Finite Group Databasebeta

← #468736

Subgroups — #468736

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2057C212yesmaximal
2#15151C211yesmaximal, characteristic
3#14155C211yesmaximal, characteristic
4#11324C212yesmaximal
5#11315C212yesmaximal
6#11256C212yesmaximal
7#11232C212yesmaximal
8#11202C211yesmaximal, characteristic
9#10884C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#107C2212yes—
12C22×#107C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C2×#398—21no—
15C2×#398—21no—
16C2×#391—21no—
17C2×#391—21no—
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#2057—28no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32#1445—21no—
33#1445—21no—
34#1445—21no—
35#1445—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—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
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#398—44no—
72#398—44no—
73#391—44no—
74#391—44no—
75#389—24no—
76#389—24no—
77#384—41no—
78#384—41no—
79C4×C4⋊C4—41no—
80C4×C4⋊C4—41no—
81C4×C4⋊C4—41no—
82C4×C4⋊C4—41no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95D411yescharacteristic
94C2×#95D411yescharacteristic
95#340—24no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340Q818yescomplemented
101#336—24no—
102#336—24no—
103#336—41no—
104C22×Q8—24no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—42no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8C2×Q814yes—
135C22×C8#4114yes—
136C4⋊Q8—44no—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139C4⋊Q8—44no—
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—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4D814yes—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×Q814yes—
181C2×C4⋊C4C2×Q814yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42C2×D412yes—
198C2×C42#4112yes—
199#107—48no—
200#107—48no—
201C4⋊C8—44no—
202#95—41no—
203#95—41no—
204#95—42no—
205#95—44no—
206#95—42no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—42no—
212#95—44no—
213#95—44no—
214C24C22×D411yescharacteristic
215C2×Q8—48no—
216C2×Q8—48no—
217C2×Q8—48no—
218C2×Q8—84no—
219C2×Q8—84no—
220C2×Q8—84no—
221C2×Q8—84no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
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—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4C2×D812yes—
278C22×C4C2×D812yes—
279C22×C4C2×D812yes—
280C22×C4AGL(1,ℤ/8)12yes—
281C22×C4AGL(1,ℤ/8)12yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×Q812yes—
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4112yes—
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C4H(5,𝔽2)11yescharacteristic
289C22×C4H(5,𝔽2)11yescharacteristic
290C22×C42−1+412yes—
291C22×C42−1+412yes—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4—88no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—42no—
333C42—42no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—84no—
339C23C2×#12111yescharacteristic
340C23#53411yescharacteristic
341C23#53511yescharacteristic
342C23#54712yes—
343C23D4×Q812yes—
344C23C22×D811yescharacteristic
345C23C2×AGL(1,ℤ/8)11yescharacteristic
346C23C2×AGL(1,ℤ/8)11yescharacteristic
347C23#57511yescharacteristic
348C23#57812yes—
349C23#57812yes—
350Q8—816no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
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—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4#45914yes—
411C2×C4#53312yes—
412C2×C4#53411yescharacteristic
413C2×C4#53411yescharacteristic
414C2×C4#53811yescharacteristic
415C2×C4#53811yescharacteristic
416C2×C4#53912yes—
417C2×C4#54112yes—
418C2×C4#54212yes—
419C2×C4#54412yes—
420C2×C4#54412yes—
421C2×C4#54812yes—
422C2×C4D4×Q812yes—
423C2×C4#55111yescharacteristic
424C2×C4#55412yes—
425C2×C4Q8211yescharacteristic
426C2×C4#56311yescharacteristic
427C2×C4#56312yes—
428C2×C4#56311yescharacteristic
429C8—44no—
430C8—44no—
431C22#242912yes—
432C22#261711yescharacteristic
433C22#271711yescharacteristic
434C22#272012yes—
435C22#273411yescharacteristic
436C22C2×#45911yescharacteristic
437C22#303611yescharacteristic
438C22#304412yes—
439C22#318611yescharacteristic
440C22#319211yescharacteristic
441C22#320211yescharacteristic
442C22#320211yescharacteristic
443C22#320511yescharacteristic
444C22#321311yescharacteristic
445C22#322312yes—
446C22#333412yes—
447C22#334312yes—
448C22#334912yes—
449C22#335312yes—
450C22Q8×D812yes—
451C22#338612yes—
452C22#338612yes—
453C22#339012yes—
454C4—24no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—84no—
461C4—84no—
462C4—84no—
463C4—162no—
464C4#258611yescharacteristic
465C4#266612yes—
466C4#272811yescharacteristic
467C4#272912yes—
468C4#273411yescharacteristic
469C4#283311yescharacteristic
470C2#1662911yescharacteristic
471C2#2544912yes—
472C2#2546312yes—
473C2#2971211yescharacteristic
474C2#2972411yescharacteristic
475C2#3225811yescharacteristic
476C2#3227111yescharacteristic
477C2#3234212yes—
478C2#3236712yes—
479C2#3268211yescharacteristic
480C2#3269311yescharacteristic
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
100. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20572
#151512
#141552
#113242
#113152
#112562
#112322
#112022
#108842