21no—
112#380D412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4C2311yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2312yes—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121#376—21no—
122C43—21no—
123#340—24no—
124#340—24no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—22no—
132C22×Q8—44no—
133C22×Q8—44no—
134C22×Q8—41no—
135C22×Q8—42no—
136C23×C4—22no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—42no—
142C2×M16—42no—
143C2×M16—42no—
144C22×C8—24no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—42no—
149C22×C8—42no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—48no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—84no—
163C4⋊Q8—84no—
164C4⋊Q8C2×Q814yes—
165#125—48no—
166#125—44no—
167#125—44no—
168C22⋊Q8—416no—
169#117—22no—
170#117—22no—
171#117—42no—
172#117—42no—
173#117SD1614yes—
174#117Q1614yes—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D414yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×Q812yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—42no—
231#103—416no—
232C22⋊C8—42no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—42no—
236C4×C8—42no—
237#95—24no—
238#95—24no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—22no—
245C24#12011yescharacteristic
246C2×Q8—24no—
247C2×Q8—48no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—416no—
251C2×Q8—42no—
252C2×Q8—44no—
253C2×Q8—88no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—82no—
257C2×Q8—82no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4C22⋊Q812yes—
280C22×C4C22⋊Q812yes—
281C22×C4C22⋊Q812yes—
282C22×C4#12412yes—
283C22×C4#12412yes—
284C22×C4#12412yes—
285C22×C4C2×SD1612yes—
286C22×C4C2×Q1612yes—
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4#13712yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—84no—
293C2×C8—24no—
294C2×C8—42no—
295C2×C8—48no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—44no—
312C4⋊C4—48no—
313C4⋊C4—48no—
314C4⋊C4—48no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—82no—
329C4⋊C4—84no—
330C4⋊C4—84no—
331C4⋊C4C2×SD1612yes—
332C4⋊C4C2×SD1612yes—
333C4⋊C4C2×Q1612yes—
334C4⋊C4C2×Q1612yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—44no—
338C42—24no—
339C42—24no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42C22⋊Q812yes—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C42#12412yes—
350C42#12412yes—
351C42#12412yes—
352C42C22×Q811yescharacteristic
353C42C2×#4111yescharacteristic
354C42C2×#4111yescharacteristic
355C42C2×#4111yescharacteristic
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23#39312yes—
361C23#45012yes—
362C23#45112yes—
363C23C2×#12011yescharacteristic
364Q8—48no—
365Q8—84no—
366Q8—88no—
367Q8—168no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—88no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—164no—
402C2×C4#39312yes—
403C2×C4#39312yes—
404C2×C4#39312yes—
405C2×C4#45012yes—
406C2×C4#45112yes—
407C2×C4#45312yes—
408C2×C4#45412yes—
409C2×C4#47514yes—
410C2×C4#47714yes—
411C2×C4#48714yes—
412C2×C4#48914yes—
413C2×C4C2×C22⋊Q811yescharacteristic
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4C2×C22⋊Q811yescharacteristic
416C2×C4C2×#12411yescharacteristic
417C2×C4C2×#12411yescharacteristic
418C2×C4C2×#12411yescharacteristic
419C2×C4C22×SD1611yescharacteristic
420C2×C4C22×Q1611yescharacteristic
421C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
422C2×C4C2×#13711yescharacteristic
423C2×C4#57712yes—
424C2×C4#57812yes—
425C8—44no—
426C8—82no—
427C8—82no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22#202414yes—
435C22C2×#39311yescharacteristic
436C22C2×#45011yescharacteristic
437C22C2×#45111yescharacteristic
438C22#300911yescharacteristic
439C22#301111yescharacteristic
440C22C2×#45311yescharacteristic
441C22C2×#45411yescharacteristic
442C4—22no—
443C4—21no—
444C4—21no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—84no—
450C4—84no—
451C4—82no—
452C4—82no—
453C4#202414yes—
454C4#202714yes—
455C4C2×#47511yescharacteristic
456C4C2×#47711yescharacteristic
457C4#308212yes—
458C4C2×#48711yescharacteristic
459C4C2×#48911yescharacteristic
460C4#313712yes—
461C2—24no—
462C2#1862111yescharacteristic
463C2#1862311yescharacteristic
464C2#1864111yescharacteristic
465C2#1864311yescharacteristic
466C2C2×#202411yescharacteristic
467C2#2345711yescharacteristic
468C2C2×#202711yescharacteristic
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. #10355C2semidirect
6. #10349C2semidirect
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
#218342
#213132
C2×#19562
#103552
#103492
#80432
#78972
#78902