Finite Group Databasebeta

← #510460

Subgroups — #510460

Order 512.

568 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
2#26255C211yesmaximal, characteristic, complemented
3#13537C211yesmaximal, characteristic, complemented
4#13481C211yesmaximal, characteristic, complemented
5#13449C211yesmaximal, characteristic
6#13364C211yesmaximal, characteristic, complemented
7#13364C211yesmaximal, characteristic, complemented
8#13327C211yesmaximal, characteristic
9#13203C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic
11#13104C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic
13#13014C211yesmaximal, characteristic
14#13013C211yesmaximal, characteristic
15#12812C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×#474—21no—
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23#2542—21no—
24#2446—21no—
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2093—21no—
36#2093—21no—
37#2056—21no—
38#2043—21no—
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1969C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1942C2211yescharacteristic
45#1878—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×Q16—21no—
60C22×Q16C2311yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C22⋊Q8—41no—
66C2×C22⋊Q8—41no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×SD32—22no—
70C2×SD32—22no—
71C2×SD32—22no—
72C2×SD32—41no—
73C2×SD32—41no—
74C2×SD32—41no—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#500—41no—
80#500—41no—
81#494—22no—
82#494—41no—
83#474—44no—
84C4×Q16—22no—
85C4×Q16—41no—
86C4×D8—41no—
87C4×D8—41no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#400—41no—
108#399—41no—
109#399—41no—
110#398—41no—
111#391—21no—
112#391—21no—
113#391—21no—
114#390—21no—
115#390—21no—
116#389—21no—
117#387—41no—
118#387—41no—
119#384—21no—
120#384—21no—
121C4×C4⋊C4D411yescharacteristic
122C4⋊C16—41no—
123#358—41no—
124#358—41no—
125#358—41no—
126#358—41no—
127#358—41no—
128#358—41no—
129#357—22no—
130#357—41no—
131C4×C16—41no—
132#340—21no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—41no—
138C22×D4—21no—
139C23×C4—41no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—42no—
152C2×Q16—42no—
153C2×Q16—81no—
154C2×Q16#4112yes—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8#4112yes—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2411yescharacteristic
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—81no—
169C22⋊Q8—84no—
170C22⋊Q8—84no—
171C4×Q8—42no—
172C4×Q8—81no—
173C4×D4—42no—
174C4×D4—42no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4D811yescharacteristic
190C2×C4⋊C4D811yescharacteristic
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C42—41no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200SD32—44no—
201SD32—44no—
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×C16—21no—
212C2×C16—21no—
213C2×C16—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—21no—
224#107—42no—
225#107—42no—
226#107—42no—
227C4⋊C8—22no—
228#103—42no—
229#103—81no—
230#103—81no—
231#103—81no—
232#102—42no—
233#102—42no—
234#102—42no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95D811yescharacteristic
249#95D811yescharacteristic
250C24—41no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—82no—
264C2×Q8—82no—
265C2×Q8—82no—
266C2×D4—22no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C22×D411yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—41no—
292Q16—41no—
293Q16—41no—
294Q16—41no—
295Q16—41no—
296Q16—41no—
297Q16—84no—
298D8—22no—
299D8—22no—
300D8—41no—
301D8—41no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8AGL(1,ℤ/8)12yes—
311C2×C8C2×#4111yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C8H(5,𝔽2)11yescharacteristic
315C2×C8H(5,𝔽2)11yescharacteristic
316C2×C82−1+411yescharacteristic
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4SD3214yescomplemented
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—82no—
350C16—41no—
351C16—41no—
352C16—41no—
353C16—41no—
354C23—42no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23C22×D811yescharacteristic
362Q8—42no—
363Q8—42no—
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368Q8—81no—
369Q8—81no—
370Q8—164no—
371D4—42no—
372D4—42no—
373D4—81no—
374D4—81no—
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—21no—
389C2×C4—42no—
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—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—164no—
411C2×C4C2×SD3212yes—
412C2×C4C2×SD3212yes—
413C2×C4C2×SD3212yes—
414C2×C4#50912yes—
415C2×C4#53511yescharacteristic
416C2×C4#54611yescharacteristic
417C2×C4#54811yescharacteristic
418C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
419C2×C4#57711yescharacteristic
420C2×C4#57711yescharacteristic
421C8—22no—
422C8—41no—
423C8—41no—
424C8#53811yescharacteristic
425C8#53911yescharacteristic
426C8#55111yescharacteristic
427C8#56211yescharacteristic
428C22—82no—
429C22—82no—
430C22—82no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22#320511yescharacteristic
436C22#329411yescharacteristic
437C22#335811yescharacteristic
438C22C22×SD3211yescharacteristic
439C22C2×#50911yescharacteristic
440C22#341711yescharacteristic
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—161no—
455C4#269611yescharacteristic
456C4#322111yescharacteristic
457C4#330611yescharacteristic
458C4#337311yescharacteristic
459C2—162no—
460C2#3163411yescharacteristic
461C2#3377411yescharacteristic
462C2#3378911yescharacteristic
463C2#3386511yescharacteristic
464C2#3387511yescharacteristic
465C2#3392611yescharacteristic
466C2#3393411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
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
2. #26255C2semidirect
3. #13537C2semidirect
4. #13481C2semidirect
6. #13364C2semidirect
7. #13364C2semidirect
9. #13203C2semidirect
11. #13104C2semidirect
15. #12812C2semidirect
337. C4⋊C4SD32semidirect

Maximal subgroups

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

subgroupindex
#285272
#262552
#135372
#134812
#134492
#133642
#133272
#132032
#131472
#131042
#130142
#130132
#128122