Finite Group Databasebeta

← #316248

Subgroups — #316248

Order 512.

786 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27648C211yesmaximal, characteristic
2#24212C211yesmaximal, characteristic
3C8×C22⋊Q8C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5#10594C212yesmaximal, complemented
6#10478C212yesmaximal, complemented
7#9566C212yesmaximal, complemented
8#9218C212yesmaximal, complemented
9#8724C211yesmaximal, characteristic
10#8596C211yesmaximal, characteristic
11#8340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#484—21no—
14C2×#462C2212yes—
15C2×#451—21no—
16C2×C8×Q8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C4×C22⋊Q8C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2083—21no—
21#2072—21no—
22#2043—21no—
23#2016—21no—
24#1990—21no—
25#1990—21no—
26#1990C2212yes—
27#1990C2212yes—
28#1984C2211yescharacteristic
29#1981C2212yes—
30#1947C2212yes—
31#1939C2211yescharacteristic
32#1916—21no—
33#1850—21no—
34#1824—21no—
35#1803C2212yes—
36#1797C2211yescharacteristic
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1787—21no—
40C8×C4⋊C4—21no—
41C8×C4⋊C4C2211yescharacteristic
42C8×C22⋊C4—21no—
43C2×#340—21no—
44C2×#340—21no—
45C2×#335C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×Q8—21no—
52C22×Q16—22no—
53C23×C8D411yescharacteristic
54C23×C8D411yescharacteristic
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8C2312yes—
61C2×C22⋊Q8D411yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42C2311yescharacteristic
69#484—44no—
70#462—24no—
71#462—42no—
72#451—44no—
73C8×Q8—42no—
74C2×#107—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—42no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391D411yescharacteristic
96#389D411yescharacteristic
97#387—21no—
98#385—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384C2312yes—
103#380D411yescharacteristic
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2311yescharacteristic
106C4×C22⋊C4—21no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#335—42no—
121#332—24no—
122#332—42no—
123C22×Q8—22no—
124C22×Q8—44no—
125C22×Q8—41no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—82no—
129C22×Q8C2×D411yescharacteristic
130C23×C4—41no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C2×Q16—44no—
136C2×Q16—44no—
137C2×Q16—82no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8—41no—
147C22×C8Q1614yescomplemented
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—42no—
159C4⋊Q8#4114yes—
160C22⋊Q8—22no—
161C22⋊Q8D814yes—
162C4×Q8—22no—
163C4×Q8—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42C2411yescharacteristic
198#107—44no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—42no—
206C4⋊C8—42no—
207#103—44no—
208#103—44no—
209#103—44no—
210#103—82no—
211C22⋊C8—42no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—41no—
217C4×C8—41no—
218C4×C8—42no—
219C4×C8—42no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225C24#12111yescharacteristic
226C2×Q8—21no—
227C2×Q8—22no—
228C2×Q8—24no—
229C2×Q8—22no—
230C2×Q8—44no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—88no—
237C2×Q8—88no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8C2×D812yes—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4C2×D812yes—
266C22×C4C2×Q1612yes—
267C22×C4#13512yes—
268C22×C4#13712yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4111yescharacteristic
272Q16—88no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8C2×Q1612yes—
298C2×C8C2×Q1612yes—
299C4⋊C4—21no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—44no—
306C4⋊C4—44no—
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—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4C2×D812yes—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C42—21no—
323C42—21no—
324C42—22no—
325C42—22no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42C2×#4112yes—
331C42H(5,𝔽2)12yes—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23#46212yes—
337C23#46612yes—
338C23#47312yes—
339C23C2×#12111yescharacteristic
340Q8—24no—
341Q8—42no—
342Q8—44no—
343Q8—44no—
344Q8—88no—
345Q8—82no—
346Q8—82no—
347C2×C4—28no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—88no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—164no—
382C2×C4#46212yes—
383C2×C4#46612yes—
384C2×C4#47112yes—
385C2×C4#52511yescharacteristic
386C2×C4#53411yescharacteristic
387C2×C4#54612yes—
388C2×C4#55112yes—
389C2×C4C22×D811yescharacteristic
390C2×C4C22×Q1611yescharacteristic
391C2×C4C2×#13511yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C2×C4#57511yescharacteristic
394C2×C4#57511yescharacteristic
395C2×C4#57712yes—
396C8—22no—
397C8—22no—
398C8—41no—
399C8—41no—
400C8—81no—
401C8—81no—
402C8—82no—
403C8—82no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22#167414yes—
411C22#253411yescharacteristic
412C22C2×#46211yescharacteristic
413C22#304011yescharacteristic
414C22C2×#46611yescharacteristic
415C22#305711yescharacteristic
416C22C2×#47111yescharacteristic
417C22C2×#47311yescharacteristic
418C4—22no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—42no—
423C4—42no—
424C4—81no—
425C4—84no—
426C4—81no—
427C4—162no—
428C4#310411yescharacteristic
429C4#311711yescharacteristic
430C4#318911yescharacteristic
431C4#319611yescharacteristic
432C4#329412yes—
433C4#329612yes—
434C2—24no—
435C2C2×#167411yescharacteristic
436C2#1920711yescharacteristic
437C2#1927611yescharacteristic
438C2#1936311yescharacteristic
439C2#2809411yescharacteristic
440C2#2812111yescharacteristic
441C2#2813511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10594C2semidirect
6. #10478C2semidirect
7. #9566C2semidirect
8. #9218C2semidirect
147. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#276482
#242122
C8×C22⋊Q82
C2×#19902
#105942
#104782
#95662
#92182
#87242
#85962
#83402