Finite Group Databasebeta

← #413843

Subgroups — #413843

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4C2×#1949C212yesmaximal
5#14276C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11219C212yesmaximal
8#11208C211yesmaximal, characteristic
9#11208C211yesmaximal, characteristic
10#10849C212yesmaximal
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16#2331—28no—
17C2×#391—22no—
18C2×#389C2212yes—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387—22no—
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×#376C2211yescharacteristic
27#2058—24no—
28#2057—24no—
29#1949—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1441—22no—
40#1441—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8D414yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42—22no—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—24no—
66C2×#107—24no—
67C2×#107—24no—
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—24no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8Q818yescomplemented
74#391—216no—
75#389—24no—
76#389—24no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#387—216no—
82#384—22no—
83#384Q814yescomplemented
84#382—24no—
85#382—24no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4D418yes—
88#376—24no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D412yes—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#336—22no—
107#336—22no—
108#336—42no—
109C22×Q8—24no—
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8—82no—
113C22×Q8—82no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2411yescharacteristic
125C22×C8—22no—
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×C8—22no—
137C22×C8C2×Q814yes—
138C22×C8C2×Q814yes—
139C4⋊Q8—216no—
140C4⋊Q8—216no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2×D414yes—
183C2×C4⋊C4C2×Q812yes—
184C2×C4⋊C4C2×Q812yes—
185C2×C4⋊C4#4114yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—28no—
193C2×C42—82no—
194C2×C42—82no—
195C2×C42C2×D412yes—
196C2×C42C2×D414yes—
197C2×C42C2×Q814yes—
198C2×C42C2×Q812yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201#107—48no—
202#106—48no—
203C4⋊C8—216no—
204#95—24no—
205#95—24no—
206#95—28no—
207#95—28no—
208#95—24no—
209#95—24no—
210#95—44no—
211#95—42no—
212#95—42no—
213#95—82no—
214#95—82no—
215C24C22×D411yescharacteristic
216C2×Q8—216no—
217C2×Q8—84no—
218C2×Q8—84no—
219C2×Q8—88no—
220C2×Q8—88no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4#12014yes—
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4C22⋊Q814yes—
278C22×C4C4⋊Q814yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D412yes—
281C22×C4C22×Q812yes—
282C22×C4C22×Q811yescharacteristic
283C22×C4C2×#4112yes—
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C42−1+411yescharacteristic
288C22×C42−1+412yes—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—216no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—88no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4—88no—
327C4⋊C4#12118yes—
328C42—24no—
329C42—24no—
330C42—42no—
331C42—42no—
332C42—88no—
333C42C22⋊Q818yes—
334C42C4⋊Q818yes—
335C23C2×#12011yescharacteristic
336C23C2×#12111yescharacteristic
337C23C2×C22⋊Q811yescharacteristic
338C23#52511yescharacteristic
339C23C2×C4⋊Q811yescharacteristic
340C23#53611yescharacteristic
341C23#53711yescharacteristic
342C23#57612yes—
343C23#57712yes—
344C23#57712yes—
345C23#57812yes—
346Q8—816no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—28no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4#39518yes—
405C2×C4C2×#12112yes—
406C2×C4C2×C22⋊Q812yes—
407C2×C4#52512yes—
408C2×C4C2×C4⋊Q812yes—
409C2×C4#53612yes—
410C2×C4#53712yes—
411C2×C4D4×Q812yes—
412C2×C4D4×Q811yescharacteristic
413C2×C4#55011yescharacteristic
414C2×C4#55411yescharacteristic
415C2×C4#55412yes—
416C2×C4#55611yescharacteristic
417C2×C4#55612yes—
418C2×C4#55712yes—
419C2×C4#55712yes—
420C2×C4#55712yes—
421C8—44no—
422C8—44no—
423C22C2×#39511yescharacteristic
424C22#253811yescharacteristic
425C22#254611yescharacteristic
426C22#258111yescharacteristic
427C22#258411yescharacteristic
428C22#261611yescharacteristic
429C22#284711yescharacteristic
430C22#301212yes—
431C22#301312yes—
432C22#304111yescharacteristic
433C22#304211yescharacteristic
434C22#304311yescharacteristic
435C22#304411yescharacteristic
436C22#308112yes—
437C22#308212yes—
438C22#311812yes—
439C22#312112yes—
440C22#316312yes—
441C22#316412yes—
442C22#324312yes—
443C22#324412yes—
444C22#325712yes—
445C22#325812yes—
446C4—28no—
447C4—42no—
448C4—48no—
449C4—42no—
450C4—82no—
451C4—84no—
452C4—82no—
453C4—82no—
454C4—161no—
455C4—161no—
456C4#253812yes—
457C4#254612yes—
458C4#258112yes—
459C4#258412yes—
460C2#1515811yescharacteristic
461C2#2355712yes—
462C2#2829011yescharacteristic
463C2#2829111yescharacteristic
464C2#2869611yescharacteristic
465C2#2869711yescharacteristic
466C2#2928112yes—
467C2#2932512yes—
468C2#2970212yes—
469C2#3291711yescharacteristic
470C2#3291811yescharacteristic
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
83. #384Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
C2×#20582
C2×#20572
C2×#19492
#142762
#112202
#112192
#112082
#108492
#108332