Finite Group Databasebeta

← #507448

Subgroups — #507448

Order 512.

524 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33216C211yesmaximal, characteristic
2#21525C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic, complemented
4#13530C211yesmaximal, characteristic, complemented
5#13453C211yesmaximal, characteristic, complemented
6#13366C211yesmaximal, characteristic, complemented
7#13112C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#13062C211yesmaximal, characteristic
10#13060C211yesmaximal, characteristic
11#12938C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12836C211yesmaximal, characteristic
14#12834C211yesmaximal, characteristic
15#12545C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2629—21no—
23#2323—21no—
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32#1992—21no—
33#1969C2211yescharacteristic
34#1943—21no—
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1935—21no—
39#1894—21no—
40#1845C2211yescharacteristic
41#1809—21no—
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45C4×#107C2211yescharacteristic
46#1763—21no—
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×#117D411yescharacteristic
66C2×M32—22no—
67C2×M32—22no—
68C2×M32—41no—
69C22×C16—22no—
70C22×C16—22no—
71C22×C16—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#499—41no—
77#499—41no—
78#467—22no—
79#467Q814yes—
80C4×Q16—41no—
81C4×Q16—41no—
82C2×#107—21no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94#405—41no—
95#405—41no—
96C2×#97D411yescharacteristic
97C2×#97D411yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#398—41no—
101#398—41no—
102#396—41no—
103#396—41no—
104#391—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—41no—
109#388—41no—
110#388—41no—
111#387—41no—
112#384—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117#367—41no—
118#366—22no—
119#366—22no—
120#366—41no—
121#358—22no—
122#358—22no—
123#358—41no—
124#358—41no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132C22×Q8—21no—
133C23×C4C2×D411yescharacteristic
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16C2×Q812yes—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—41no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×Q812yes—
151C22×C8#4112yes—
152C22×C8C2411yescharacteristic
153#125—42no—
154#125—42no—
155C22⋊Q8—24no—
156C22⋊Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159#117—41no—
160#117—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4D811yescharacteristic
178C2×C4⋊C4D811yescharacteristic
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4D811yescharacteristic
184C2×C22⋊C4D811yescharacteristic
185C2×C42—41no—
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189M32—44no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—44no—
201C2×C16#4112yes—
202C2×C16#4112yes—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
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—42no—
217#107C2×Q812yes—
218#106—42no—
219#103—24no—
220#103—42no—
221#103—42no—
222#97—41no—
223#97—41no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237C24C2×D811yescharacteristic
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C22⋊Q812yes—
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4#13712yes—
263C22×C4C22×D411yescharacteristic
264Q16—22no—
265Q16—22no—
266Q16—41no—
267Q16—41no—
268M16—44no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8C22⋊Q812yes—
281C2×C8#13512yes—
282C2×C8C22×Q811yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8H(5,𝔽2)11yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—44no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—161no—
309C4⋊C4—161no—
310C4⋊C4—161no—
311C4⋊C4—161no—
312C4⋊C4—161no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—44no—
316C22⋊C4—161no—
317C22⋊C4—161no—
318C22⋊C4—161no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—82no—
324C42—161no—
325C16—22no—
326C16—22no—
327C16—41no—
328C16—41no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23#47412yes—
336C23C22×D811yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—81no—
340Q8—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
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—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
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#47412yes—
384C2×C4C2×C22⋊Q811yescharacteristic
385C2×C4#53211yescharacteristic
386C2×C4#53411yescharacteristic
387C2×C4C2×#13511yescharacteristic
388C2×C4C2×#13711yescharacteristic
389C2×C4#57511yescharacteristic
390C8—22no—
391C8—22no—
392C8—41no—
393C8#55111yescharacteristic
394C8#55111yescharacteristic
395C8#55411yescharacteristic
396C8#55411yescharacteristic
397C22—22no—
398C22—22no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22C2×#47411yescharacteristic
405C22#316911yescharacteristic
406C22#318611yescharacteristic
407C22#341511yescharacteristic
408C22#341611yescharacteristic
409C22#341711yescharacteristic
410C22#341711yescharacteristic
411C4—22no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—82no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4#256411yescharacteristic
425C4#307611yescharacteristic
426C4#318111yescharacteristic
427C4#319011yescharacteristic
428C2—42no—
429C2#2915711yescharacteristic
430C2#3363211yescharacteristic
431C2#3363311yescharacteristic
432C2#3374111yescharacteristic
433C2#3374411yescharacteristic
434C2#3376711yescharacteristic
435C2#3376911yescharacteristic
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
3. #13537C2semidirect
4. #13530C2semidirect
5. #13453C2semidirect
6. #13366C2semidirect
7. #13112C2semidirect
8. #13109C2semidirect
11. #12938C2semidirect
12. #12936C2semidirect

Maximal subgroups

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

subgroupindex
#332162
#215252
#135372
#135302
#134532
#133662
#131122
#131092
#130622
#130602
#129382
#129362
#128362
#128342
#125452