Finite Group Databasebeta

← #367411

Subgroups — #367411

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#13779C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11315C212yesmaximal
8#10842C211yesmaximal, characteristic
9#10807C212yesmaximal
10#10806C212yesmaximal
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4C2212yes—
19C2×#376C2211yescharacteristic
20#1993C418yescomplemented
21#1990—24no—
22#1836—24no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1446—22no—
34#1445—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C23×Q8—21no—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C414yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42—22no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#97—22no—
65C2×#97—22no—
66C2×#97C2×C414yes—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70#384—22no—
71#384D414yes—
72#382—22no—
73#382—22no—
74#382—22no—
75#382C2×C414yes—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—24no—
78#376—22no—
79#376—22no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D412yes—
92C2×#95D412yes—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340C2×C418yes—
100#340C2×C418yes—
101#336—24no—
102#336—24no—
103C22×Q8—24no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8—82no—
107C22×Q8—82no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8C22×C414yes—
132C22×C8#4114yes—
133C4⋊Q8—28no—
134C4⋊Q8—28no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4C2×D414yes—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—44no—
184C2×C42—82no—
185C2×C42—82no—
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42#4114yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193#97—28no—
194C4×C8—44no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—42no—
200#95—42no—
201#95—44no—
202#95—82no—
203#95—82no—
204#95—82no—
205#95—82no—
206#95—82no—
207#95—82no—
208C24C22×D411yescharacteristic
209C2×Q8—48no—
210C2×Q8—88no—
211C2×Q8—88no—
212C2×Q8—84no—
213C2×Q8—84no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C4×D412yes—
266C22×C4C4×D412yes—
267C22×C4#12112yes—
268C22×C4#12112yes—
269C22×C4#12312yes—
270C22×C4#12312yes—
271C22×C4#12414yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4112yes—
278C22×C4C2×#4112yes—
279C22×C4C2×#4111yescharacteristic
280C22×C42−1+412yes—
281C22×C42−1+412yes—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—48no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C42—24no—
320C42—24no—
321C42—42no—
322C42—42no—
323C42—88no—
324C42#11714yes—
325C42C4×D414yes—
326C42#12414yes—
327C42#12614yes—
328C23C2×C4×D411yescharacteristic
329C23C2×#12111yescharacteristic
330C23C2×#12311yescharacteristic
331C23C2×#12411yescharacteristic
332C23#53211yescharacteristic
333C23#53611yescharacteristic
334C23#53611yescharacteristic
335C23#57612yes—
336C23#57712yes—
337C23#57712yes—
338C23#57812yes—
339Q8—816no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
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#38814yes—
397C2×C4#38814yes—
398C2×C4C2×#11711yescharacteristic
399C2×C4C2×C4×D411yescharacteristic
400C2×C4C4×#4112yes—
401C2×C4#51912yes—
402C2×C4C2×#12411yescharacteristic
403C2×C4C2×#12611yescharacteristic
404C2×C4#52912yes—
405C2×C4#54112yes—
406C2×C4#54712yes—
407C2×C4#54812yes—
408C2×C4#55212yes—
409C2×C4#55212yes—
410C2×C4#55212yes—
411C2×C4#55612yes—
412C2×C4#55612yes—
413C2×C4#55612yes—
414C8—44no—
415C8—44no—
416C22C2×#38811yescharacteristic
417C22C4×#12411yescharacteristic
418C22#232711yescharacteristic
419C22#249911yescharacteristic
420C22#250311yescharacteristic
421C22#254511yescharacteristic
422C22#254711yescharacteristic
423C22#294612yes—
424C22#294712yes—
425C22#304111yescharacteristic
426C22#304211yescharacteristic
427C22#304311yescharacteristic
428C22#304411yescharacteristic
429C22#309612yes—
430C22#309712yes—
431C22#313612yes—
432C22#313712yes—
433C22#317312yes—
434C22#317512yes—
435C22#324112yes—
436C22#324212yes—
437C22#324312yes—
438C22#324412yes—
439C4—24no—
440C4—24no—
441C4—44no—
442C4—84no—
443C4—82no—
444C4—82no—
445C4—84no—
446C4—82no—
447C4—161no—
448C4—161no—
449C4#235612yes—
450C4#236212yes—
451C4#248912yes—
452C4#251412yes—
453C2#1400311yescharacteristic
454C2#2106311yescharacteristic
455C2#2106411yescharacteristic
456C2#2165412yes—
457C2#2225512yes—
458C2#2788612yes—
459C2#2797112yes—
460C2#2866211yescharacteristic
461C2#2866311yescharacteristic
462C2#2872811yescharacteristic
463C2#2872911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
20. #1993C4semidirect

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×#18362
#142822
#137792
#113162
#113152
#108422
#108072
#108062
#106942