Finite Group Databasebeta

← #501762

Subgroups — #501762

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2225C211yesmaximal, characteristic, complemented
2C2×#2151C211yesmaximal, characteristic
3#33215C211yesmaximal, characteristic
4#21525C211yesmaximal, characteristic
5#13516C211yesmaximal, characteristic, complemented
6#13516C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13475C211yesmaximal, characteristic, complemented
9#13276C211yesmaximal, characteristic, complemented
10#13178C211yesmaximal, characteristic, complemented
11#13062C211yesmaximal, characteristic
12#13060C211yesmaximal, characteristic
13#12836C211yesmaximal, characteristic
14#12835C211yesmaximal, characteristic
15#12814C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2628—21no—
23#2323—21no—
24#2225—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29#2151—24no—
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38#1993—21no—
39#1969C2211yescharacteristic
40#1944—21no—
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1935—21no—
45#1894—21no—
46#1846C2211yescharacteristic
47#1809—21no—
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1763—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×#117D411yescharacteristic
68C2×M32—22no—
69C2×M32—22no—
70C2×M32—41no—
71C22×C16—22no—
72C22×C16—22no—
73C22×C16—41no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#500—22no—
79#500—22no—
80#467—22no—
81#467Q814yescomplemented
82C4×Q16—22no—
83C4×Q16—22no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96#405—22no—
97#405—22no—
98C2×#97D411yescharacteristic
99C2×#97D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#400—41no—
103#399—41no—
104#395—41no—
105#393—41no—
106#391—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—41no—
111#388—41no—
112#388—41no—
113#387—41no—
114#382—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119#367—22no—
120#367—22no—
121#367—41no—
122#366—41no—
123C4⋊C16—22no—
124C4⋊C16—22no—
125C4⋊C16—22no—
126C4⋊C16D414yes—
127#358—22no—
128#358—22no—
129#358—41no—
130#358—41no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138C22×Q8—21no—
139C22×Q8—41no—
140C23×C4C2×D411yescharacteristic
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16C2×Q812yes—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—41no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×Q812yes—
158C22×C8#4112yes—
159C22×C8C2411yescharacteristic
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C22⋊Q8—24no—
163C22⋊Q8—42no—
164C4×Q8—42no—
165C4×Q8—42no—
166#117—22no—
167#117—22no—
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—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4D811yescharacteristic
182C2×C4⋊C4D811yescharacteristic
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4D811yescharacteristic
188C2×C22⋊C4D811yescharacteristic
189C2×C42—41no—
190C2×C42—41no—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193M32—44no—
194C2×C16—21no—
195C2×C16—21no—
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—44no—
205C2×C16C2×D412yes—
206C2×C16C2×D412yes—
207#107—21no—
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—21no—
220#107—42no—
221#107C2×Q812yes—
222#106—42no—
223#103—24no—
224#103—42no—
225#103—42no—
226#97—22no—
227#97—22no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8C2×D412yes—
231C4×C8#4112yes—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243C24C2×D811yescharacteristic
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—82no—
251C2×Q8—161no—
252C2×Q8—161no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C22⋊Q812yes—
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4#13712yes—
272C22×C4C22×D411yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—41no—
276Q16—41no—
277M16—44no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8#12112yes—
290C2×C8C22⋊Q812yes—
291C2×C8C22×D411yescharacteristic
292C2×C8C22×Q811yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8H(5,𝔽2)11yescharacteristic
296C2×C82−1+411yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
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—82no—
315C4⋊C4—82no—
316C4⋊C4—161no—
317C4⋊C4—161no—
318C4⋊C4—161no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—44no—
322C22⋊C4—161no—
323C22⋊C4—161no—
324C22⋊C4—161no—
325C42—21no—
326C42—21no—
327C42—82no—
328C42—161no—
329C42#12112yes—
330C42#13512yes—
331C16—22no—
332C16—22no—
333C16—41no—
334C16—41no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23#47412yes—
342C23C22×D811yescharacteristic
343Q8—42no—
344Q8—42no—
345Q8—81no—
346Q8—81no—
347Q8—164no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4#46612yes—
389C2×C4#46612yes—
390C2×C4#47412yes—
391C2×C4C2×#12111yescharacteristic
392C2×C4C2×C22⋊Q811yescharacteristic
393C2×C4#53711yescharacteristic
394C2×C4C2×#13511yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4#57511yescharacteristic
397C8—22no—
398C8—22no—
399C8—41no—
400C8#54611yescharacteristic
401C8D4×Q811yescharacteristic
402C8#55211yescharacteristic
403C8#55411yescharacteristic
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22C2×#46611yescharacteristic
412C22C2×#47411yescharacteristic
413C22#324511yescharacteristic
414C22#341611yescharacteristic
415C22#341611yescharacteristic
416C22#341611yescharacteristic
417C22#341711yescharacteristic
418C4—22no—
419C4—22no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—82no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4#241711yescharacteristic
432C4#305611yescharacteristic
433C4#307611yescharacteristic
434C4#324811yescharacteristic
435C2—42no—
436C2#2483511yescharacteristic
437C2#3360911yescharacteristic
438C2#3361311yescharacteristic
439C2#3363211yescharacteristic
440C2#3363411yescharacteristic
441C2#3382411yescharacteristic
442C2#3382911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2225C2semidirect
5. #13516C2semidirect
6. #13516C2semidirect
7. #13477C2semidirect
8. #13475C2semidirect
9. #13276C2semidirect
10. #13178C2semidirect
15. #12814C2semidirect
81. #467Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22252
C2×#21512
#332152
#215252
#135162
#134772
#134752
#132762
#131782
#130622
#130602
#128362
#128352
#128142