Finite Group Databasebeta

← #99523

Subgroups — #99523

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22799C211yesmaximal, characteristic, complemented
2C4×#459C211yesmaximal, characteristic
3#9674C211yesmaximal, characteristic, complemented
4#9638C211yesmaximal, characteristic, complemented
5C4×D4⋊C8C211yesmaximal, characteristic, complemented
6#7031C211yesmaximal, characteristic, complemented
7#7015C211yesmaximal, characteristic, complemented
8C2×#459C411yescharacteristic
9C2×C4×D8—21no—
10C2×#432—21no—
11C4×#127C2211yescharacteristic
12C42×D4C2211yescharacteristic
13#1982C411yescharacteristic
14#1954—21no—
15#1947C411yescharacteristic
16#1925—21no—
17#1924—22no—
18#1879—21no—
19#1842C2211yescharacteristic
20#1802C411yescharacteristic
21#1797—21no—
22#1770—21no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×#102—21no—
27C42×C8C2211yescharacteristic
28C2×D4⋊C8—22no—
29C2×D4⋊C8—21no—
30C2×D4⋊C8—21no—
31C2×D4⋊C8—21no—
32#1279—21no—
33C22×D8—21no—
34C2×#127C2×C411yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4—21no—
37C2×C4×D4—21no—
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C22×C42—21no—
42#459—22no—
43#459—22no—
44C4×D8—42no—
45C4×D8—42no—
46#432—42no—
47#432—42no—
48C2×#107—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8C2×C411yescharacteristic
53C2×C4⋊C8C2×C411yescharacteristic
54C2×C4⋊C8D411yescharacteristic
55C2×C4⋊C8D411yescharacteristic
56C2×C4⋊C8D412yes—
57C2×#102—21no—
58C2×#102—21no—
59C2×#102—21no—
60C2×#102—21no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×#97—22no—
64C2×C4×C8—21no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70#390—21no—
71#390—21no—
72#390—21no—
73#384C2×C411yescharacteristic
74C4×C4⋊C4C2×C411yescharacteristic
75C4×C4⋊C4D411yescharacteristic
76C4×C4⋊C4D411yescharacteristic
77C4×C22⋊C4—21no—
78C4×C22⋊C4—21no—
79C4×C22⋊C4—21no—
80C43C2311yescharacteristic
81#340—21no—
82#340—21no—
83#336—22no—
84#336—21no—
85#336—21no—
86#336—21no—
87D4⋊C8—24no—
88D4⋊C8—44no—
89D4⋊C8—42no—
90D4⋊C8—42no—
91C22×D4—21no—
92C22×D4—21no—
93C22×D4—21no—
94C22×D4C22⋊C411yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—41no—
99C2×D8—24no—
100C2×D8—42no—
101C2×D8—42no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8—41no—
109#127—21no—
110#127—21no—
111#127—21no—
112#127M1612yes—
113C4×D4—22no—
114C4×D4—21no—
115C4×D4—21no—
116C4×D4—21no—
117C4×D4—21no—
118C4×D4—21no—
119C4×D4—21no—
120C4×D4—21no—
121C4×D4—21no—
122C4×D4—41no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C22⋊C411yescharacteristic
138C2×C4⋊C4C22⋊C411yescharacteristic
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42C22×C411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159#107—42no—
160C4⋊C8—22no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—42no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8M1612yes—
174C4⋊C8D812yescomplemented
175#102—24no—
176#102—42no—
177#102—42no—
178#102—42no—
179C22⋊C8—44no—
180C22⋊C8—82no—
181#97—44no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—41no—
186C4×C8—41no—
187C4×C8—41no—
188C4×C8—41no—
189C4×C8—41no—
190C4×C8—41no—
191C4×C8—41no—
192C4×C8—41no—
193C4×C8—41no—
194#95—21no—
195#95—21no—
196#95—21no—
197C24—21no—
198C24—41no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#10412yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12311yescharacteristic
254D8—48no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—81no—
275C2×C8—81no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—82no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42C2×M1611yescharacteristic
311C42C2×M1611yescharacteristic
312C42#13111yescharacteristic
313C42#13111yescharacteristic
314C42C2×D811yescharacteristic
315C42#13511yescharacteristic
316C42#13511yescharacteristic
317C42#13511yescharacteristic
318C42AGL(1,ℤ/8)11yescharacteristic
319C42AGL(1,ℤ/8)11yescharacteristic
320C42AGL(1,ℤ/8)11yescharacteristic
321C42AGL(1,ℤ/8)11yescharacteristic
322C23—22no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—42no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23#38611yescharacteristic
337D4—22no—
338D4—22no—
339D4—44no—
340D4—42no—
341D4—42no—
342D4—41no—
343D4—42no—
344D4—42no—
345D4—41no—
346D4—82no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—161no—
383C2×C4—161no—
384C2×C4—161no—
385C2×C4—161no—
386C2×C4#40711yescharacteristic
387C2×C4#40811yescharacteristic
388C2×C4#41611yescharacteristic
389C2×C4#41811yescharacteristic
390C2×C4C2×#10411yescharacteristic
391C2×C4#42111yescharacteristic
392C2×C4#43511yescharacteristic
393C2×C4#43511yescharacteristic
394C2×C4#43611yescharacteristic
395C2×C4#43611yescharacteristic
396C2×C4C4×D811yescharacteristic
397C2×C4#44211yescharacteristic
398C2×C4#44311yescharacteristic
399C2×C4#44411yescharacteristic
400C2×C4#44711yescharacteristic
401C2×C4#44911yescharacteristic
402C2×C4#44911yescharacteristic
403C2×C4#44911yescharacteristic
404C2×C4#45311yescharacteristic
405C2×C4#45411yescharacteristic
406C2×C4#45911yescharacteristic
407C2×C4#46311yescharacteristic
408C2×C4#48211yescharacteristic
409C2×C4#48211yescharacteristic
410C8—44no—
411C8—42no—
412C8—42no—
413C8—82no—
414C8—82no—
415C8—81no—
416C8—81no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22#187011yescharacteristic
432C22#187111yescharacteristic
433C22#187511yescharacteristic
434C22#187611yescharacteristic
435C22#187911yescharacteristic
436C22#188111yescharacteristic
437C22#188311yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—41no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—161no—
449C4#148811yescharacteristic
450C4#149011yescharacteristic
451C4#149411yescharacteristic
452C4#149611yescharacteristic
453C4#158111yescharacteristic
454C4#158611yescharacteristic
455C4#158911yescharacteristic
456C4#159411yescharacteristic
457C4#162111yescharacteristic
458C4#162411yescharacteristic
459C4#162511yescharacteristic
460C4#162811yescharacteristic
461C2—82no—
462C2—162no—
463C2#894311yescharacteristic
464C2#895111yescharacteristic
465C2#895611yescharacteristic
466C2#896911yescharacteristic
467C2#898011yescharacteristic
468C2#899111yescharacteristic
469C2#899611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC23#386
prime radicalC2×#459C4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22799C2semidirect
3. #9674C2semidirect
4. #9638C2semidirect
5. C4×D4⋊C8C2semidirect
6. #7031C2semidirect
7. #7015C2semidirect
174. C4⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#227992
C4×#4592
#96742
#96382
C4×D4⋊C82
#70312
#70152