Finite Group Databasebeta

← #416523

Subgroups — #416523

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic
4C2×#1960C211yesmaximal, characteristic
5C2×#1960C211yesmaximal, characteristic
6#14707C211yesmaximal, characteristic
7#11190C214yesmaximal
8#11067C214yesmaximal
9#10693C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×#103C2214yes—
13C22×#103C2214yes—
14C22×#97C2211yescharacteristic
15C22×#97C2211yescharacteristic
16#2862—24no—
17#2862—24no—
18C2×#391—22no—
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#389C2212yes—
22C2×#382—22no—
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C43C2211yescharacteristic
26#1993—24no—
27#1993—24no—
28#1960—24no—
29#1960—24no—
30C2×#340C2214yes—
31C2×#340C2214yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1436—22no—
35#1436—22no—
36C4⋊#95C2212yes—
37C23×Q8—21no—
38C23×Q8C2312yes—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8D414yes—
52C2×C4⋊Q8D414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#103—28no—
66C2×#103—28no—
67C2×#103—28no—
68C2×#103—28no—
69C2×#103—44no—
70C2×#103—44no—
71C2×#97—22no—
72C2×#97—22no—
73C2×#97—22no—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97—22no—
77#391—216no—
78#391—24no—
79#391—24no—
80#391—24no—
81#391—24no—
82#389—44no—
83#382—216no—
84#382—22no—
85#382—22no—
86#382—22no—
87#382D414yes—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C43—22no—
98C43—22no—
99#340—28no—
100#340—28no—
101#336—24no—
102#336—24no—
103C22×Q8—24no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—24no—
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8—44no—
114C22×Q8—44no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8C2×D414yes—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
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×C8—22no—
138C22×C8—22no—
139C22×C8#4114yes—
140C22×C8#4114yes—
141C4⋊Q8—28no—
142C4⋊Q8—28no—
143C4⋊Q8—28no—
144C4⋊Q8—28no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C2×D414yes—
174C2×C4⋊C4C2×D414yes—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194C2×C42#4112yes—
195#103—416no—
196#103—416no—
197#97—44no—
198#97—44no—
199#95—216no—
200#95—216no—
201#95—24no—
202#95—24no—
203#95—44no—
204#95—44no—
205#95—42no—
206#95—42no—
207#95—44no—
208C24C22×D411yescharacteristic
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—28no—
212C2×Q8—28no—
213C2×Q8—416no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—84no—
228C2×Q8—84no—
229C2×Q8—88no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
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—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
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—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4#12012yes—
273C22×C4#12012yes—
274C22×C4#12112yes—
275C22×C4#12112yes—
276C22×C4#12412yes—
277C22×C4#12412yes—
278C22×C4#13714yes—
279C22×C4#13714yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4C2×#4112yes—
287C22×C4C2×#4112yes—
288C22×C4H(5,𝔽2)12yes—
289C22×C42−1+412yes—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—28no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—88no—
316C42—216no—
317C42—48no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42#12114yes—
323C42#12114yes—
324C42#12314yes—
325C42#12314yes—
326C23C2×#12011yescharacteristic
327C23C2×#12111yescharacteristic
328C23C2×#12411yescharacteristic
329C23#53211yescharacteristic
330C23#53211yescharacteristic
331C23#53511yescharacteristic
332C23#53611yescharacteristic
333C23C2×#13712yes—
334C23C2×#13712yes—
335C23#57512yes—
336C23#57512yes—
337Q8—48no—
338Q8—48no—
339Q8—816no—
340Q8—84no—
341Q8—84no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—44no—
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—84no—
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#39414yes—
391C2×C4#39414yes—
392C2×C4C2×#12111yescharacteristic
393C2×C4C2×#12111yescharacteristic
394C2×C4C2×#12311yescharacteristic
395C2×C4C2×#12311yescharacteristic
396C2×C4#52912yes—
397C2×C4#52912yes—
398C2×C4#54212yes—
399C2×C4#54212yes—
400C2×C4#54612yes—
401C2×C4#54612yes—
402C2×C4#54712yes—
403C2×C4#54812yes—
404C2×C4#54812yes—
405C2×C4#55012yes—
406C2×C4#55612yes—
407C2×C4#55912yes—
408C8—44no—
409C8—44no—
410C22C2×#39411yescharacteristic
411C22#243111yescharacteristic
412C22#243211yescharacteristic
413C22#254411yescharacteristic
414C22#254511yescharacteristic
415C22#264011yescharacteristic
416C22#286111yescharacteristic
417C22#300412yes—
418C22#300512yes—
419C22#303912yes—
420C22#304012yes—
421C22#313911yescharacteristic
422C22#313911yescharacteristic
423C22#314011yescharacteristic
424C22#314011yescharacteristic
425C22#317212yes—
426C22#317212yes—
427C22#318112yes—
428C22#318112yes—
429C22#321512yes—
430C22#321612yes—
431C22#323612yes—
432C22#323712yes—
433C4—24no—
434C4—24no—
435C4—44no—
436C4—44no—
437C4—84no—
438C4—84no—
439C4—84no—
440C4—162no—
441C4#242612yes—
442C4#242712yes—
443C4#244712yes—
444C4#244812yes—
445C2#1522611yescharacteristic
446C2#2350611yescharacteristic
447C2#2350711yescharacteristic
448C2#2560312yes—
449C2#2567712yes—
450C2#2862012yes—
451C2#2866012yes—
452C2#3021411yescharacteristic
453C2#3021511yescharacteristic
454C2#3328111yescharacteristic
455C2#3328111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#28622
C2×#19932
C2×#19602
#147072
#111902
#110672
#106932