Finite Group Databasebeta

← #238227

Subgroups — #238227

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5#10357C214yesmaximal, complemented
6#10350C214yesmaximal, complemented
7#8039C211yesmaximal, characteristic
8#7893C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15#2857—22no—
16#2659—22no—
17#2552—21no—
18#2552C2212yes—
19#2551—21no—
20#2551C2212yes—
21#2550—22no—
22#2550—22no—
23#2316—21no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990C2212yes—
28#1990C2212yes—
29#1988—22no—
30#1988C2214yes—
31#1987—22no—
32#1981C2214yes—
33#1956—21no—
34#1956—21no—
35#1824—22no—
36#1787—22no—
37C2×#340—21no—
38C2×#340—21no—
39#1296C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—21no—
44C23×C8—21no—
45#544—44no—
46#544—44no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8Q814yescomplemented
57C2×C4⋊Q8C2312yes—
58C2×#125—22no—
59C2×#125—22no—
60C2×C22⋊Q8—24no—
61C2×#117—21no—
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#431—22no—
70#431—22no—
71#431—22no—
72#431—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—24no—
80C2×#103—24no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2312yes—
85#398—24no—
86#398—24no—
87#395—24no—
88#395—24no—
89#391—24no—
90#389—22no—
91#389—22no—
92#389—24no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#384—22no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#382—21no—
103#382—21no—
104#380—24no—
105#380—21no—
106#380—21no—
107#380D412yes—
108#380D412yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C4×C4⋊C4C2312yes—
117C2×#95—21no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—22no—
125C22×Q8—41no—
126C22×Q8—42no—
127C22×Q8—82no—
128C22×Q8—82no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—44no—
148C4⋊Q8—48no—
149C4⋊Q8—44no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—84no—
154C4⋊Q8C2×Q814yes—
155#125—44no—
156#125—48no—
157#125—44no—
158C22⋊Q8—416no—
159#117—42no—
160#117—42no—
161#117—42no—
162#117—42no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4SD1614yes—
189C2×C4⋊C4Q1614yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—82no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×Q812yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—42no—
216C4⋊C8—42no—
217#103—84no—
218#103—84no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—42no—
224#95—24no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24#12011yescharacteristic
233C2×Q8—24no—
234C2×Q8—48no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—88no—
240C2×Q8—82no—
241C2×Q8—82no—
242C2×Q8—88no—
243C2×Q8—84no—
244C2×Q8—84no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4C22⋊Q812yes—
269C22×C4C22⋊Q812yes—
270C22×C4C22⋊Q812yes—
271C22×C4#12412yes—
272C22×C4#12412yes—
273C22×C4#12412yes—
274C22×C4C2×SD1612yes—
275C22×C4C2×Q1612yes—
276C22×C4AGL(1,ℤ/8)12yes—
277C22×C4#13712yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—84no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—48no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—81no—
317C4⋊C4—81no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4C2×SD1614yes—
321C4⋊C4C2×Q1614yes—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C42—42no—
325C42—42no—
326C42—44no—
327C42—44no—
328C42—82no—
329C42C22⋊Q812yes—
330C42C22⋊Q812yes—
331C42C22⋊Q812yes—
332C42#12412yes—
333C42#12412yes—
334C42#12412yes—
335C42C22×Q811yescharacteristic
336C42C2×#4111yescharacteristic
337C42C2×#4111yescharacteristic
338C42C2×#4111yescharacteristic
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23#39312yes—
344C23#45012yes—
345C23#45112yes—
346C23C2×#12011yescharacteristic
347Q8—48no—
348Q8—88no—
349Q8—84no—
350Q8—168no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—88no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4#39312yes—
386C2×C4#39312yes—
387C2×C4#39312yes—
388C2×C4#45012yes—
389C2×C4#45112yes—
390C2×C4#47514yes—
391C2×C4#47714yes—
392C2×C4#48714yes—
393C2×C4#48914yes—
394C2×C4C2×C22⋊Q811yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×#12411yescharacteristic
398C2×C4C2×#12411yescharacteristic
399C2×C4C2×#12411yescharacteristic
400C2×C4C22×SD1611yescharacteristic
401C2×C4C22×Q1611yescharacteristic
402C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C2×C4#57612yes—
405C2×C4#57712yes—
406C8—82no—
407C8—82no—
408C8—82no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22#202414yes—
416C22C2×#39311yescharacteristic
417C22C2×#45011yescharacteristic
418C22C2×#45111yescharacteristic
419C22#300811yescharacteristic
420C22#301011yescharacteristic
421C22#302211yescharacteristic
422C22#302211yescharacteristic
423C4—22no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—44no—
428C4—44no—
429C4—84no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—82no—
434C4#202414yes—
435C4C2×#47511yescharacteristic
436C4C2×#47711yescharacteristic
437C4#308112yes—
438C4C2×#48711yescharacteristic
439C4C2×#48911yescharacteristic
440C4#313612yes—
441C2—24no—
442C2#1890311yescharacteristic
443C2#1890511yescharacteristic
444C2#1892311yescharacteristic
445C2#1892511yescharacteristic
446C2C2×#202411yescharacteristic
447C2#2345611yescharacteristic
448C2#2346411yescharacteristic
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. #10357C2semidirect
6. #10350C2semidirect
56. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#218342
#218042
C2×#19882
#103572
#103502
#80392
#78932
#78922