Finite Group Databasebeta

← #501745

Subgroups — #501745

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C211yesmaximal, characteristic, complemented
2C2×#2150C211yesmaximal, characteristic
3#33208C211yesmaximal, characteristic
4C4×#467C211yesmaximal, characteristic
5#13516C211yesmaximal, characteristic, complemented
6#13515C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13475C211yesmaximal, characteristic, complemented
9#13277C211yesmaximal, characteristic, complemented
10#13178C211yesmaximal, characteristic, complemented
11#13062C211yesmaximal, characteristic
12#13062C211yesmaximal, characteristic
13#12836C211yesmaximal, characteristic
14#12836C211yesmaximal, characteristic
15#12814C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2851—21no—
23C4×C22⋊Q8—21no—
24#2229—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29#2150—24no—
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38#1991—21no—
39#1969C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1941—21no—
44#1933—21no—
45#1892—21no—
46#1846C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1807—21no—
51C4×#107C2211yescharacteristic
52C4×#103—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C23×C8D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C22×C42D411yescharacteristic
68C2×M32—22no—
69C2×M32—22no—
70C2×M32—22no—
71C2×M32—22no—
72C2×M32—41no—
73C2×M32—41no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#499—22no—
79#499—22no—
80#467—22no—
81#467Q814yes—
82C4×Q16—22no—
83C4×Q16—22no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×#106—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8D411yescharacteristic
99C2×C4×C8D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#391—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#387—41no—
107#387—41no—
108#387—41no—
109#385—41no—
110#385—41no—
111#385—41no—
112#382—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—41no—
118C4×C22⋊C4—41no—
119#367—22no—
120#367—41no—
121#366—22no—
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—
139C23×C4D811yescharacteristic
140C23×C4D811yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16C2×Q812yes—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—41no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×Q812yes—
159C22×C8#4112yes—
160C22×C8C2411yescharacteristic
161#125—42no—
162#125—42no—
163C22⋊Q8—24no—
164C22⋊Q8—42no—
165C4×Q8—42no—
166C4×Q8—42no—
167C2×C4⋊C4—21no—
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—41no—
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×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42D811yescharacteristic
190C2×C42D811yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193M32—44no—
194M32—44no—
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—21no—
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#107C2×Q812yes—
221#106—42no—
222#106—42no—
223#103—24no—
224#103—42no—
225#103—42no—
226C4×C8—22no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—22no—
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—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
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×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C22⋊Q812yes—
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4#13712yes—
275C22×C4C22×D411yescharacteristic
276Q16—22no—
277Q16—22no—
278Q16—41no—
279Q16—41no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—44no—
292C2×C8#12112yes—
293C2×C8C22⋊Q812yes—
294C2×C8C22×D411yescharacteristic
295C2×C8C22×Q811yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8C2×#4111yescharacteristic
298C2×C8H(5,𝔽2)11yescharacteristic
299C2×C82−1+411yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—44no—
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—41no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C22⋊C4—44no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—21no—
326C42—21no—
327C42—22no—
328C42—22no—
329C42—82no—
330C42—82no—
331C42#12112yes—
332C42#13512yes—
333C16—22no—
334C16—22no—
335C16—41no—
336C16—41no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23#47412yes—
344C23C22×D811yescharacteristic
345Q8—42no—
346Q8—42no—
347Q8—81no—
348Q8—81no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
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—81no—
389C2×C4—81no—
390C2×C4#46612yes—
391C2×C4#46612yes—
392C2×C4#47412yes—
393C2×C4#51012yes—
394C2×C4C2×#12111yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4#53711yescharacteristic
397C2×C4C2×#13511yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4#57511yescharacteristic
400C8—22no—
401C8—22no—
402C8—41no—
403C8#54611yescharacteristic
404C8D4×Q811yescharacteristic
405C8#55211yescharacteristic
406C8#55411yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22C2×#46611yescharacteristic
415C22C2×#47411yescharacteristic
416C22#324511yescharacteristic
417C22C2×#51011yescharacteristic
418C22#341411yescharacteristic
419C22#341411yescharacteristic
420C22#341411yescharacteristic
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—41no—
426C4—82no—
427C4—82no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4#241711yescharacteristic
435C4#305611yescharacteristic
436C4#307611yescharacteristic
437C4#324811yescharacteristic
438C2—42no—
439C2#2483511yescharacteristic
440C2#3360411yescharacteristic
441C2#3360611yescharacteristic
442C2#3362811yescharacteristic
443C2#3363011yescharacteristic
444C2#3381811yescharacteristic
445C2#3382211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2229C2semidirect
5. #13516C2semidirect
6. #13515C2semidirect
7. #13477C2semidirect
8. #13475C2semidirect
9. #13277C2semidirect
10. #13178C2semidirect
15. #12814C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22292
C2×#21502
#332082
C4×#4672
#135162
#135152
#134772
#134752
#132772
#131782
#130622
#128362
#128142