Finite Group Databasebeta

← #468739

Subgroups — #468739

Order 512.

1006 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×#2057C212yesmaximal
2#15196C211yesmaximal, characteristic
3#14161C211yesmaximal, characteristic
4#11324C212yesmaximal
5#11316C212yesmaximal
6#11256C212yesmaximal
7#11232C212yesmaximal
8#11202C211yesmaximal, characteristic
9#10885C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#107C2212yes—
12C22×#107C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C2×#389C2212yes—
15C2×#384C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#2057—28no—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28#1446—21no—
29#1446—21no—
30#1444—21no—
31#1444—21no—
32#1441—21no—
33#1441—21no—
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C23×Q8—21no—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8D414yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#107—24no—
64C2×#107—24no—
65C2×#107—42no—
66C2×#107—42no—
67C2×#107—42no—
68C2×#107—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—41no—
71#389—24no—
72#389—24no—
73#384—41no—
74#384—41no—
75C4×C4⋊C4—41no—
76C4×C4⋊C4—41no—
77C4×C4⋊C4—41no—
78C4×C4⋊C4—41no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D411yescharacteristic
90C2×#95D411yescharacteristic
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340Q818yescomplemented
97#336—24no—
98#336—24no—
99#336—41no—
100C22×Q8—24no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8—82no—
104C22×Q8—82no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
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×C8C2×Q814yes—
129C22×C8#4114yes—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
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—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4Q1614yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2×Q814yes—
173C2×C4⋊C4C2×Q814yes—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42C2×D412yes—
190C2×C42#4112yes—
191#107—48no—
192#107—48no—
193C4⋊C8—44no—
194#95—41no—
195#95—41no—
196#95—42no—
197#95—42no—
198#95—44no—
199#95—42no—
200#95—82no—
201#95—82no—
202#95—82no—
203#95—82no—
204#95—82no—
205#95—82no—
206C24C22×D411yescharacteristic
207C2×Q8—48no—
208C2×Q8—88no—
209C2×Q8—84no—
210C2×Q8—84no—
211C2×Q8—88no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
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—42no—
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—42no—
265C22×C4#12112yes—
266C22×C4#12112yes—
267C22×C4C2×Q1612yes—
268C22×C4C2×Q1612yes—
269C22×C4C2×Q1612yes—
270C22×C4#13712yes—
271C22×C4#13712yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×Q812yes—
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C4H(5,𝔽2)11yescharacteristic
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—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—48no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—88no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—84no—
329C23C2×#12111yescharacteristic
330C23#53411yescharacteristic
331C23#53511yescharacteristic
332C23#54712yes—
333C23D4×Q812yes—
334C23C22×Q1611yescharacteristic
335C23C2×#13711yescharacteristic
336C23C2×#13711yescharacteristic
337C23#57511yescharacteristic
338C23#57612yes—
339C23#57612yes—
340Q8—816no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
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—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#46214yes—
401C2×C4#53312yes—
402C2×C4#53411yescharacteristic
403C2×C4#53411yescharacteristic
404C2×C4#53811yescharacteristic
405C2×C4#53811yescharacteristic
406C2×C4#53912yes—
407C2×C4#54112yes—
408C2×C4#54212yes—
409C2×C4#54412yes—
410C2×C4#54412yes—
411C2×C4#54812yes—
412C2×C4D4×Q812yes—
413C2×C4#55111yescharacteristic
414C2×C4#55412yes—
415C2×C4Q8211yescharacteristic
416C2×C4#56311yescharacteristic
417C2×C4#56312yes—
418C2×C4#56311yescharacteristic
419C8—44no—
420C8—44no—
421C22#242912yes—
422C22#261711yescharacteristic
423C22#271711yescharacteristic
424C22#272012yes—
425C22#273411yescharacteristic
426C22C2×#46211yescharacteristic
427C22#303711yescharacteristic
428C22#304112yes—
429C22#318911yescharacteristic
430C22#319511yescharacteristic
431C22#319911yescharacteristic
432C22#319911yescharacteristic
433C22#320911yescharacteristic
434C22#321511yescharacteristic
435C22#322012yes—
436C22#333812yes—
437C22#334612yes—
438C22#334712yes—
439C22#335712yes—
440C22Q8×Q1612yes—
441C22#338512yes—
442C22#338512yes—
443C22#338912yes—
444C4—24no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—81no—
450C4—84no—
451C4—84no—
452C4—84no—
453C4—162no—
454C4#258611yescharacteristic
455C4#266612yes—
456C4#272811yescharacteristic
457C4#272912yes—
458C4#273411yescharacteristic
459C4#283311yescharacteristic
460C2#1662911yescharacteristic
461C2#2545512yes—
462C2#2546212yes—
463C2#2971511yescharacteristic
464C2#2972111yescharacteristic
465C2#3226111yescharacteristic
466C2#3226811yescharacteristic
467C2#3234812yes—
468C2#3237312yes—
469C2#3268411yescharacteristic
470C2#3269511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
96. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20572
#151962
#141612
#113242
#113162
#112562
#112322
#112022
#108852