Finite Group Databasebeta

← #504131

Subgroups — #504131

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2224C211yesmaximal, characteristic, complemented
2C2×C4×D16C211yesmaximal, characteristic
3#26484C211yesmaximal, characteristic, complemented
4C4⋊C4×D8C211yesmaximal, characteristic, complemented
5#13526C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13451C211yesmaximal, characteristic, complemented
9#13165C211yesmaximal, characteristic, complemented
10#13104C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic
12#13012C211yesmaximal, characteristic
13#12935C211yesmaximal, characteristic, complemented
14#12935C211yesmaximal, characteristic, complemented
15#12935C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2452—21no—
23D4×C4⋊C4—21no—
24#2224—24no—
25C4×D16—24no—
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#2067—21no—
40#2023—21no—
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1862—21no—
48#1862—21no—
49#1862—21no—
50#1845C2211yescharacteristic
51#1802—21no—
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C22⋊Q8—41no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C22×C4⋊C4—41no—
70C2×D16—22no—
71C2×D16—22no—
72C2×D16Q814yescomplemented
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#500—22no—
78#500—41no—
79C4×D8—22no—
80C4×D8—22no—
81C4×D8—22no—
82C4×D8—22no—
83C4×D8—22no—
84C4×D8—41no—
85C4×D8—41no—
86C4×D8—41no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#396—41no—
106#393—41no—
107#390—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#385—41no—
114#385—41no—
115#385—41no—
116#384—21no—
117#380—41no—
118C4×C4⋊C4D411yescharacteristic
119#366—22no—
120#366—22no—
121#366—22no—
122#366—22no—
123#366—41no—
124C4⋊C16—22no—
125#357—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—41no—
130#357D414yes—
131C4×C16—22no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×Q8—41no—
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
161C4⋊Q8—42no—
162C4⋊Q8—81no—
163C22⋊Q8—84no—
164C4×D4—24no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—81no—
170C4×D4—81no—
171C4×D4—81no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
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×C16C2×D412yes—
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—42no—
223#107—81no—
224#107C2×D412yes—
225#107#4112yes—
226C4⋊C8—41no—
227#102—24no—
228#102—42no—
229#102—42no—
230#102—42no—
231#102—42no—
232#102—81no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95D811yescharacteristic
244#95D811yescharacteristic
245C24—41no—
246C24—41no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8—161no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—161no—
278C22×C4—161no—
279C22×C4—161no—
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C22×D411yescharacteristic
287D8—22no—
288D8—22no—
289D8—22no—
290D8—22no—
291D8—22no—
292D8—41no—
293D8—41no—
294D8C22⋊Q814yescomplemented
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8C22×D411yescharacteristic
305C2×C8C22×Q811yescharacteristic
306C2×C8C2×#4111yescharacteristic
307C2×C8C2×#4111yescharacteristic
308C2×C8H(5,𝔽2)11yescharacteristic
309C2×C82−1+411yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—22no—
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—82no—
329C4⋊C4—161no—
330C4⋊C4D1614yescomplemented
331C22⋊C4—44no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—161no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C16—22no—
346C16—22no—
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
364Q8—164no—
365D4—24no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—81no—
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—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—88no—
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—81no—
407C2×C4—164no—
408C2×C4—164no—
409C2×C4—164no—
410C2×C4C2×D1612yes—
411C2×C4C2×D1612yes—
412C2×C4C2×D1612yes—
413C2×C4#52511yescharacteristic
414C2×C4D4211yescharacteristic
415C2×C4D4×Q811yescharacteristic
416C2×C4#57511yescharacteristic
417C2×C4#57611yescharacteristic
418C2×C4#57811yescharacteristic
419C8—41no—
420C8—41no—
421C8—41no—
422C8C2×C22⋊Q811yescharacteristic
423C8#53911yescharacteristic
424C8#54711yescharacteristic
425C8#55411yescharacteristic
426C22—44no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22#310011yescharacteristic
441C22D4×D811yescharacteristic
442C22Q8×D811yescharacteristic
443C22C22×D1611yescharacteristic
444C22#341411yescharacteristic
445C22#341511yescharacteristic
446C22#341711yescharacteristic
447C4—22no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—82no—
454C4—82no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4#244911yescharacteristic
460C4#312411yescharacteristic
461C4#326911yescharacteristic
462C4#338911yescharacteristic
463C2—84no—
464C2—162no—
465C2#2636811yescharacteristic
466C2#3366011yescharacteristic
467C2#3368411yescharacteristic
468C2#3383511yescharacteristic
469C2D4×D1611yescharacteristic
470C2Q8×D1611yescharacteristic
471C2#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×#2224C2semidirect
3. #26484C2semidirect
4. C4⋊C4×D8C2semidirect
5. #13526C2semidirect
6. #13505C2semidirect
7. #13474C2semidirect
8. #13451C2semidirect
9. #13165C2semidirect
10. #13104C2semidirect
13. #12935C2semidirect
14. #12935C2semidirect
15. #12935C2semidirect
72. C2×D16Q8semidirect
294. D8C22⋊Q8semidirect
330. C4⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
C2×#22242
C2×C4×D162
#264842
C4⋊C4×D82
#135262
#135052
#134742
#134512
#131652
#131042
#130122
#129352