Finite Group Databasebeta

← #501716

Subgroups — #501716

Order 512.

639 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×#2177C211yesmaximal, characteristic
3#28534C211yesmaximal, characteristic, complemented
4C4⋊C4×D8C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13506C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13284C211yesmaximal, characteristic, complemented
10#13160C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic
12#13012C211yesmaximal, characteristic
13#12813C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#478—21no—
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23#2566—21no—
24D4×C4⋊C4—21no—
25#2228—24no—
26#2177—24no—
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#346C2211yescharacteristic
40#2094—21no—
41#2092—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1862—21no—
47#1862—21no—
48#1846C2211yescharacteristic
49#1845C2211yescharacteristic
50#1845C2211yescharacteristic
51#1802—21no—
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#125—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C22×C4⋊C4—41no—
69C2×D16—22no—
70C2×D16—22no—
71C2×D16Q814yes—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#499—22no—
77#499—22no—
78#478—44no—
79C4×D8—22no—
80C4×D8—22no—
81C4×D8—22no—
82C4×D8—22no—
83C4×D8—22no—
84C4×D8—41no—
85C4×D8—41no—
86C4×D8D414yes—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#106—21no—
96C2×#106—21no—
97C2×C4⋊C8D411yescharacteristic
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#396—41no—
108#396—41no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#385—41no—
114#385—41no—
115#385—41no—
116#385—41no—
117#384—21no—
118#384—21no—
119#382—21no—
120#380—41no—
121C4×C4⋊C4D411yescharacteristic
122#367—22no—
123#367—22no—
124#366—22no—
125C4⋊C16—22no—
126C4⋊C16—41no—
127C4⋊C16—41no—
128#357—22no—
129#357—22no—
130#357—22no—
131#357—22no—
132#357—41no—
133#357—41no—
134#346—22no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C22×D4—21no—
140C22×D4—21no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4—41no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8C2×D412yes—
154C2×D8C2×Q812yes—
155C2×D8C2×Q812yes—
156C2×D8#4112yes—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2411yescharacteristic
161#125—42no—
162#125—42no—
163C4×D4—24no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—81no—
170C4×D4—81no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
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—41no—
185C2×C4⋊C4—84no—
186C2×C4⋊C4D811yescharacteristic
187C2×C4⋊C4D811yescharacteristic
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199D16—28no—
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×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16C2×Q812yes—
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—21no—
222#107—21no—
223#107—42no—
224#107—81no—
225#107C2×D412yes—
226#106—42no—
227#106—42no—
228C4⋊C8—22no—
229#102—24no—
230#102—42no—
231#102—42no—
232#102—42no—
233#102—42no—
234#102—42no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8C2×D412yes—
240C4×C8#4112yes—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95D811yescharacteristic
246#95D811yescharacteristic
247C24—41no—
248C24—41no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—161no—
276C22×C4—161no—
277C22×C4—161no—
278C22×C4—161no—
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C22×D411yescharacteristic
286D8—22no—
287D8—22no—
288D8—22no—
289D8—22no—
290D8—22no—
291D8—41no—
292D8—41no—
293D8C22⋊Q814yes—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8#12112yes—
303C2×C8C22×D411yescharacteristic
304C2×C8C22×Q811yescharacteristic
305C2×C8C2×#4111yescharacteristic
306C2×C8C2×#4111yescharacteristic
307C2×C8H(5,𝔽2)11yescharacteristic
308C2×C82−1+411yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—161no—
329C4⋊C4—161no—
330C4⋊C4—161no—
331C22⋊C4—44no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—161no—
338C22⋊C4—161no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42#12112yes—
344C42AGL(1,ℤ/8)12yes—
345C16—22no—
346C16—41no—
347C16—41no—
348C16—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23C22×D811yescharacteristic
364D4—24no—
365D4—42no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—81no—
371D4—81no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—42no—
388C2×C4—88no—
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—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—164no—
407C2×C4—164no—
408C2×C4—164no—
409C2×C4#45912yes—
410C2×C4#45912yes—
411C2×C4#50912yes—
412C2×C4C2×#12111yescharacteristic
413C2×C4#54611yescharacteristic
414C2×C4D4×Q811yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4#57711yescharacteristic
417C2×C4#57811yescharacteristic
418C8—22no—
419C8—41no—
420C8—41no—
421C8C2×C22⋊Q811yescharacteristic
422C8#53711yescharacteristic
423C8#55211yescharacteristic
424C8#55411yescharacteristic
425C22—44no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22C2×#45911yescharacteristic
440C22#329411yescharacteristic
441C22Q8×D811yescharacteristic
442C22C2×#50911yescharacteristic
443C22#341411yescharacteristic
444C22#341511yescharacteristic
445C22#341611yescharacteristic
446C4—22no—
447C4—22no—
448C4—22no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—82no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4#241711yescharacteristic
459C4#304611yescharacteristic
460C4#333011yescharacteristic
461C4#338411yescharacteristic
462C2—84no—
463C2—162no—
464C2#2481111yescharacteristic
465C2#3358011yescharacteristic
466C2#3358911yescharacteristic
467C2#3387411yescharacteristic
468C2#3388511yescharacteristic
469C2#3395111yescharacteristic
470C2#3395611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊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
3. #28534C2semidirect
4. C4⋊C4×D8C2semidirect
5. #13507C2semidirect
6. #13506C2semidirect
7. #13476C2semidirect
8. #13474C2semidirect
9. #13284C2semidirect
10. #13160C2semidirect
13. #12813C2semidirect
14. #12811C2semidirect
15. #12811C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#21772
#285342
C4⋊C4×D82
#135072
#135062
#134762
#134742
#132842
#131602
#130122
#128132
#128112