Finite Group Databasebeta

← #474727

Subgroups — #474727

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15151C211yesmaximal, characteristic
2#15128C211yesmaximal, characteristic
3#11384C212yesmaximal
4#11315C212yesmaximal
5#11292C212yesmaximal
6#11227C212yesmaximal
7#11207C212yesmaximal
8#10880C212yesmaximal
9#10834C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13C2×#398—21no—
14C2×#398—21no—
15C2×#395—21no—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#384C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1446—21no—
31#1445—21no—
32#1445—21no—
33#1444—21no—
34#1441—21no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C23×Q8—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—42no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8—41no—
63C2×C4×C8—22no—
64C2×C4×C8—41no—
65C2×C4×C8—41no—
66#398—44no—
67#398—44no—
68#395—44no—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—42no—
73#384—41no—
74#384—41no—
75C4×C4⋊C4—41no—
76C4×C4⋊C4—41no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D411yescharacteristic
88C2×#95D411yescharacteristic
89#340—24no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340Q818yescomplemented
97#336—42no—
98#336—42no—
99C22×Q8—44no—
100C22×Q8—42no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8—82no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—22no—
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—
129C4⋊Q8—44no—
130C4⋊Q8—48no—
131C4⋊Q8—44no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
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—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—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4D814yes—
172C2×C4⋊C4SD1614yes—
173C2×C4⋊C4C2×Q814yes—
174C2×C4⋊C4C2×Q814yes—
175C2×C4⋊C4#4114yes—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42#4112yes—
189C2×C42#4112yes—
190C4⋊C8—44no—
191C4×C8—44no—
192#95—41no—
193#95—41no—
194#95—44no—
195#95—42no—
196#95—44no—
197#95—42no—
198#95—44no—
199#95—42no—
200#95—44no—
201#95—42no—
202#95—44no—
203#95—82no—
204C24C22×D411yescharacteristic
205C2×Q8—48no—
206C2×Q8—88no—
207C2×Q8—84no—
208C2×Q8—88no—
209C2×Q8—84no—
210C22×C4—21no—
211C22×C4—21no—
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—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
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—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#12412yes—
263C22×C4#12412yes—
264C22×C4C2×D812yes—
265C22×C4C2×D812yes—
266C22×C4C2×D812yes—
267C22×C4C2×SD1612yes—
268C22×C4C2×SD1612yes—
269C22×C4C2×SD1612yes—
270C22×C4C22×Q812yes—
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4112yes—
273C22×C4C2×#4111yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C4H(5,𝔽2)11yescharacteristic
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C4H(5,𝔽2)12yes—
278C22×C42−1+411yescharacteristic
279C22×C42−1+412yes—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—48no—
303C4⋊C4—42no—
304C4⋊C4—42no—
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—88no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4—88no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—42no—
322C42—42no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C23C2×#12411yescharacteristic
328C23#53411yescharacteristic
329C23#53711yescharacteristic
330C23#54612yes—
331C23D4×Q812yes—
332C23C22×D811yescharacteristic
333C23C22×SD1611yescharacteristic
334C23#57511yescharacteristic
335C23#57511yescharacteristic
336C23#57712yes—
337C23#57812yes—
338Q8—816no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
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—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4#48614yes—
399C2×C4#53312yes—
400C2×C4#53811yescharacteristic
401C2×C4#53812yes—
402C2×C4#53811yescharacteristic
403C2×C4#53911yescharacteristic
404C2×C4#53911yescharacteristic
405C2×C4#54112yes—
406C2×C4#54212yes—
407C2×C4#54412yes—
408C2×C4#54412yes—
409C2×C4#55111yescharacteristic
410C2×C4#55412yes—
411C2×C4#55412yes—
412C2×C4#55612yes—
413C2×C4Q8211yescharacteristic
414C2×C4#56011yescharacteristic
415C2×C4#56212yes—
416C2×C4#56311yescharacteristic
417C8—44no—
418C8—82no—
419C22#249212yes—
420C22#263911yescharacteristic
421C22#269711yescharacteristic
422C22#270412yes—
423C22#273411yescharacteristic
424C22C2×#48611yescharacteristic
425C22#313511yescharacteristic
426C22#313712yes—
427C22#318611yescharacteristic
428C22#318711yescharacteristic
429C22#320011yescharacteristic
430C22#320211yescharacteristic
431C22#324511yescharacteristic
432C22#324611yescharacteristic
433C22#325812yes—
434C22#329412yes—
435C22#329812yes—
436C22#332312yes—
437C22#332712yes—
438C22Q8×D812yes—
439C22Q8×SD1612yes—
440C22#338812yes—
441C22#339012yes—
442C4—24no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—84no—
449C4—84no—
450C4—84no—
451C4—162no—
452C4#258611yescharacteristic
453C4#266612yes—
454C4#278812yes—
455C4#278911yescharacteristic
456C4#279211yescharacteristic
457C4#283011yescharacteristic
458C2#1700411yescharacteristic
459C2#2781312yes—
460C2#2782012yes—
461C2#3018511yescharacteristic
462C2#3019311yescharacteristic
463C2#3166611yescharacteristic
464C2#3166811yescharacteristic
465C2#3186612yes—
466C2#3186812yes—
467C2#3268211yescharacteristic
468C2#3268311yescharacteristic
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
#151512
#151282
#113842
#113152
#112922
#112272
#112072
#108802
#108342