Finite Group Databasebeta

← #504115

Subgroups — #504115

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×#2159C211yesmaximal, characteristic, complemented
3#26066C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic
5#13527C211yesmaximal, characteristic, complemented
6#13516C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13177C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#13061C211yesmaximal, characteristic
12#12992C211yesmaximal, characteristic
13#12937C211yesmaximal, characteristic, complemented
14#12920C211yesmaximal, characteristic, complemented
15#12913C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2441—21no—
23#2361—21no—
24#2228—24no—
25#2159—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39#2084—21no—
40#2059—21no—
41#1970C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1925—21no—
46#1874—21no—
47#1862—21no—
48#1854—21no—
49#1845C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#125—21no—
64C2×#121—21no—
65C2×C4×D4—21no—
66C2×C4×D4—41no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C2×M32—22no—
70C2×M32—22no—
71C2×M32—22no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#499—22no—
77#499—41no—
78#466—22no—
79#466Q814yes—
80C4×D8—22no—
81C4×D8—41no—
82#427—22no—
83#427—41no—
84C2×#107—22no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—41no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#106—21no—
96C2×#106—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#399—41no—
104#396—41no—
105#390—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—41no—
110#388—41no—
111#387—41no—
112#386—41no—
113#385—41no—
114#384—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—22no—
120#367—22no—
121#367—41no—
122#366—22no—
123#366D414yescomplemented
124C4⋊C16—22no—
125C4⋊C16—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—41no—
130C22⋊C16—22no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×D4—21no—
139C22×D4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4C2×D411yescharacteristic
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8C2×Q812yes—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×Q812yes—
157C22×C8#4112yes—
158C22×C8C2411yescharacteristic
159#125—42no—
160#125—81no—
161#121—24no—
162#121—42no—
163C4×D4—42no—
164C4×D4—81no—
165C4×D4—84no—
166#117—42no—
167#117—81no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—81no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4D811yescharacteristic
192C2×C22⋊C4D811yescharacteristic
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197M32—28no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16C2×D412yes—
209C2×C16#4112yes—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—42no—
222#107—81no—
223#107C2×D412yes—
224#107C2×D412yes—
225#107C2×Q812yes—
226#106—42no—
227#106—42no—
228#102—24no—
229#102—42no—
230#102—81no—
231C22⋊C8—22no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95D811yescharacteristic
242#95D811yescharacteristic
243C24—41no—
244C24C2×D811yescharacteristic
245C2×D4—22no—
246C2×D4—44no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—82no—
253C2×D4—82no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—161no—
270C22×C4—161no—
271C22×C4C22⋊Q812yes—
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4#13712yes—
278C22×C4C22×D411yescharacteristic
279D8—22no—
280D8—22no—
281D8—41no—
282D8—41no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8C22⋊Q812yes—
296C2×C8C22×D411yescharacteristic
297C2×C8C22×Q811yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C2×C8C2×#4111yescharacteristic
300C2×C8H(5,𝔽2)11yescharacteristic
301C2×C82−1+411yescharacteristic
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—161no—
323C4⋊C4—161no—
324C22⋊C4—22no—
325C22⋊C4—41no—
326C22⋊C4—44no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—161no—
331C22⋊C4—161no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—82no—
337C42—82no—
338C16—22no—
339C16—22no—
340C16—41no—
341C16—41no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—42no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23#47412yes—
356C23C22×D811yescharacteristic
357D4—42no—
358D4—42no—
359D4—88no—
360D4—81no—
361D4—81no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—164no—
401C2×C4—164no—
402C2×C4#47412yes—
403C2×C4#50912yes—
404C2×C4C2×C22⋊Q811yescharacteristic
405C2×C4D4211yescharacteristic
406C2×C4#54711yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C2×C4#57611yescharacteristic
409C2×C4#57811yescharacteristic
410C8—22no—
411C8—41no—
412C8—41no—
413C8#52511yescharacteristic
414C8#53911yescharacteristic
415C8D4×Q811yescharacteristic
416C8#55411yescharacteristic
417C22—22no—
418C22—22no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22C2×#47411yescharacteristic
432C22D4×D811yescharacteristic
433C22#333411yescharacteristic
434C22C2×#50911yescharacteristic
435C22#341411yescharacteristic
436C22#341611yescharacteristic
437C22#341611yescharacteristic
438C4—22no—
439C4—22no—
440C4—41no—
441C4—41no—
442C4—41no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4#244911yescharacteristic
451C4#308311yescharacteristic
452C4#327011yescharacteristic
453C4#334611yescharacteristic
454C2—42no—
455C2—162no—
456C2#2633611yescharacteristic
457C2#3362711yescharacteristic
458C2#3363411yescharacteristic
459C2#3383711yescharacteristic
460C2#3383811yescharacteristic
461C2#3391211yescharacteristic
462C2#3391811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2228C2semidirect
2. C2×#2159C2semidirect
3. #26066C2semidirect
5. #13527C2semidirect
6. #13516C2semidirect
7. #13474C2semidirect
8. #13453C2semidirect
9. #13177C2semidirect
10. #13109C2semidirect
13. #12937C2semidirect
14. #12920C2semidirect
15. #12913C2semidirect
123. #366D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#21592
#260662
#230252
#135272
#135162
#134742
#134532
#131772
#131092
#130612
#129922
#129372
#129202
#129132