Finite Group Databasebeta

← #404729

Subgroups — #404729

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic
2C2×#1918C211yesmaximal, characteristic
3#15143C211yesmaximal, characteristic
4#13810C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11322C211yesmaximal, characteristic
7#11267C212yesmaximal
8#11232C212yesmaximal
9#11227C211yesmaximal, characteristic
10#10849C212yesmaximal
11#10806C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C2×#395—21no—
17C2×#395—21no—
18C2×#389C2211yescharacteristic
19C2×#384C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×#376—21no—
23#1988—24no—
24#1918—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1444—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—22no—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42—22no—
63C22×C42C2311yescharacteristic
64C22×C42C2312yes—
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8—42no—
73#395—44no—
74#395—44no—
75#389—22no—
76#389—22no—
77#384—24no—
78#384—42no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4D414yes—
82C4×C4⋊C4Q818yescomplemented
83#376—44no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95D411yescharacteristic
94C2×#95D412yes—
95C2×#95D411yescharacteristic
96#340—22no—
97#340—22no—
98#340—22no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#336—22no—
106#336—24no—
107C22×Q8—44no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8—82no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2411yescharacteristic
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
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—21no—
138C22×C8—21no—
139C22×C8#4112yes—
140C4⋊Q8—28no—
141C4⋊Q8—48no—
142C4⋊Q8—44no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×Q814yes—
183C2×C4⋊C4C2×Q814yes—
184C2×C4⋊C4#4114yes—
185C2×C4⋊C4#4112yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42—28no—
194C2×C42—42no—
195C2×C42—44no—
196C2×C42—42no—
197C2×C42—44no—
198C2×C42C2×D412yes—
199C2×C42C2×Q814yes—
200C2×C42#4112yes—
201C2×C42#4114yes—
202#107—28no—
203C4⋊C8—48no—
204C4⋊C8—44no—
205#95—24no—
206#95—28no—
207#95—22no—
208#95—24no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—44no—
213#95—44no—
214#95—82no—
215#95—81no—
216#95D814yes—
217C24C22×D411yescharacteristic
218C2×Q8—48no—
219C2×Q8—88no—
220C2×Q8—88no—
221C2×Q8—84no—
222C2×Q8—84no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4#12112yes—
278C22×C4#12112yes—
279C22×C4C22⋊Q814yes—
280C22×C4C22⋊Q814yes—
281C22×C4C2×D812yes—
282C22×C4C2×D814yes—
283C22×C4AGL(1,ℤ/8)12yes—
284C22×C4#13714yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×Q812yes—
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4112yes—
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4112yes—
291C22×C4C2×#4111yescharacteristic
292C22×C4H(5,𝔽2)12yes—
293C22×C42−1+411yescharacteristic
294C22×C42−1+411yescharacteristic
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—48no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—88no—
335C4⋊C4—88no—
336C4⋊C4—84no—
337C4⋊C4C4⋊Q814yes—
338C42—22no—
339C42—24no—
340C42—22no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—48no—
346C42—416no—
347C23C2×#12111yescharacteristic
348C23C2×C22⋊Q812yes—
349C23#54612yes—
350C23#54711yescharacteristic
351C23#54811yescharacteristic
352C23C22×D811yescharacteristic
353C23C2×AGL(1,ℤ/8)11yescharacteristic
354C23C2×#13712yes—
355C23#57711yescharacteristic
356C23#57712yes—
357C23#57811yescharacteristic
358Q8—816no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4#45914yes—
420C2×C4#47418yes—
421C2×C4#52511yescharacteristic
422C2×C4#52511yescharacteristic
423C2×C4C2×C4⋊Q811yescharacteristic
424C2×C4#53211yescharacteristic
425C2×C4#53312yes—
426C2×C4#53312yes—
427C2×C4#53312yes—
428C2×C4#53511yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#54412yes—
431C2×C4#54412yes—
432C2×C4#55112yes—
433C2×C4#55112yes—
434C2×C4#55212yes—
435C2×C4#55211yescharacteristic
436C2×C4#55512yes—
437C2×C4#55612yes—
438C2×C4#55611yescharacteristic
439C8—42no—
440C8—81no—
441C8—82no—
442C22#241111yescharacteristic
443C22#241511yescharacteristic
444C22#243812yes—
445C22#243912yes—
446C22#285711yescharacteristic
447C22C2×#45911yescharacteristic
448C22#303911yescharacteristic
449C22#304311yescharacteristic
450C22#304611yescharacteristic
451C22C2×#47412yes—
452C22#307912yes—
453C22#308212yes—
454C22#308412yes—
455C22#329412yes—
456C22#331012yes—
457C22#331212yes—
458C22#332912yes—
459C22#333411yescharacteristic
460C22#334412yes—
461C22#335411yescharacteristic
462C22#335811yescharacteristic
463C22#336711yescharacteristic
464C22#336912yes—
465C4—28no—
466C4—24no—
467C4—44no—
468C4—44no—
469C4—42no—
470C4—82no—
471C4—82no—
472C4—84no—
473C4—84no—
474C4—161no—
475C4#239911yescharacteristic
476C4#260411yescharacteristic
477C4#260512yes—
478C4#261011yescharacteristic
479C4#261212yes—
480C4#266011yescharacteristic
481C2#1490111yescharacteristic
482C2#2447611yescharacteristic
483C2#2449811yescharacteristic
484C2#2467611yescharacteristic
485C2#2469811yescharacteristic
486C2#2598812yes—
487C2#2600912yes—
488C2#2602012yes—
489C2#2603812yes—
490C2#3319511yescharacteristic
491C2#3320411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
82. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19182
#151432
#138102
#113842
#113222
#112672
#112322
#112272
#108492
#108062