Finite Group Databasebeta

← #497893

Subgroups — #497893

Order 512.

604 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2179C211yesmaximal, characteristic
2#33460C211yesmaximal, characteristic
3#33459C211yesmaximal, characteristic
4#22692C211yesmaximal, characteristic, complemented
5#22690C211yesmaximal, characteristic, complemented
6#13172C211yesmaximal, characteristic, complemented
7#13171C211yesmaximal, characteristic, complemented
8#12946C211yesmaximal, characteristic, complemented
9#12946C211yesmaximal, characteristic, complemented
10#12945C211yesmaximal, characteristic, complemented
11#12945C211yesmaximal, characteristic, complemented
12#12908C211yesmaximal, characteristic, complemented
13#12821C211yesmaximal, characteristic, complemented
14#12820C211yesmaximal, characteristic, complemented
15#12665C211yesmaximal, characteristic, complemented
16C2×#508C2211yescharacteristic
17C2×#443C2211yescharacteristic
18C2×#443C2211yescharacteristic
19#2951C2211yescharacteristic
20#2951C2211yescharacteristic
21#2951C2211yescharacteristic
22#2951C2211yescharacteristic
23#2913—21no—
24#2912—21no—
25#2179C418yescomplemented
26#2177—21no—
27#2176—21no—
28#2175—21no—
29#2175—21no—
30C4×Q32—21no—
31C4×SD32—21no—
32C4×SD32—21no—
33C4×D16—21no—
34C2×#368C2211yescharacteristic
35C2×#367C2211yescharacteristic
36C2×#367C2211yescharacteristic
37C2×#366C2211yescharacteristic
38C2×#366C2211yescharacteristic
39C2×#364C2211yescharacteristic
40C2×#359C2211yescharacteristic
41C2×#359C2211yescharacteristic
42#2139C2211yescharacteristic
43#2139C2211yescharacteristic
44#2139C2211yescharacteristic
45#2139C2211yescharacteristic
46#2108C2211yescharacteristic
47#2108C2211yescharacteristic
48C2×#346C2211yescharacteristic
49#1952C2211yescharacteristic
50#1952C2211yescharacteristic
51#1952C2211yescharacteristic
52#1952C2211yescharacteristic
53#1860—21no—
54#1860—21no—
55#1859—21no—
56#1859—21no—
57#1846C2211yescharacteristic
58#1845C2211yescharacteristic
59#1816—21no—
60#1815—21no—
61#1772C2211yescharacteristic
62#1387C2211yescharacteristic
63#1387C2211yescharacteristic
64#1387C2211yescharacteristic
65#1387C2211yescharacteristic
66#1379C2211yescharacteristic
67#1379C2211yescharacteristic
68C2×#135C2311yescharacteristic
69C2×#135C2311yescharacteristic
70C2×#131—21no—
71C2×#131—21no—
72#520—21no—
73#520—21no—
74#520—21no—
75#520—21no—
76#508C2×C414yes—
77#508Q814yes—
78C2×Q32D411yescharacteristic
79C2×SD32D411yescharacteristic
80C2×SD32D411yescharacteristic
81C2×D16D411yescharacteristic
82C2×M32C2311yescharacteristic
83C2×M32C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86#443—22no—
87#443—22no—
88#443C2×C414yescomplemented
89#443C2×C414yescomplemented
90#440—41no—
91#440—41no—
92#440—41no—
93#440—41no—
94C4×Q16—21no—
95C4×Q16—21no—
96C4×Q16—21no—
97C4×Q16—21no—
98C4×D8—21no—
99C4×D8—21no—
100C4×D8—21no—
101C4×D8—21no—
102C2×#108D411yescharacteristic
103C2×#108C2311yescharacteristic
104C2×#108C2311yescharacteristic
105#428—21no—
106#428—21no—
107#428—21no—
108#428—21no—
109#428—41no—
110#428—41no—
111#428—41no—
112#428—41no—
113#427—41no—
114#427—41no—
115#427D411yescharacteristic
116C2×#107—21no—
117C2×#107—41no—
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#107C2311yescharacteristic
121C2×#107C2311yescharacteristic
122C2×#106—21no—
123C2×#106—41no—
124C2×#104—21no—
125C2×#104—21no—
126#417—21no—
127#417—21no—
128#417—21no—
129#417—21no—
130#405C2311yescharacteristic
131#405C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#384—21no—
134#382—21no—
135#368C2×C414yescomplemented
136#367—21no—
137#367—21no—
138#367—41no—
139#366—21no—
140#366—21no—
141#366—41no—
142#364—22no—
143#359C2×C414yescomplemented
144#359C2×C414yescomplemented
145#358—21no—
146#358—21no—
147#358—21no—
148#358—21no—
149#357—21no—
150#357—21no—
151#357—21no—
152#357—21no—
153#346—21no—
154#346—21no—
155#346C2×C414yescomplemented
156C4×C16—21no—
157C4×C16—21no—
158#343—21no—
159#343—21no—
160#343—21no—
161#343—21no—
162#339D411yescharacteristic
163#339D411yescharacteristic
164C2×#41—21no—
165C2×#41—21no—
166#135—21no—
167#135—21no—
168#135—21no—
169#135—21no—
170#135C22×C412yes—
171#135C22×C412yes—
172#135C2×Q812yes—
173#135C2×Q812yes—
174C2×Q16C2×D411yescharacteristic
175C2×Q16C2×D411yescharacteristic
176C2×SD16—21no—
177C2×SD16—21no—
178C2×D8C2×D411yescharacteristic
179C2×D8C2×D411yescharacteristic
180#131—24no—
181#131—24no—
182#131—42no—
183#131—42no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—41no—
187C2×M16—41no—
188C2×M16C2×D411yescharacteristic
189C2×M16C2×D411yescharacteristic
190C22×C8—41no—
191C22×C8—41no—
192C22×C8C2×D411yescharacteristic
193C22×C8C2411yescharacteristic
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—81no—
203C4×D4—81no—
204C4×D4—81no—
205C4×D4—81no—
206#117—41no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117C2×D411yescharacteristic
211#117C2×D411yescharacteristic
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—81no—
221C2×C42C2×D411yescharacteristic
222Q32C4⋊C414yescomplemented
223SD32C4⋊C414yescomplemented
224SD32C4⋊C414yescomplemented
225D16C4⋊C414yescomplemented
226M32—21no—
227M32—21no—
228M32—21no—
229M32—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16C22×C412yes—
235C2×C16C22×C412yes—
236C2×C16C2×D411yescharacteristic
237C2×C16C2×D411yescharacteristic
238C2×C16C2×Q812yes—
239#108—22no—
240#108—21no—
241#108—21no—
242#108C22×C412yes—
243#108C22×C412yes—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—41no—
253#107—41no—
254#107—41no—
255#107—41no—
256#107—41no—
257#107—41no—
258#107D811yescharacteristic
259#107D811yescharacteristic
260#106—41no—
261#106—41no—
262#106—41no—
263#106—41no—
264#106—41no—
265#106—41no—
266#106D811yescharacteristic
267#106D811yescharacteristic
268#104—24no—
269#104—24no—
270#103—41no—
271#103—41no—
272#103—41no—
273#103—41no—
274#102—41no—
275#102—41no—
276#102—41no—
277#102—41no—
278#97—21no—
279#97—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8C22×C412yes—
283C4×C8#4112yes—
284#95—81no—
285#41—22no—
286#41—22no—
287#41—22no—
288#41—22no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4C22×D411yescharacteristic
311Q16—21no—
312Q16—21no—
313Q16C2×C4⋊C412yes—
314Q16C2×C4⋊C412yes—
315SD16—22no—
316SD16—22no—
317SD16—22no—
318SD16—22no—
319D8—21no—
320D8—21no—
321D8C2×C4⋊C412yes—
322D8C2×C4⋊C412yes—
323M16—22no—
324M16—22no—
325M16—21no—
326M16—21no—
327M16—21no—
328M16—21no—
329M16—41no—
330M16—41no—
331M16—44no—
332M16—44no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8C4×D412yes—
340C2×C8C2×D811yescharacteristic
341C2×C8C2×D811yescharacteristic
342C2×C8#13512yes—
343C2×C8C23×C411yescharacteristic
344C2×C8C22×D411yescharacteristic
345C2×C8C22×Q811yescharacteristic
346C2×C8C2×#4111yescharacteristic
347C2×C8H(5,𝔽2)11yescharacteristic
348C2×C82−1+411yescharacteristic
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—162no—
362C4⋊C4C2×D811yescharacteristic
363C4⋊C4C2×D811yescharacteristic
364C4⋊C4C2×D811yescharacteristic
365C4⋊C4C2×D811yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—81no—
369C22⋊C4—81no—
370C22⋊C4—81no—
371C22⋊C4—81no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—81no—
376C42—81no—
377C42—81no—
378C42—81no—
379C42C4×D412yes—
380C16—22no—
381C16—41no—
382C16—41no—
383C16C2×C4⋊C412yes—
384C16C2×C4⋊C412yes—
385C23—41no—
386C23—41no—
387C23#57611yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—41no—
391Q8—41no—
392D4—22no—
393D4—22no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—41no—
399D4—41no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—161no—
418C2×C4—161no—
419C2×C4—161no—
420C2×C4—161no—
421C2×C4—321no—
422C2×C4—321no—
423C2×C4—321no—
424C2×C4—321no—
425C2×C4C2×C4×D411yescharacteristic
426C2×C4D4211yescharacteristic
427C2×C4D4×Q811yescharacteristic
428C2×C4C22×D811yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4#57811yescharacteristic
431C8—42no—
432C8—42no—
433C8—41no—
434C8—41no—
435C8C4×D812yes—
436C8C4×D812yes—
437C8C22×C4⋊C411yescharacteristic
438C8#52011yescharacteristic
439C8#54711yescharacteristic
440C8#55411yescharacteristic
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22#295111yescharacteristic
450C22D4×D811yescharacteristic
451C22#338011yescharacteristic
452C4—42no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—42no—
458C4—42no—
459C4—161no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4D4×C4⋊C411yescharacteristic
464C4C2×C4×D811yescharacteristic
465C4#328311yescharacteristic
466C4Q8×D811yescharacteristic
467C2—21no—
468C2—21no—
469C2—82no—
470C2—82no—
471C2C4⋊C4×D811yescharacteristic
472C2#3385411yescharacteristic
473C2#3385511yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#508C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1772 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22692C2semidirect
5. #22690C2semidirect
6. #13172C2semidirect
7. #13171C2semidirect
8. #12946C2semidirect
9. #12946C2semidirect
10. #12945C2semidirect
11. #12945C2semidirect
12. #12908C2semidirect
13. #12821C2semidirect
14. #12820C2semidirect
15. #12665C2semidirect
25. #2179C4semidirect
88. #443C2×C4semidirect
89. #443C2×C4semidirect
135. #368C2×C4semidirect
143. #359C2×C4semidirect
144. #359C2×C4semidirect
155. #346C2×C4semidirect
222. Q32C4⋊C4semidirect
223. SD32C4⋊C4semidirect
224. SD32C4⋊C4semidirect
225. D16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21792
#334602
#334592
#226922
#226902
#131722
#131712
#129462
#129452
#129082
#128212
#128202
#126652