Finite Group Databasebeta

← #381963

Subgroups — #381963

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic
2C2×#1942C211yesmaximal, characteristic, complemented
3C2×#1750C211yesmaximal, characteristic
4#14264C212yesmaximal
5#11379C211yesmaximal, characteristic, complemented
6#11328C211yesmaximal, characteristic, complemented
7#11293C212yesmaximal, complemented
8#11200C212yesmaximal
9#11096C211yesmaximal, characteristic
10#10881C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23#2075C418yescomplemented
24#1942—24no—
25#1750—24no—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1444—22no—
37#1442—22no—
38#1439—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C24×C4C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—42no—
50C2×#117—42no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D411yescharacteristic
62C22×C22⋊C4D411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—42no—
69C2×#107C2×C414yes—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8C2×C414yes—
76C2×#97—22no—
77C2×#97—42no—
78C2×C4×C8—42no—
79C2×C4×C8D414yes—
80#389—24no—
81#389—24no—
82#389—42no—
83#389—42no—
84#380—24no—
85#380—42no—
86#380—42no—
87#380C2×C418yes—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96#340—24no—
97#340—24no—
98#340—22no—
99#340—42no—
100#340—42no—
101#340—42no—
102#340C2×C414yes—
103#340C2×C418yes—
104#336—22no—
105#336—22no—
106C25C2×D411yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4C22×C412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4#4112yes—
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C22×C412yes—
142C22×C8C22×C412yes—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C22×C8#4112yes—
146C22×C8#4112yes—
147#117—44no—
148#117—44no—
149#117—48no—
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—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—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C22×C414yes—
183C2×C4⋊C4C22×C414yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42C2×D412yes—
201C2×C42#4112yes—
202#107—28no—
203#107—44no—
204C22⋊C8—28no—
205C22⋊C8—28no—
206#97—44no—
207C4×C8—44no—
208#95—28no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—82no—
213#95—82no—
214#95—82no—
215#95—82no—
216C24—21no—
217C24—21no—
218C24—22no—
219C24—22no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—21no—
226C24C4×D412yes—
227C24C22×D411yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
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—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
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—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C4×D412yes—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4H(5,𝔽2)12yes—
306C22×C4H(5,𝔽2)12yes—
307C22×C42−1+412yes—
308C22×C42−1+412yes—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8C4×D414yes—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—88no—
348C4⋊C4—88no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—88no—
352C22⋊C4—88no—
353C42—22no—
354C42—22no—
355C42—24no—
356C42—41no—
357C42—41no—
358C42—42no—
359C23—24no—
360C23—24no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23C2×C4×D411yescharacteristic
383C23C2×#12111yescharacteristic
384C23#53211yescharacteristic
385C23#53412yes—
386C23#53612yes—
387C23#57611yescharacteristic
388C23#57611yescharacteristic
389C23#57611yescharacteristic
390C23#57712yes—
391C23#57811yescharacteristic
392C23#57812yes—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C4×#4111yescharacteristic
454C2×C4#52012yes—
455C2×C4#52012yes—
456C2×C4#52511yescharacteristic
457C2×C4#52911yescharacteristic
458C2×C4#53411yescharacteristic
459C2×C4#53511yescharacteristic
460C2×C4#53611yescharacteristic
461C2×C4#53611yescharacteristic
462C2×C4#53812yes—
463C2×C4#53912yes—
464C2×C4#53912yes—
465C2×C4#53912yes—
466C2×C4#53912yes—
467C2×C4#54112yes—
468C2×C4#54212yes—
469C2×C4#54212yes—
470C8—24no—
471C8—42no—
472C8—42no—
473C8—42no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—44no—
478C22—42no—
479C22—81no—
480C22—81no—
481C22—82no—
482C22—82no—
483C22—81no—
484C22—81no—
485C22C4×#12111yescharacteristic
486C22#232212yes—
487C22#261711yescharacteristic
488C22#262111yescharacteristic
489C22#266312yes—
490C22#294712yes—
491C22#294811yescharacteristic
492C22#295111yescharacteristic
493C22#305811yescharacteristic
494C22#305911yescharacteristic
495C22#306011yescharacteristic
496C22#306111yescharacteristic
497C22#317511yescharacteristic
498C22#317611yescharacteristic
499C22#317812yes—
500C22#319812yes—
501C22#319812yes—
502C22#320212yes—
503C22#320312yes—
504C22#324112yes—
505C22#324212yes—
506C22#324312yes—
507C22#324312yes—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—84no—
512C4—81no—
513C4—81no—
514C4—84no—
515C4—162no—
516C4—162no—
517C4#232211yescharacteristic
518C4#232611yescharacteristic
519C4#233412yes—
520C4#263311yescharacteristic
521C4#263811yescharacteristic
522C4#265212yes—
523C2—82no—
524C2#1431311yescharacteristic
525C2#2152411yescharacteristic
526C2#2153711yescharacteristic
527C2#2209312yes—
528C2#2209412yes—
529C2#2973611yescharacteristic
530C2#2973911yescharacteristic
531C2#2988011yescharacteristic
532C2#2988211yescharacteristic
533C2#3036912yes—
534C2#3037012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1942C2semidirect
5. #11379C2semidirect
6. #11328C2semidirect
7. #11293C2semidirect
10. #10881C2semidirect
11. #10820C2semidirect
23. #2075C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
C2×#19422
C2×#17502
#142642
#113792
#113282
#112932
#112002
#110962
#108812
#108202