Finite Group Databasebeta

← #510331

Subgroups — #510331

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28527C211yesmaximal, characteristic, complemented
2#26484C211yesmaximal, characteristic, complemented
3#13526C211yesmaximal, characteristic, complemented
4#13505C211yesmaximal, characteristic, complemented
5#13474C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13363C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic, complemented
9#13282C211yesmaximal, characteristic, complemented
10#13235C211yesmaximal, characteristic
11#13104C211yesmaximal, characteristic, complemented
12#13012C211yesmaximal, characteristic
13#13012C211yesmaximal, characteristic
14#12935C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#474—21no—
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2542—21no—
23#2452—21no—
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2093—21no—
36#2093—21no—
37#2067—21no—
38#2023—21no—
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1862—21no—
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C22⋊Q8—41no—
64C2×C22⋊Q8—41no—
65C2×C22⋊Q8—41no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×D16—41no—
69C2×D16—41no—
70C2×D16Q814yescomplemented
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#500—22no—
76#500—41no—
77#500—41no—
78#500—41no—
79#474—44no—
80C4×D8—22no—
81C4×D8—41no—
82C4×D8—41no—
83C4×D8—41no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#399—41no—
105#399—41no—
106#396—41no—
107#393—41no—
108#390—21no—
109#390—21no—
110#390—21no—
111#390—21no—
112#389—21no—
113#389—21no—
114#385—41no—
115#384—21no—
116#384—21no—
117C4×C4⋊C4D411yescharacteristic
118#366—41no—
119#366—41no—
120#366—41no—
121C4⋊C16—41no—
122#357—22no—
123#357—41no—
124#357—41no—
125#357—41no—
126#357—41no—
127#357—41no—
128C4×C16—41no—
129#340—21no—
130#340—21no—
131#336D411yescharacteristic
132#336D411yescharacteristic
133C22×Q8—41no—
134C22×Q8—41no—
135C22×D4—21no—
136C22×D4—21no—
137C23×C4—41no—
138C23×C4—41no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8C2×Q812yes—
150C2×D8C2×Q812yes—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2411yescharacteristic
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—81no—
159C22⋊Q8—84no—
160C22⋊Q8—84no—
161C22⋊Q8—84no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—81no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4D811yescharacteristic
181C2×C4⋊C4D811yescharacteristic
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
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⋊C4—41no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195D16—44no—
196C2×C16—21no—
197C2×C16—21no—
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×C16C2×Q812yes—
208#107—21no—
209#107—21no—
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—42no—
220#107—42no—
221#107—42no—
222#107—42no—
223#107#4112yes—
224C4⋊C8—22no—
225#102—42no—
226#102—42no—
227#102—42no—
228#102—42no—
229#102—42no—
230#102—81no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237#95—41no—
238#95—41no—
239#95D811yescharacteristic
240#95D811yescharacteristic
241C24—41no—
242C24—41no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×Q8—161no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
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—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—161no—
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C22×D411yescharacteristic
280D8—22no—
281D8—22no—
282D8—22no—
283D8—22no—
284D8—41no—
285D8—41no—
286D8—41no—
287D8—41no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8AGL(1,ℤ/8)12yes—
297C2×C8C22×Q811yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C2×C8H(5,𝔽2)11yescharacteristic
300C2×C8H(5,𝔽2)11yescharacteristic
301C2×C8H(5,𝔽2)11yescharacteristic
302C2×C82−1+411yescharacteristic
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—82no—
323C4⋊C4—82no—
324C4⋊C4D1614yescomplemented
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—161no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C16—22no—
342C16—41no—
343C16—41no—
344C16—41no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23C22×D811yescharacteristic
360Q8—164no—
361Q8—164no—
362D4—42no—
363D4—42no—
364D4—42no—
365D4—42no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
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—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
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—81no—
401C2×C4—81no—
402C2×C4—164no—
403C2×C4—164no—
404C2×C4C2×D1612yes—
405C2×C4C2×D1612yes—
406C2×C4C2×D1612yes—
407C2×C4#50912yes—
408C2×C4#53511yescharacteristic
409C2×C4#54611yescharacteristic
410C2×C4D4×Q811yescharacteristic
411C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
412C2×C4#57711yescharacteristic
413C2×C4#57811yescharacteristic
414C8—22no—
415C8—41no—
416C8—41no—
417C8#54211yescharacteristic
418C8#54311yescharacteristic
419C8#55411yescharacteristic
420C8#56211yescharacteristic
421C22—82no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22#320511yescharacteristic
436C22#329411yescharacteristic
437C22Q8×D811yescharacteristic
438C22C22×D1611yescharacteristic
439C22C2×#50911yescharacteristic
440C22#341611yescharacteristic
441C22#341711yescharacteristic
442C4—22no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—82no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4#269211yescharacteristic
455C4#322511yescharacteristic
456C4#330811yescharacteristic
457C4#338411yescharacteristic
458C2—162no—
459C2—162no—
460C2#3151811yescharacteristic
461C2#3377111yescharacteristic
462C2#3379111yescharacteristic
463C2#3386111yescharacteristic
464C2#3387511yescharacteristic
465C2Q8×D1611yescharacteristic
466C2#3395111yescharacteristic
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. #28527C2semidirect
2. #26484C2semidirect
3. #13526C2semidirect
4. #13505C2semidirect
5. #13474C2semidirect
6. #13451C2semidirect
7. #13363C2semidirect
8. #13363C2semidirect
9. #13282C2semidirect
11. #13104C2semidirect
14. #12935C2semidirect
15. #12811C2semidirect
70. C2×D16Q8semidirect
324. C4⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
#285272
#264842
#135262
#135052
#134742
#134512
#133632
#132822
#132352
#131042
#130122
#129352
#128112