Finite Group Databasebeta

← #417082

Subgroups — #417082

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1984C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic
3#13925C211yesmaximal, characteristic
4#13925C211yesmaximal, characteristic
5#11316C212yesmaximal
6#11316C212yesmaximal
7#11315C212yesmaximal
8#11315C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10850C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1984—24no—
20#1835—24no—
21C2×#340C2211yescharacteristic
22C2×#340C2211yescharacteristic
23C2×#340C2211yescharacteristic
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1446—21no—
34#1446—21no—
35#1445—21no—
36#1445—21no—
37#1436—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8D411yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C42—21no—
62C22×C42—21no—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8D414yes—
71#382—22no—
72#382—22no—
73#382—22no—
74#382—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—24no—
104#336—24no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—42no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
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×D412yes—
137C22×C8C2×D412yes—
138C4×Q8—44no—
139C4×Q8—44no—
140C4×Q8—44no—
141C4×Q8—44no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4#4114yes—
175C2×C4⋊C4#4114yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—81no—
185C2×C42—81no—
186C2×C42—81no—
187C2×C42—81no—
188C2×C42C2×D412yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194C4⋊C8—44no—
195C4×C8—28no—
196#95—44no—
197#95—44no—
198#95—42no—
199#95—42no—
200#95—42no—
201#95—42no—
202#95—42no—
203#95—42no—
204#95—44no—
205#95—44no—
206#95—82no—
207#95—82no—
208#95—82no—
209#95—82no—
210C24C22×D411yescharacteristic
211C2×Q8—28no—
212C2×Q8—48no—
213C2×Q8—48no—
214C2×Q8—44no—
215C2×Q8—44no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
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—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4#12112yes—
270C22×C4#12112yes—
271C22×C4#12312yes—
272C22×C4#12312yes—
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4#13514yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C42−1+412yes—
285C22×C42−1+412yes—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
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—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—48no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—84no—
330C42—84no—
331C42#12314yes—
332C42#12414yes—
333C42#12614yes—
334C42#12614yes—
335C23C2×#12111yescharacteristic
336C23C2×#12311yescharacteristic
337C23C2×#12311yescharacteristic
338C23#54812yes—
339C23#54812yes—
340C23C2×#13512yes—
341C23#57512yes—
342C23#57611yescharacteristic
343C23#57711yescharacteristic
344C23#57711yescharacteristic
345C23#57811yescharacteristic
346Q8—416no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
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—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4#39714yes—
406C2×C4#39714yes—
407C2×C4C2×#12311yescharacteristic
408C2×C4C2×#12411yescharacteristic
409C2×C4C2×#12611yescharacteristic
410C2×C4C2×#12611yescharacteristic
411C2×C4#52912yes—
412C2×C4#52912yes—
413C2×C4#53211yescharacteristic
414C2×C4#53211yescharacteristic
415C2×C4#53611yescharacteristic
416C2×C4#53611yescharacteristic
417C2×C4#54112yes—
418C2×C4#54112yes—
419C2×C4#55212yes—
420C2×C4#55212yes—
421C2×C4#55612yes—
422C2×C4#55612yes—
423C2×C4#55612yes—
424C2×C4#55612yes—
425C8—42no—
426C8—42no—
427C8—82no—
428C22C2×#39711yescharacteristic
429C22#242112yes—
430C22#242112yes—
431C22#254511yescharacteristic
432C22#254511yescharacteristic
433C22#305011yescharacteristic
434C22#305411yescharacteristic
435C22#306011yescharacteristic
436C22#306111yescharacteristic
437C22#309112yes—
438C22#309112yes—
439C22#309611yescharacteristic
440C22#309611yescharacteristic
441C22#309711yescharacteristic
442C22#309711yescharacteristic
443C22#336012yes—
444C22#336112yes—
445C22#336412yes—
446C22#336512yes—
447C22#337012yes—
448C22#337112yes—
449C22#337612yes—
450C22#337712yes—
451C4—24no—
452C4—24no—
453C4—44no—
454C4—82no—
455C4—84no—
456C4—84no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4—161no—
461C4#249911yescharacteristic
462C4#249911yescharacteristic
463C4#251412yes—
464C4#251412yes—
465C4#264211yescharacteristic
466C4#286111yescharacteristic
467C2#1524111yescharacteristic
468C2#2363811yescharacteristic
469C2#2364111yescharacteristic
470C2#2499812yes—
471C2#2499912yes—
472C2#2500012yes—
473C2#2500112yes—
474C2#2864111yescharacteristic
475C2#2864211yescharacteristic
476C2#2864311yescharacteristic
477C2#2864411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19842
C2×#18352
#139252
#113162
#113152
#108502
C4×#3402