Finite Group Databasebeta

← #415384

Subgroups — #415384

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3#14282C211yesmaximal, characteristic
4#13779C211yesmaximal, characteristic
5#11357C212yesmaximal
6#11316C212yesmaximal
7#11315C212yesmaximal
8#11268C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10850C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1988—24no—
22#1987—24no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1446—22no—
34#1445—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×Q8—21no—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8D414yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42—21no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69#384—22no—
70#384Q814yescomplemented
71#382—22no—
72#382—22no—
73#382—22no—
74#382—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79#376—22no—
80#376—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D412yes—
93C2×#95D412yes—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#340—42no—
102#336—24no—
103#336—42no—
104C22×Q8—24no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—82no—
108C22×Q8—82no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
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×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C4⋊Q8—28no—
135C4⋊Q8—28no—
136C4⋊Q8—44no—
137C4⋊Q8—44no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×Q812yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C4⋊C4#4112yes—
179C2×C4⋊C4#4112yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—82no—
193C2×C42—82no—
194C2×C42C2×D412yes—
195C2×C42C2×Q812yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202C4⋊C8—44no—
203C4⋊C8—44no—
204#95—28no—
205#95—24no—
206#95—24no—
207#95—42no—
208#95—42no—
209#95—42no—
210#95—42no—
211#95—82no—
212#95—82no—
213#95—82no—
214#95—82no—
215#95—82no—
216#95—82no—
217C24C22×D411yescharacteristic
218C2×Q8—48no—
219C2×Q8—84no—
220C2×Q8—84no—
221C2×Q8—88no—
222C2×Q8—88no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
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#12112yes—
275C22×C4#12112yes—
276C22×C4C22⋊Q814yes—
277C22×C4#12312yes—
278C22×C4#12312yes—
279C22×C4#12412yes—
280C22×C4#12412yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×Q811yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4H(5,𝔽2)11yescharacteristic
292C22×C42−1+411yescharacteristic
293C22×C42−1+411yescharacteristic
294C22×C42−1+411yescharacteristic
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—48no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—88no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—84no—
344C42—84no—
345C42#12314yes—
346C42#12314yes—
347C42#12614yes—
348C42#12614yes—
349C23C2×#12111yescharacteristic
350C23C2×C22⋊Q811yescharacteristic
351C23C2×#12311yescharacteristic
352C23#52511yescharacteristic
353C23C2×#12411yescharacteristic
354C23#53611yescharacteristic
355C23#53711yescharacteristic
356C23#57612yes—
357C23#57712yes—
358C23#57712yes—
359C23#57812yes—
360Q8—816no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
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—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4#39714yes—
418C2×C4#39714yes—
419C2×C4C2×#12311yescharacteristic
420C2×C4C2×#12311yescharacteristic
421C2×C4C2×#12611yescharacteristic
422C2×C4C2×#12611yescharacteristic
423C2×C4#52911yescharacteristic
424C2×C4#52911yescharacteristic
425C2×C4#53311yescharacteristic
426C2×C4#53311yescharacteristic
427C2×C4#54111yescharacteristic
428C2×C4#54111yescharacteristic
429C2×C4#54211yescharacteristic
430C2×C4#54211yescharacteristic
431C2×C4#54811yescharacteristic
432C2×C4#54811yescharacteristic
433C2×C4D4×Q811yescharacteristic
434C2×C4#55011yescharacteristic
435C2×C4#55211yescharacteristic
436C2×C4#55211yescharacteristic
437C2×C4#55411yescharacteristic
438C2×C4#55611yescharacteristic
439C2×C4#55611yescharacteristic
440C2×C4#55611yescharacteristic
441C2×C4#55611yescharacteristic
442C2×C4#55611yescharacteristic
443C2×C4#55711yescharacteristic
444C2×C4#55711yescharacteristic
445C2×C4#55911yescharacteristic
446C2×C4#55911yescharacteristic
447C8—82no—
448C8—82no—
449C22C2×#39711yescharacteristic
450C22#253611yescharacteristic
451C22#253811yescharacteristic
452C22#256811yescharacteristic
453C22#259211yescharacteristic
454C22#262511yescharacteristic
455C22#285511yescharacteristic
456C22#304111yescharacteristic
457C22#304211yescharacteristic
458C22#304311yescharacteristic
459C22#304411yescharacteristic
460C22#308112yes—
461C22#308212yes—
462C22#309612yes—
463C22#309712yes—
464C22#312412yes—
465C22#312512yes—
466C22#313612yes—
467C22#313712yes—
468C22#324312yes—
469C22#324412yes—
470C22#325712yes—
471C22#325812yes—
472C4—24no—
473C4—24no—
474C4—44no—
475C4—82no—
476C4—82no—
477C4—84no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4—162no—
482C4#244611yescharacteristic
483C4#244611yescharacteristic
484C4#244811yescharacteristic
485C4#244811yescharacteristic
486C4#251111yescharacteristic
487C4#251111yescharacteristic
488C4#251411yescharacteristic
489C4#251411yescharacteristic
490C2#1519211yescharacteristic
491C2#2363211yescharacteristic
492C2#2363311yescharacteristic
493C2#2821111yescharacteristic
494C2#2821211yescharacteristic
495C2#2829011yescharacteristic
496C2#2829111yescharacteristic
497C2#2922912yes—
498C2#2936312yes—
499C2#2994512yes—
500C2#3315912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
70. #384Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19872
#142822
#137792
#113572
#113162
#113152
#112682
#108502
#108332