Finite Group Databasebeta

← #218186

Subgroups — #218186

Order 512.

555 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23778C211yesmaximal, characteristic
2C8×#125C211yesmaximal, characteristic, complemented
3#21266C211yesmaximal, characteristic
4#9867C211yesmaximal, characteristic, complemented
5#9606C211yesmaximal, characteristic, complemented
6#9596C211yesmaximal, characteristic, complemented
7#9595C211yesmaximal, characteristic, complemented
8#9556C211yesmaximal, characteristic
9#9556C211yesmaximal, characteristic
10#9542C211yesmaximal, characteristic
11#9541C211yesmaximal, characteristic
12#9534C211yesmaximal, characteristic, complemented
13#9533C211yesmaximal, characteristic, complemented
14#8569C211yesmaximal, characteristic
15#8261C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17C4×#125C2211yescharacteristic
18#2293C2211yescharacteristic
19#2101C411yescharacteristic
20#2092C411yescharacteristic
21#2088C411yescharacteristic
22#2087C411yescharacteristic
23#2067C411yescharacteristic
24#2065C411yescharacteristic
25#2059C411yescharacteristic
26#2025C411yescharacteristic
27#1954C2211yescharacteristic
28#1954C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1923—21no—
34#1923—21no—
35#1916—21no—
36#1916—21no—
37#1831C2211yescharacteristic
38C4×#107C2211yescharacteristic
39C4×#106C2211yescharacteristic
40C8×C4⋊C4—21no—
41C8×C4⋊C4—21no—
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1767C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1750—21no—
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×#125C2×C411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70#433—44no—
71C2×#107C2×C411yescharacteristic
72C2×#107C2×C411yescharacteristic
73C2×#106C2×C411yescharacteristic
74C2×#106C2×C411yescharacteristic
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2×C411yescharacteristic
81C2×C4⋊C8C2×C411yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×#102C2×C411yescharacteristic
87C2×#102C2×C411yescharacteristic
88C2×#102C2×C411yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8D411yescharacteristic
100C2×C4×C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#390C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#389C2×C411yescharacteristic
110#389C2×C411yescharacteristic
111#389C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#388—21no—
114#388—21no—
115#387—21no—
116#387—21no—
117#382C2×C411yescharacteristic
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C22⋊C4—21no—
124C43C2311yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336C2×C411yescharacteristic
144#336C2×C411yescharacteristic
145#335—41no—
146#334—41no—
147D4⋊C8—41no—
148D4⋊C8—41no—
149#322—41no—
150#322—41no—
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C22×C411yescharacteristic
160C22×C8C22×C411yescharacteristic
161C22×C8C22×C411yescharacteristic
162C22×C8C22×C411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167#125—22no—
168#125—22no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173#117—42no—
174#117—42no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42C2411yescharacteristic
203#107—22no—
204#107—22no—
205#106—22no—
206#106—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—42no—
218C4⋊C8—42no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221C4⋊C8—42no—
222#102—22no—
223#102—22no—
224#102—22no—
225#102—22no—
226C22⋊C8—42no—
227C22⋊C8—42no—
228C22⋊C8—42no—
229C22⋊C8—42no—
230#97—42no—
231C4×C8—22no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—41no—
245C4×C8—41no—
246C4×C8—41no—
247C4×C8—42no—
248C4×C8—42no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258C24—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8#13112yes—
325C2×C8#13112yes—
326C2×C8AGL(1,ℤ/8)12yes—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4#13112yes—
346C4⋊C4#13112yes—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—42no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—22no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42H(5,𝔽2)11yescharacteristic
370C422−1+411yescharacteristic
371C422−1+411yescharacteristic
372C422−1+411yescharacteristic
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23C2×C4×D411yescharacteristic
381D4—41no—
382D4—41no—
383D4—41no—
384D4—41no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4#44412yes—
424C2×C4#51911yescharacteristic
425C2×C4#52011yescharacteristic
426C2×C4#53611yescharacteristic
427C2×C4#53711yescharacteristic
428C2×C4#54111yescharacteristic
429C2×C4#54211yescharacteristic
430C2×C4C2×#13111yescharacteristic
431C2×C4C2×#13111yescharacteristic
432C2×C4C2×#13111yescharacteristic
433C2×C4C2×#13111yescharacteristic
434C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
435C2×C4#57511yescharacteristic
436C2×C4#57511yescharacteristic
437C2×C4#57511yescharacteristic
438C8—22no—
439C8—22no—
440C8—22no—
441C8—41no—
442C8—41no—
443C8—41no—
444C8—41no—
445C8—41no—
446C8—81no—
447C8—81no—
448C8—81no—
449C8—81no—
450C8—81no—
451C8—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22#232311yescharacteristic
460C22#292911yescharacteristic
461C22#293111yescharacteristic
462C22#294311yescharacteristic
463C22C4×AGL(1,ℤ/8)11yescharacteristic
464C22C2×#44411yescharacteristic
465C22#295511yescharacteristic
466C4—22no—
467C4—21no—
468C4—21no—
469C4—41no—
470C4—41no—
471C4—42no—
472C4—41no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4#298611yescharacteristic
480C4#298611yescharacteristic
481C4#299411yescharacteristic
482C4#299411yescharacteristic
483C4#323411yescharacteristic
484C4#324011yescharacteristic
485C4#325111yescharacteristic
486C4#325611yescharacteristic
487C2—161no—
488C2#1790911yescharacteristic
489C2#1791711yescharacteristic
490C2#1796311yescharacteristic
491C2#1796411yescharacteristic
492C2#2209911yescharacteristic
493C2#2212011yescharacteristic
494C2#2213111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C8×#125C2semidirect
4. #9867C2semidirect
5. #9606C2semidirect
6. #9596C2semidirect
7. #9595C2semidirect
12. #9534C2semidirect
13. #9533C2semidirect
15. #8261C2semidirect

Maximal subgroups

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

subgroupindex
#237782
C8×#1252
#212662
#98672
#96062
#95962
#95952
#95562
#95422
#95412
#95342
#95332
#85692
#82612