Finite Group Databasebeta

← #367397

Subgroups — #367397

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1991C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4#14160C211yesmaximal, characteristic
5#13955C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11315C212yesmaximal
8#10850C211yesmaximal, characteristic
9#10820C212yesmaximal
10#10820C212yesmaximal
11C4×#340C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4×C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1991—24no—
21#1989C418yescomplemented
22C8×C4⋊C4—24no—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1442—21no—
35#1442—21no—
36#1441—21no—
37#1441—21no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43#1434—22no—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125C2×C414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C42—21no—
64C22×C42—21no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8C2×C414yes—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8—24no—
73C2×C4×C8—24no—
74C2×C4×C8—24no—
75C2×C4×C8D414yes—
76#382—22no—
77#382—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4C2×C414yes—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95D412yes—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340C2×C418yes—
106#340C2×C418yes—
107#336—24no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8C22×C414yes—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8#4114yes—
136#125—28no—
137#125—28no—
138#125—44no—
139#125—44no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
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—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—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C414yes—
181C2×C4⋊C4#4114yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—42no—
190C2×C42—81no—
191C2×C42—81no—
192C2×C42—81no—
193C2×C42—81no—
194C2×C42C22×C412yes—
195C2×C42C2×D412yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C4⋊C8—28no—
201C4×C8—28no—
202C4×C8—48no—
203#95—24no—
204#95—24no—
205#95—42no—
206#95—42no—
207#95—42no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—82no—
213#95—82no—
214#95—82no—
215#95—82no—
216C24C22×D411yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
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—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
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—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12112yes—
274C22×C4#12112yes—
275C22×C4#12412yes—
276C22×C4#12412yes—
277C22×C4#13514yes—
278C22×C4#13514yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4112yes—
285C22×C4C2×#4112yes—
286C22×C4C2×#4111yescharacteristic
287C22×C42−1+412yes—
288C22×C42−1+412yes—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8#12414yes—
306C4⋊C4—28no—
307C4⋊C4—28no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—84no—
337C42—84no—
338C42#11714yes—
339C42C4×D414yes—
340C42#12314yes—
341C42#12614yes—
342C23C2×C4×D411yescharacteristic
343C23C2×#12111yescharacteristic
344C23C2×#12411yescharacteristic
345C23#54712yes—
346C23#54812yes—
347C23C2×#13512yes—
348C23C2×#13512yes—
349C23#57611yescharacteristic
350C23#57711yescharacteristic
351C23#57711yescharacteristic
352C23#57811yescharacteristic
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
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—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4#38814yes—
414C2×C4#38814yes—
415C2×C4C2×#11711yescharacteristic
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C4×#4112yes—
418C2×C4#51912yes—
419C2×C4C2×#12311yescharacteristic
420C2×C4C2×#12411yescharacteristic
421C2×C4C2×#12611yescharacteristic
422C2×C4#52912yes—
423C2×C4#53211yescharacteristic
424C2×C4#53611yescharacteristic
425C2×C4#53611yescharacteristic
426C2×C4#54112yes—
427C2×C4#55212yes—
428C2×C4#55212yes—
429C2×C4#55212yes—
430C2×C4#55612yes—
431C2×C4#55612yes—
432C2×C4#55612yes—
433C8—24no—
434C8—42no—
435C8—44no—
436C22C2×#38811yescharacteristic
437C22#236212yes—
438C22#248912yes—
439C22#254511yescharacteristic
440C22#254711yescharacteristic
441C22C4×#13512yes—
442C22#294611yescharacteristic
443C22#294711yescharacteristic
444C22#305011yescharacteristic
445C22#305011yescharacteristic
446C22#306011yescharacteristic
447C22#306111yescharacteristic
448C22#313112yes—
449C22#313611yescharacteristic
450C22#313711yescharacteristic
451C22#333612yes—
452C22#333712yes—
453C22#333712yes—
454C22#333912yes—
455C22#336012yes—
456C22#336112yes—
457C22#336412yes—
458C22#336512yes—
459C4—24no—
460C4—24no—
461C4—44no—
462C4—44no—
463C4—84no—
464C4—84no—
465C4—82no—
466C4—82no—
467C4—161no—
468C4—161no—
469C4C4×#12411yescharacteristic
470C4#232711yescharacteristic
471C4#235612yes—
472C4#249911yescharacteristic
473C4#250311yescharacteristic
474C4#251412yes—
475C2#1400311yescharacteristic
476C2#2108311yescharacteristic
477C2#2109211yescharacteristic
478C2#2306412yes—
479C2#2306512yes—
480C2#2773112yes—
481C2#2773212yes—
482C2#2864111yescharacteristic
483C2#2864211yescharacteristic
484C2#2870711yescharacteristic
485C2#2870811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
21. #1989C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19912
C2×#19892
C2×C8×C4⋊C42
#141602
#139552
#113162
#113152
#108502
#108202
C4×#3402