Finite Group Databasebeta

← #99873

Subgroups — #99873

Order 512.

572 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#459C211yesmaximal, characteristic
2#21264C211yesmaximal, characteristic, complemented
3#9638C211yesmaximal, characteristic, complemented
4#8563C211yesmaximal, characteristic, complemented
5#7569C211yesmaximal, characteristic, complemented
6#7031C211yesmaximal, characteristic, complemented
7#7015C211yesmaximal, characteristic, complemented
8C2×#459C411yescharacteristic
9C2×C4×D8—21no—
10C2×#431—21no—
11C4×#127C2211yescharacteristic
12C42×D4C2211yescharacteristic
13#1982C411yescharacteristic
14#1954—21no—
15#1947C411yescharacteristic
16#1939—21no—
17#1925—21no—
18#1923—22no—
19#1879—21no—
20#1842C2211yescharacteristic
21#1802C411yescharacteristic
22C4×C4⋊C8C2211yescharacteristic
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C4×#102—21no—
26C4×C22⋊C8—21no—
27C42×C8C2211yescharacteristic
28C2×D4⋊C8—21no—
29C2×D4⋊C8—21no—
30C2×D4⋊C8—21no—
31#1279—21no—
32#1270—22no—
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#431—42no—
47#431—42no—
48C2×#107—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C411yescharacteristic
56C2×C4⋊C8C2×C411yescharacteristic
57C2×#102—21no—
58C2×#102—21no—
59C2×#102—21no—
60C2×#102—21no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C4×C8—21no—
64C2×C4×C8—21no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
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#336D412yes—
87D4⋊C8—42no—
88D4⋊C8—42no—
89D4⋊C8—42no—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4—21no—
93C22×D4C22⋊C411yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—41no—
98C2×D8—24no—
99C2×D8—42no—
100C2×D8—42no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—41no—
107C22×C8—41no—
108#127—21no—
109#127—21no—
110#127—21no—
111#127M1612yes—
112C4×D4—21no—
113C4×D4—21no—
114C4×D4—21no—
115C4×D4—21no—
116C4×D4—21no—
117C4×D4—21no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121C4×D4—41no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—81no—
129C4×D4—81no—
130C4×D4D812yes—
131C4×D4SD1612yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C22⋊C411yescharacteristic
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158#107—42no—
159C4⋊C8—22no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—44no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—81no—
172C4⋊C8M1612yes—
173#102—24no—
174#102—42no—
175#102—42no—
176#102—42no—
177C22⋊C8—44no—
178C22⋊C8—44no—
179C4×C8—21no—
180C4×C8—21no—
181C4×C8—21no—
182C4×C8—41no—
183C4×C8—41no—
184C4×C8—42no—
185C4×C8—41no—
186C4×C8—41no—
187C4×C8—41no—
188C4×C8—41no—
189C4×C8—41no—
190C4×C8—41no—
191C4×C8D812yescomplemented
192#95—21no—
193#95—21no—
194#95—21no—
195C24—21no—
196C24—41no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—81no—
217C2×D4—81no—
218C2×D4#10212yes—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
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—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12311yescharacteristic
252D8—48no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—81no—
270C2×C8—81no—
271C2×C8—81no—
272C2×C8—81no—
273C2×C8—81no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4#10212yes—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—82no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42C2×M1611yescharacteristic
309C42C2×M1611yescharacteristic
310C42#13111yescharacteristic
311C42#13111yescharacteristic
312C42C2×D811yescharacteristic
313C42C2×D811yescharacteristic
314C42C2×SD1611yescharacteristic
315C42#13511yescharacteristic
316C42#13511yescharacteristic
317C42#13511yescharacteristic
318C42AGL(1,ℤ/8)11yescharacteristic
319C42AGL(1,ℤ/8)11yescharacteristic
320C23—22no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23#38611yescharacteristic
335D4—22no—
336D4—22no—
337D4—42no—
338D4—42no—
339D4—41no—
340D4—42no—
341D4—41no—
342D4—42no—
343D4—82no—
344D4—82no—
345D4—81no—
346D4—81no—
347D4—81no—
348D4—81no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—161no—
381C2×C4—161no—
382C2×C4—161no—
383C2×C4—161no—
384C2×C4#40711yescharacteristic
385C2×C4#40811yescharacteristic
386C2×C4C2×#10211yescharacteristic
387C2×C4#41611yescharacteristic
388C2×C4#42111yescharacteristic
389C2×C4#42111yescharacteristic
390C2×C4#43511yescharacteristic
391C2×C4#43511yescharacteristic
392C2×C4#43611yescharacteristic
393C2×C4#43611yescharacteristic
394C2×C4C4×D811yescharacteristic
395C2×C4#44211yescharacteristic
396C2×C4#44311yescharacteristic
397C2×C4#44411yescharacteristic
398C2×C4#44711yescharacteristic
399C2×C4#44911yescharacteristic
400C2×C4#45911yescharacteristic
401C2×C4#46311yescharacteristic
402C2×C4#46511yescharacteristic
403C2×C4#46511yescharacteristic
404C2×C4#46611yescharacteristic
405C2×C4#46611yescharacteristic
406C2×C4#47111yescharacteristic
407C2×C4#47211yescharacteristic
408C8—42no—
409C8—42no—
410C8—42no—
411C8—82no—
412C8—82no—
413C8—81no—
414C8—161no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22#187011yescharacteristic
430C22#187611yescharacteristic
431C22#187911yescharacteristic
432C22#189311yescharacteristic
433C22#189711yescharacteristic
434C22#190611yescharacteristic
435C22#190711yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—42no—
443C4—41no—
444C4—81no—
445C4—81no—
446C4—161no—
447C4#148011yescharacteristic
448C4#148211yescharacteristic
449C4#148611yescharacteristic
450C4#148811yescharacteristic
451C4#158111yescharacteristic
452C4#158611yescharacteristic
453C4#158911yescharacteristic
454C4#159411yescharacteristic
455C4#166511yescharacteristic
456C4#166711yescharacteristic
457C4#167711yescharacteristic
458C4#168111yescharacteristic
459C2—82no—
460C2—162no—
461C2#895111yescharacteristic
462C2#907611yescharacteristic
463C2#909711yescharacteristic
464C2#920811yescharacteristic
465C2#921311yescharacteristic
466C2#922111yescharacteristic
467C2#923711yescharacteristic
subgroupG/H
centerC22×C4#121
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
2. #21264C2semidirect
3. #9638C2semidirect
4. #8563C2semidirect
5. #7569C2semidirect
6. #7031C2semidirect
7. #7015C2semidirect
191. C4×C8D8semidirect

Maximal subgroups

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

subgroupindex
C4×#4592
#212642
#96382
#85632
#75692
#70312
#70152