Finite Group Databasebeta

← #499575

Subgroups — #499575

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#466C211yesmaximal, characteristic
2#20818C211yesmaximal, characteristic, complemented
3#13539C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic, complemented
7#13493C211yesmaximal, characteristic, complemented
8#13482C211yesmaximal, characteristic, complemented
9#13381C211yesmaximal, characteristic
10#13380C211yesmaximal, characteristic
11C4×#367C211yesmaximal, characteristic, complemented
12C4×#366C211yesmaximal, characteristic, complemented
13C4×#357C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic, complemented
15C4×C22⋊C16C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C4×#121—21no—
20#2283—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1970C2211yescharacteristic
34#1942C2211yescharacteristic
35#1934—21no—
36#1893—21no—
37#1846C2211yescharacteristic
38#1846C2211yescharacteristic
39#1835—21no—
40#1808C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1807—21no—
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#106—21no—
48C4×#102—21no—
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—21no—
63C2×C4×D4—21no—
64C2×C4×D4—41no—
65C22×C42D411yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#466—22no—
71#466—22no—
72C4×D8—22no—
73C4×D8—22no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×#106—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8D411yescharacteristic
89C2×C4×C8D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#390—21no—
93#389—21no—
94#388—41no—
95#386—41no—
96#383—41no—
97#382—21no—
98#382—21no—
99#381—41no—
100#380—21no—
101#380—21no—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C22⋊C4—41no—
107C4×C22⋊C4—41no—
108#376—41no—
109#367—22no—
110#367—22no—
111#366—22no—
112#366—22no—
113#357—22no—
114#357—22no—
115#357—22no—
116#357—22no—
117C22⋊C16—22no—
118C22⋊C16—22no—
119C4×C16—22no—
120C4×C16—22no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×D4—21no—
129C22×D4—41no—
130C23×C4—41no—
131C23×C4D811yescharacteristic
132C23×C4D811yescharacteristic
133C23×C4C2×D411yescharacteristic
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8#4112yes—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—41no—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2411yescharacteristic
153#121—24no—
154#121—42no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—84no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42D811yescharacteristic
179C2×C42D811yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16#4112yes—
193C2×C16#4112yes—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107#4112yes—
208#106—42no—
209#106—42no—
210#102—24no—
211#102—42no—
212#102—42no—
213C4×C8—22no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—22no—
217C4×C8#4112yes—
218C4×C8#4112yes—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230C24—41no—
231C24C2×D811yescharacteristic
232C2×D4—22no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—82no—
240C2×D4—82no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C22×D411yescharacteristic
269D8—22no—
270D8—22no—
271D8—41no—
272D8—41no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—44no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8#12312yes—
286C2×C8#13512yes—
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—82no—
308C22⋊C4—44no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—21no—
316C42—21no—
317C42—22no—
318C42—22no—
319C42—82no—
320C42—82no—
321C42—82no—
322C42#12312yes—
323C42#13712yes—
324C16—22no—
325C16—22no—
326C16—41no—
327C16—41no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23#50812yes—
342C23C22×D811yescharacteristic
343D4—42no—
344D4—42no—
345D4—88no—
346D4—81no—
347D4—81no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
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—44no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
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—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—164no—
387C2×C4#48012yes—
388C2×C4#48012yes—
389C2×C4#50812yes—
390C2×C4#50912yes—
391C2×C4C2×#12311yescharacteristic
392C2×C4#52511yescharacteristic
393C2×C4#53211yescharacteristic
394C2×C4C2×#13511yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4#57511yescharacteristic
397C8—22no—
398C8—22no—
399C8—41no—
400C8#52911yescharacteristic
401C8#52911yescharacteristic
402C8#52911yescharacteristic
403C8#52911yescharacteristic
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22C2×#48011yescharacteristic
419C22#310011yescharacteristic
420C22#316911yescharacteristic
421C22C2×#50811yescharacteristic
422C22C2×#50811yescharacteristic
423C22C2×#50911yescharacteristic
424C22#341411yescharacteristic
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#223414yes—
437C4#239511yescharacteristic
438C4#309111yescharacteristic
439C4#310711yescharacteristic
440C4#317211yescharacteristic
441C2—42no—
442C2—162no—
443C2#2375411yescharacteristic
444C2C2×#223411yescharacteristic
445C2#3364511yescharacteristic
446C2#3366211yescharacteristic
447C2#3366511yescharacteristic
448C2#3373511yescharacteristic
449C2#3373711yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutator, Frattini, cosocleC22×C8C24
socleC23C22×D8
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20818C2semidirect
3. #13539C2semidirect
4. #13528C2semidirect
5. #13507C2semidirect
6. #13494C2semidirect
7. #13493C2semidirect
8. #13482C2semidirect
11. C4×#367C2semidirect
12. C4×#366C2semidirect
13. C4×#357C2semidirect
14. C4×#357C2semidirect
15. C4×C22⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
C4×#4662
#208182
#135392
#135282
#135072
#134942
#134932
#134822
#133812
#133802
C4×#3672
C4×#3662
C4×#3572
C4×C22⋊C162