Finite Group Databasebeta

← #415151

Subgroups — #415151

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1847C211yesmaximal, characteristic
4C2×#1768C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic
6#14276C211yesmaximal, characteristic
7#11232C212yesmaximal
8#11231C212yesmaximal
9#11220C212yesmaximal
10#11219C212yesmaximal
11#10842C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1993—24no—
24#1990—24no—
25#1847—24no—
26#1768—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1446—22no—
35#1445—22no—
36#1441—22no—
37#1441—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8Q814yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—24no—
66C2×#107—24no—
67C2×#107—42no—
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—42no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8Q818yescomplemented
74C2×#97—22no—
75C2×#97—22no—
76C2×#97D414yes—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80#389—42no—
81#389—42no—
82#384—22no—
83#384D414yes—
84#382—22no—
85#382—22no—
86#382—22no—
87#382—22no—
88#376—22no—
89#376—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D412yes—
102C2×#95D412yes—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#336—24no—
110C22×Q8—24no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—82no—
114C22×Q8—82no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8C2×D414yes—
137C22×C8C2×Q814yes—
138C22×C8C2×Q814yes—
139C22×C8#4114yes—
140C4⋊Q8—28no—
141C4⋊Q8—28no—
142C4⋊Q8—44no—
143C4⋊Q8—44no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
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—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×Q812yes—
185C2×C4⋊C4C2×Q812yes—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×Q814yes—
196C2×C42C2×Q812yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200#107—48no—
201#106—48no—
202C4⋊C8—216no—
203#97—28no—
204C4×C8—44no—
205#95—24no—
206#95—24no—
207#95—44no—
208#95—42no—
209#95—42no—
210#95—42no—
211#95—42no—
212#95—82no—
213#95—82no—
214#95—82no—
215#95—82no—
216#95—82no—
217#95—82no—
218C24C22×D411yescharacteristic
219C2×Q8—48no—
220C2×Q8—88no—
221C2×Q8—84no—
222C2×Q8—84no—
223C2×Q8—88no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4#12112yes—
278C22×C4#12112yes—
279C22×C4C22⋊Q812yes—
280C22×C4C22⋊Q812yes—
281C22×C4C4⋊Q814yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×Q811yescharacteristic
285C22×C4C22×Q812yes—
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4H(5,𝔽2)12yes—
291C22×C42−1+412yes—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8C22⋊Q818yes—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—48no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4—88no—
330C42—24no—
331C42—24no—
332C42—42no—
333C42—42no—
334C42#12114yes—
335C42C22⋊Q814yes—
336C42C22⋊Q814yes—
337C42#12614yes—
338C23C2×#12111yescharacteristic
339C23C2×#12111yescharacteristic
340C23C2×C22⋊Q811yescharacteristic
341C23C2×C4⋊Q811yescharacteristic
342C23#53211yescharacteristic
343C23#53511yescharacteristic
344C23#53611yescharacteristic
345C23#57612yes—
346C23#57712yes—
347C23#57712yes—
348C23#57812yes—
349Q8—816no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
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—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#39614yes—
407C2×C4#39614yes—
408C2×C4C2×#12111yescharacteristic
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×C22⋊Q812yes—
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×#12611yescharacteristic
413C2×C4#53312yes—
414C2×C4#53712yes—
415C2×C4#54212yes—
416C2×C4#54612yes—
417C2×C4D4×Q812yes—
418C2×C4#55212yes—
419C2×C4#55412yes—
420C2×C4#55712yes—
421C2×C4#55712yes—
422C2×C4Q8212yes—
423C2×C4#55912yes—
424C8—28no—
425C8—44no—
426C22C2×#39611yescharacteristic
427C22#254411yescharacteristic
428C22#254611yescharacteristic
429C22#256511yescharacteristic
430C22#256711yescharacteristic
431C22#262011yescharacteristic
432C22#285711yescharacteristic
433C22#304111yescharacteristic
434C22#304211yescharacteristic
435C22#304311yescharacteristic
436C22#304411yescharacteristic
437C22#306212yes—
438C22#306312yes—
439C22#308112yes—
440C22#308212yes—
441C22#316312yes—
442C22#316412yes—
443C22#317812yes—
444C22#317912yes—
445C22#322412yes—
446C22#322512yes—
447C22#324312yes—
448C22#324412yes—
449C4—24no—
450C4—24no—
451C4—44no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—84no—
456C4—162no—
457C4—161no—
458C4—161no—
459C4#241712yes—
460C4#245112yes—
461C4#251112yes—
462C4#258212yes—
463C2#1518411yescharacteristic
464C2#2361411yescharacteristic
465C2#2361511yescharacteristic
466C2#2862711yescharacteristic
467C2#2862811yescharacteristic
468C2#2869611yescharacteristic
469C2#2869711yescharacteristic
470C2#2918912yes—
471C2#2921712yes—
472C2#2985712yes—
473C2#3321912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
73. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19932
C2×#19902
C2×#18472
C2×#17682
#142822
#142762
#112322
#112312
#112202
#112192
#108422