Finite Group Databasebeta

← #380202

Subgroups — #380202

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#13797C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11315C212yesmaximal
8#11257C212yesmaximal
9#11256C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1988—24no—
22#1987C418yescomplemented
23#1949—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1446—21no—
35#1446—21no—
36#1445—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41#1434—22no—
42#1434—21no—
43#1434—21no—
44C23×Q8—21no—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42—21no—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×#107—22no—
66C2×#107—42no—
67C2×#106—22no—
68C2×#106—42no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2×C414yes—
75#382—22no—
76#382—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4C2×C414yes—
80C4×C4⋊C4D414yes—
81#376—22no—
82#376—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95D412yes—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#340C2×C418yes—
105#340C2×C418yes—
106#336—42no—
107#336—42no—
108C22×Q8—24no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8—82no—
112C22×Q8—82no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
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—22no—
138C22×C8—22no—
139C22×C8C22×C412yes—
140C22×C8C22×C412yes—
141C4⋊Q8—28no—
142C4⋊Q8—28no—
143C4⋊Q8—44no—
144C4⋊Q8—44no—
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—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C22×C414yes—
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—82no—
185C2×C42—81no—
186C2×C42—81no—
187C2×C42C22×C412yes—
188C2×C42C2×D412yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193#107—44no—
194#106—44no—
195C4⋊C8—28no—
196C4⋊C8—44no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—22no—
201#95—22no—
202#95—42no—
203#95—42no—
204#95—82no—
205#95—82no—
206#95—82no—
207#95—82no—
208#95—82no—
209#95—82no—
210#95—82no—
211#95—82no—
212C24C22×D411yescharacteristic
213C2×Q8—48no—
214C2×Q8—88no—
215C2×Q8—88no—
216C2×Q8—84no—
217C2×Q8—84no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
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—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4C4×D412yes—
273C22×C4C4×D412yes—
274C22×C4#12112yes—
275C22×C4#12112yes—
276C22×C4#12312yes—
277C22×C4#12312yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4H(5,𝔽2)12yes—
285C22×C42−1+412yes—
286C22×C42−1+412yes—
287C22×C42−1+412yes—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
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—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—48no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—84no—
331C42—84no—
332C42#11714yes—
333C42C4×D414yes—
334C42#12414yes—
335C42#12614yes—
336C23C2×C4×D411yescharacteristic
337C23C2×#12111yescharacteristic
338C23C2×#12311yescharacteristic
339C23#53612yes—
340C23#53712yes—
341C23#57612yes—
342C23#57712yes—
343C23#57711yescharacteristic
344C23#57711yescharacteristic
345C23#57811yescharacteristic
346C23#57811yescharacteristic
347Q8—816no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
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#38814yes—
407C2×C4#38814yes—
408C2×C4C2×#11711yescharacteristic
409C2×C4C2×C4×D411yescharacteristic
410C2×C4#51912yes—
411C2×C4#52012yes—
412C2×C4C2×#12411yescharacteristic
413C2×C4C2×#12611yescharacteristic
414C2×C4#53511yescharacteristic
415C2×C4#53611yescharacteristic
416C2×C4#53611yescharacteristic
417C2×C4#53611yescharacteristic
418C2×C4#54112yes—
419C2×C4#54112yes—
420C2×C4#54112yes—
421C2×C4#54112yes—
422C2×C4#54212yes—
423C2×C4#54212yes—
424C2×C4#54212yes—
425C2×C4#54212yes—
426C8—42no—
427C8—42no—
428C8—82no—
429C22C2×#38811yescharacteristic
430C22#232312yes—
431C22#262011yescharacteristic
432C22#262111yescharacteristic
433C22#262512yes—
434C22#294612yes—
435C22#294911yescharacteristic
436C22#295011yescharacteristic
437C22#304111yescharacteristic
438C22#304311yescharacteristic
439C22#304611yescharacteristic
440C22#304611yescharacteristic
441C22#309612yes—
442C22#309711yescharacteristic
443C22#309711yescharacteristic
444C22#324112yes—
445C22#324212yes—
446C22#324312yes—
447C22#324412yes—
448C22#325712yes—
449C22#325812yes—
450C22#326212yes—
451C22#326312yes—
452C4—24no—
453C4—24no—
454C4—84no—
455C4—84no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—161no—
461C4—161no—
462C4#232012yes—
463C4#232611yescharacteristic
464C4#232711yescharacteristic
465C4#264011yescharacteristic
466C4#264211yescharacteristic
467C4#264512yes—
468C2#1426111yescharacteristic
469C2#2106311yescharacteristic
470C2#2106611yescharacteristic
471C2#2212312yes—
472C2#2212512yes—
473C2#2983911yescharacteristic
474C2#2984111yescharacteristic
475C2#2987211yescharacteristic
476C2#2987311yescharacteristic
477C2#2994512yes—
478C2#2994712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
22. #1987C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19872
C2×#19492
#142822
#137972
#113162
#113152
#112572
#112562
#108502
#108332