Finite Group Databasebeta

← #499576

Subgroups — #499576

Order 512.

536 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#467C211yesmaximal, characteristic
2#20818C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13529C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic
7#13492C211yesmaximal, characteristic
8#13483C211yesmaximal, characteristic, complemented
9#13383C211yesmaximal, characteristic
10#13382C211yesmaximal, characteristic
11C4×#367C211yesmaximal, characteristic, complemented
12C4×#366C211yesmaximal, characteristic, complemented
13C4×#358C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15C4×C22⋊C16C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C4×C22⋊Q8—21no—
20#2283—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1969C2211yescharacteristic
34#1942C2211yescharacteristic
35#1933—21no—
36#1892—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×#103—21no—
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C22×C42D411yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#467—22no—
70#467—22no—
71C4×Q16—22no—
72C4×Q16—22no—
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#106—21no—
81C2×#106—21no—
82C2×#103—21no—
83C2×#103—21no—
84C2×#103—21no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#391—21no—
92#389—21no—
93#387—41no—
94#385—41no—
95#383—41no—
96#382—21no—
97#382—21no—
98#381—41no—
99#380—21no—
100#380—21no—
101#380—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—41no—
106C4×C22⋊C4—41no—
107C4×C22⋊C4—41no—
108#376—41no—
109#367—22no—
110#367—22no—
111#366—22no—
112#366—22no—
113#358—22no—
114#358—22no—
115#358—22no—
116#358—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×Q8—21no—
129C23×C4D811yescharacteristic
130C23×C4D811yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16#4112yes—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—41no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2411yescharacteristic
151C22⋊Q8—24no—
152C22⋊Q8—42no—
153C4×Q8—42no—
154C4×Q8—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42D811yescharacteristic
176C2×C42D811yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16#4112yes—
190C2×C16#4112yes—
191#107—21no—
192#107—21no—
193#107—21no—
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#4112yes—
205#106—42no—
206#106—42no—
207#103—24no—
208#103—42no—
209#103—42no—
210C4×C8—22no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—22no—
214C4×C8#4112yes—
215C4×C8#4112yes—
216#95—41no—
217#95—41no—
218#95—41no—
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—
229C24C2×D811yescharacteristic
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C22×D411yescharacteristic
262Q16—22no—
263Q16—22no—
264Q16—41no—
265Q16—41no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—44no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8#12312yes—
279C2×C8#13512yes—
280C2×C8C2×#4111yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C22⋊C4—44no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C42—21no—
308C42—21no—
309C42—22no—
310C42—22no—
311C42—82no—
312C42—82no—
313C42—82no—
314C42—82no—
315C42#12312yes—
316C42#13712yes—
317C16—22no—
318C16—22no—
319C16—41no—
320C16—41no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23#50812yes—
328C23C22×D811yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—81no—
332Q8—81no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—44no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
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#48012yes—
375C2×C4#48012yes—
376C2×C4#50812yes—
377C2×C4#51012yes—
378C2×C4C2×#12311yescharacteristic
379C2×C4#52511yescharacteristic
380C2×C4#53211yescharacteristic
381C2×C4C2×#13511yescharacteristic
382C2×C4C2×#13711yescharacteristic
383C2×C4#57511yescharacteristic
384C8—22no—
385C8—22no—
386C8—41no—
387C8#52911yescharacteristic
388C8#52911yescharacteristic
389C8#52911yescharacteristic
390C8#52911yescharacteristic
391C22—22no—
392C22—22no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22C2×#48011yescharacteristic
399C22#310011yescharacteristic
400C22#316911yescharacteristic
401C22C2×#50811yescharacteristic
402C22C2×#50811yescharacteristic
403C22C2×#51011yescharacteristic
404C22#341411yescharacteristic
405C4—22no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—82no—
410C4—82no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4#223714yes—
418C4#239511yescharacteristic
419C4#309111yescharacteristic
420C4#310711yescharacteristic
421C4#317211yescharacteristic
422C2—42no—
423C2#2375411yescharacteristic
424C2C2×#223711yescharacteristic
425C2#3364511yescharacteristic
426C2#3366211yescharacteristic
427C2#3366811yescharacteristic
428C2#3373511yescharacteristic
429C2#3373811yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutator, Frattini, cosocleC22×C8C24
socleC23C22×D8
prime radicalC24C2×D8
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
3. #13539C2semidirect
4. #13529C2semidirect
5. #13507C2semidirect
8. #13483C2semidirect
11. C4×#367C2semidirect
12. C4×#366C2semidirect
13. C4×#358C2semidirect
14. C4×#358C2semidirect

Maximal subgroups

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

subgroupindex
C4×#4672
#208182
#135392
#135292
#135072
#134942
#134922
#134832
#133832
#133822
C4×#3672
C4×#3662
C4×#3582
C4×C22⋊C162