Finite Group Databasebeta

← #380612

Subgroups — #380612

Order 512.

2052 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2657C211yesmaximal, characteristic
2C2×#1961C211yesmaximal, characteristic, complemented
3C2×#1959C211yesmaximal, characteristic, complemented
4C2×#1958C211yesmaximal, characteristic, complemented
5C2×#1957C211yesmaximal, characteristic, complemented
6C2×#1746C211yesmaximal, characteristic, complemented
7#14348C211yesmaximal, characteristic, complemented
8#11009C214yesmaximal, complemented
9#11008C214yesmaximal, complemented
10C22×#127C2211yescharacteristic
11C22×#127C2212yes—
12C22×#124C2211yescharacteristic
13C22×#124C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2214yes—
19C22×#102C2214yes—
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2657—24no—
23#2657C418yescomplemented
24C2×#392—24no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#1961—24no—
29#1959—24no—
30#1958—24no—
31#1957—24no—
32#1746—24no—
33C2×#336C2214yes—
34#1442—24no—
35C4⋊#95C2214yes—
36C23×Q8C2311yescharacteristic
37C23×D4—24no—
38C23×D4—22no—
39C23×D4C2311yescharacteristic
40C23×D4C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#127—22no—
44C2×#127—22no—
45C2×#127—24no—
46C2×#127—22no—
47C2×#127—22no—
48C2×#127—22no—
49C2×#127—44no—
50C2×#127—44no—
51C2×#127C2×C414yes—
52C2×#127C2×C418yes—
53C2×#124—24no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—44no—
60C2×#124—44no—
61C2×#124C2×C414yes—
62C2×#124C2×C418yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—22no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—44no—
73C2×#103—44no—
74C2×#102—28no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—28no—
78C2×#102—44no—
79C2×#102—44no—
80C2×#102—44no—
81C2×#102—44no—
82C2×#102—44no—
83C2×#102—44no—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97D414yescomplemented
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8D414yescomplemented
90#392—232no—
91#384—22no—
92#384—22no—
93#382—22no—
94#382—22no—
95#376C2×C414yes—
96#376D414yes—
97C2×#95—22no—
98C2×#95—28no—
99C2×#95D414yes—
100#336—28no—
101C25—21no—
102C25—22no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—24no—
109C22×Q8—24no—
110C22×Q8C22×C412yes—
111C22×D4—216no—
112C22×D4—28no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—48no—
130C22×D4—48no—
131C22×D4—44no—
132C22×D4—44no—
133C22×D4—84no—
134C22×D4—84no—
135C22×D4—88no—
136C22×D4—88no—
137C22×D4C22×C412yes—
138C22×D4C22×C414yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D414yes—
143C23×C4C2×D412yes—
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—24no—
150C22×C8—24no—
151C22×C8—24no—
152C22×C8—24no—
153C22×C8C2×D414yes—
154C22×C8C2×D414yes—
155#127—28no—
156#127—28no—
157#127—416no—
158#127—416no—
159#127—44no—
160#127—44no—
161#124—28no—
162#124—28no—
163#124—416no—
164#124—416no—
165#124—44no—
166#124—44no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—84no—
177C2×C22⋊C4—216no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—48no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42C22×C414yes—
186C2×C42C22×C412yes—
187C2×C42C2×D414yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193#103—48no—
194#103—48no—
195#102—416no—
196#102—416no—
197#102—48no—
198#102—48no—
199#97—28no—
200C4×C8—28no—
201#95—44no—
202#95—88no—
203#95C22⋊C4116yes—
204C24—24no—
205C24—22no—
206C24—41no—
207C24—42no—
208C24—41no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24—42no—
216C24—48no—
217C24—44no—
218C24—48no—
219C24—44no—
220C24C22×D411yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—28no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—44no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—28no—
240C2×D4—432no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—416no—
263C2×D4—816no—
264C2×D4—832no—
265C2×D4—816no—
266C2×D4—88no—
267C2×D4—88no—
268C2×D4—816no—
269C2×D4—816no—
270C2×D4—832no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—44no—
289C22×C4C2×C22⋊C418yes—
290C22×C4C2×C22⋊C414yes—
291C22×C4#12414yes—
292C22×C4#12714yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4H(5,𝔽2)14yes—
300C22×C4H(5,𝔽2)14yes—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—88no—
318C22⋊C4—432no—
319C22⋊C4—416no—
320C42—28no—
321C42—28no—
322C42C4×D414yes—
323C42C4×D414yes—
324C42#12114yes—
325C42#12114yes—
326C23—44no—
327C23—44no—
328C23—44no—
329C23—48no—
330C23—416no—
331C23—42no—
332C23—44no—
333C23—88no—
334C23—84no—
335C23—82no—
336C23—82no—
337C23—82no—
338C23—84no—
339C23—82no—
340C23—84no—
341C23—82no—
342C23—81no—
343C23—82no—
344C23—81no—
345C23—82no—
346C23—84no—
347C23—84no—
348C23—84no—
349C23—84no—
350C23—84no—
351C23—84no—
352C23—88no—
353C23—88no—
354C23—88no—
355C23C22×C22⋊C411yescharacteristic
356C23C2×#12411yescharacteristic
357C23C2×#12711yescharacteristic
358C23#53512yes—
359C23#53512yes—
360C23#57614yes—
361C23#57714yes—
362Q8—44no—
363Q8—44no—
364Q8—82no—
365Q8—82no—
366D4—44no—
367D4—44no—
368D4—48no—
369D4—48no—
370D4—832no—
371D4—864no—
372D4—82no—
373D4—82no—
374D4—84no—
375D4—84no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—28no—
381C2×C4—216no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—88no—
396C2×C4—88no—
397C2×C4#39018yes—
398C2×C4C2×C4×D411yescharacteristic
399C2×C4C2×C4×D411yescharacteristic
400C2×C4#51814yes—
401C2×C4C2×#12111yescharacteristic
402C2×C4C2×#12111yescharacteristic
403C2×C4#53512yes—
404C2×C4#53512yes—
405C2×C4#53512yes—
406C2×C4#53512yes—
407C2×C4#53514yes—
408C2×C4#54014yes—
409C2×C4#54014yes—
410C8—44no—
411C8—44no—
412C22—88no—
413C22—88no—
414C22—88no—
415C22—84no—
416C22—82no—
417C22—88no—
418C22—84no—
419C22—164no—
420C22—164no—
421C22—164no—
422C22—162no—
423C22—161no—
424C22—162no—
425C22—161no—
426C22—164no—
427C22—162no—
428C22C2×#39011yescharacteristic
429C22#230912yes—
430C22#264112yes—
431C22#265711yescharacteristic
432C22#265711yescharacteristic
433C22#289714yes—
434C22#313614yes—
435C22#315412yes—
436C22#315512yes—
437C22#322014yes—
438C22#322214yes—
439C22#322418yes—
440C4—28no—
441C4—48no—
442C4—82no—
443C4—162no—
444C4—162no—
445C4#232812yes—
446C4#232812yes—
447C4#261912yes—
448C4#261912yes—
449C2—162no—
450C2—164no—
451C2#1427211yescharacteristic
452C2#2120612yes—
453C2#2181314yes—
454C2#3022214yes—
455C2#3027011yescharacteristic
456C2#3027111yescharacteristic
457C2#3027412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2657C2
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
2. C2×#1961C2semidirect
3. C2×#1959C2semidirect
4. C2×#1958C2semidirect
5. C2×#1957C2semidirect
6. C2×#1746C2semidirect
7. #14348C2semidirect
8. #11009C2semidirect
9. #11008C2semidirect
23. #2657C4semidirect
86. C2×#97D4semidirect
89. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26572
C2×#19612
C2×#19592
C2×#19582
C2×#19572
C2×#17462
#143482
#110092
#110082