Finite Group Databasebeta

← #468758

Subgroups — #468758

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2086C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic
3#15202C212yesmaximal
4#11328C212yesmaximal
5#11316C212yesmaximal
6#11292C212yesmaximal
7#11232C211yesmaximal, characteristic
8#11219C212yesmaximal
9#10881C212yesmaximal
10C22×C4×Q8C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#389C2212yes—
18#2086—24no—
19#1985—24no—
20C2×#340C2212yes—
21C2×#340C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1446—22no—
31#1444—22no—
32#1444—22no—
33#1444—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C23×Q8D411yescharacteristic
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40C23×C8C2312yes—
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—41no—
44C2×C4×Q8—41no—
45C2×C4×Q8—41no—
46C2×C4×Q8—41no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D412yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C2×#107—22no—
59C2×#107—24no—
60C2×#107—41no—
61C2×#107—41no—
62C2×#107—41no—
63C2×C4⋊C8—41no—
64C2×C4⋊C8—41no—
65C2×C4⋊C8D414yes—
66C2×#97—22no—
67C2×#97—41no—
68C2×#97—41no—
69#389—24no—
70#389—24no—
71#389—42no—
72#389—42no—
73#389—42no—
74#389—42no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84#340—22no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—41no—
89#340—42no—
90#340—42no—
91#340Q818yescomplemented
92#336—24no—
93#336—24no—
94C22×Q8—22no—
95C22×Q8—24no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8—42no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2411yescharacteristic
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8C2×D412yes—
123C22×C8C2×D412yes—
124C22×C8C2×Q814yes—
125C4×Q8—44no—
126C4×Q8—44no—
127C4×Q8—44no—
128C4×Q8—44no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
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—24no—
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—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C2×Q814yes—
166C2×C4⋊C4C2×Q814yes—
167C2×C4⋊C4#4114yes—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42C2×D412yes—
179C2×C42#4112yes—
180#107—44no—
181#107—44no—
182C4⋊C8—44no—
183#97—44no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—82no—
191#95—82no—
192#95—82no—
193C24C22×D411yescharacteristic
194C2×Q8—44no—
195C2×Q8—48no—
196C2×Q8—44no—
197C2×Q8—48no—
198C2×Q8—44no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
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#12112yes—
251C22×C4#12112yes—
252C22×C4#13714yes—
253C22×C4C22×D411yescharacteristic
254C22×C4C22×Q812yes—
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4112yes—
257C22×C4H(5,𝔽2)11yescharacteristic
258C22×C4H(5,𝔽2)11yescharacteristic
259C22×C4H(5,𝔽2)11yescharacteristic
260C22×C4H(5,𝔽2)11yescharacteristic
261C22×C42−1+412yes—
262C22×C42−1+412yes—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—48no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—88no—
299C4⋊C4—88no—
300C4⋊C4—88no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—42no—
309C23C2×#12111yescharacteristic
310C23#53411yescharacteristic
311C23#53411yescharacteristic
312C23#54812yes—
313C23D4×Q812yes—
314C23C2×#13712yes—
315C23#57512yes—
316C23#57611yescharacteristic
317C23#57611yescharacteristic
318C23#57611yescharacteristic
319C23#57811yescharacteristic
320Q8—416no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4#53312yes—
376C2×C4#53411yescharacteristic
377C2×C4#53511yescharacteristic
378C2×C4#53811yescharacteristic
379C2×C4#53811yescharacteristic
380C2×C4#53912yes—
381C2×C4#54112yes—
382C2×C4#54212yes—
383C2×C4#54412yes—
384C2×C4#54412yes—
385C2×C4#54712yes—
386C2×C4D4×Q812yes—
387C2×C4#55111yescharacteristic
388C2×C4#55412yes—
389C2×C4Q8211yescharacteristic
390C2×C4#56311yescharacteristic
391C2×C4#56312yes—
392C2×C4#56311yescharacteristic
393C8—42no—
394C8—42no—
395C8—44no—
396C22#242912yes—
397C22#261711yescharacteristic
398C22#272811yescharacteristic
399C22#272912yes—
400C22#273411yescharacteristic
401C22#305612yes—
402C22#305811yescharacteristic
403C22#306111yescharacteristic
404C22#319511yescharacteristic
405C22#319612yes—
406C22#319711yescharacteristic
407C22#319911yescharacteristic
408C22#319911yescharacteristic
409C22#320311yescharacteristic
410C22#320311yescharacteristic
411C22#337012yes—
412C22#337012yes—
413C22#337412yes—
414C22#337512yes—
415C22#338512yes—
416C22#338512yes—
417C22#338912yes—
418C22#339012yes—
419C4—42no—
420C4—42no—
421C4—44no—
422C4—82no—
423C4—84no—
424C4—84no—
425C4—84no—
426C4—162no—
427C4#258611yescharacteristic
428C4#266612yes—
429C4#271711yescharacteristic
430C4#272012yes—
431C4#273411yescharacteristic
432C4#283311yescharacteristic
433C2#1662911yescharacteristic
434C2#2548112yes—
435C2#2548412yes—
436C2#2973811yescharacteristic
437C2#2974311yescharacteristic
438C2#3255811yescharacteristic
439C2#3256311yescharacteristic
440C2#3258012yes—
441C2#3258712yes—
442C2#3269111yescharacteristic
443C2#3269511yescharacteristic
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

Complements

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

NH ≅ G/Ntype
91. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20862
C2×#19852
#152022
#113282
#113162
#112922
#112322
#112192
#108812