Finite Group Databasebeta

← #6805749

Subgroups — #6805749

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#33225C211yesmaximal, characteristic
4#30814C218yesmaximal
5#30206C212yesmaximal
6#29206C211yesmaximal, characteristic
7#28748C211yesmaximal, characteristic
8#28656C212yesmaximal
9#28515C212yesmaximal
10#26461C214yesmaximal
11#26317C214yesmaximal
12#22218C212yesmaximal
13C4×#500C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×#500C2211yescharacteristic
16C2×#487C2212yes—
17C2×#487C2214yes—
18C2×#479C2218yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2212yes—
22#2859C2212yes—
23#2582—21no—
24#2582—21no—
25#2571—21no—
26#2570—22no—
27#2568C2212yes—
28#2567—21no—
29#2552C2212yes—
30#2550C2214yes—
31#2511—22no—
32#2507—22no—
33#2451—22no—
34#2446C2214yes—
35#2369—21no—
36#2365C2212yes—
37#2355—21no—
38#2355—21no—
39#2355C2212yes—
40Q8×C4⋊C4—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C42×Q8C2211yescharacteristic
46#2097C2218yes—
47#2074C2218yes—
48#2068C2218yes—
49#2058C2214yes—
50#2043C2214yes—
51#2026C2218yes—
52#1990C2212yes—
53#1990C2211yescharacteristic
54#1990C2211yescharacteristic
55#1984C2212yes—
56#1981C2214yes—
57#1956C2212yes—
58#1956C2214yes—
59#1956C2212yes—
60#1956C2212yes—
61#1948C2214yes—
62#1942C2214yes—
63#1939C2212yes—
64#1928C2218yes—
65#1927C2218yes—
66#1916C2214yes—
67#1865C2218yes—
68#1845C2211yescharacteristic
69#1844C2212yes—
70C4×#107C2214yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×C4⋊C8C2212yes—
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2312yes—
83C2×#125C2314yes—
84C2×C4×Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—22no—
87C2×C4×Q8D411yescharacteristic
88C2×C4×Q8D411yescharacteristic
89C2×C4×Q8C2314yes—
90C2×C4×Q8C2311yescharacteristic
91#500—24no—
92#487—28no—
93#487—48no—
94#479—416no—
95C2×#107C2314yes—
96C2×#107C2314yes—
97C2×#106C2318yes—
98C2×C4⋊C8C2314yes—
99C2×C4⋊C8C2312yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104C2×C4×C8C2312yes—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2312yes—
107C2×C4×C8C2312yes—
108C2×C4×C8C2311yescharacteristic
109#400—24no—
110#400—24no—
111#398—24no—
112#398—24no—
113#398—24no—
114#395—24no—
115#391—22no—
116#391—22no—
117#391—22no—
118#391—21no—
119#391—21no—
120#391—22no—
121#391—22no—
122#391—24no—
123#391C2314yes—
124#391C2314yes—
125#391C2314yes—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—21no—
130#389—22no—
131#389—22no—
132#389—21no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—24no—
138#389C2312yes—
139#389C2314yes—
140#389C2312yes—
141#387—24no—
142#387—24no—
143#387—24no—
144#387—24no—
145#384—22no—
146#384—21no—
147#384—21no—
148#384—21no—
149#384—21no—
150#384—21no—
151#384—21no—
152#384C2311yescharacteristic
153#384C2311yescharacteristic
154#384C2312yes—
155#384C2312yes—
156#382—22no—
157#382—22no—
158#382—22no—
159#382C2314yes—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4D412yes—
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4D412yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C43C2312yes—
177C43C2311yescharacteristic
178#340C2318yes—
179#340C2314yes—
180#340C2314yes—
181#336C2312yes—
182#336C2314yes—
183#336C2314yes—
184#336C2314yes—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8C2×D411yescharacteristic
189C22×Q8C2414yes—
190C22×C8C2412yes—
191C22×C8C2414yes—
192C22×C8C2412yes—
193C4⋊Q8—22no—
194C4⋊Q8—24no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—28no—
198C4⋊Q8—24no—
199C4⋊Q8—44no—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C4⋊Q8—82no—
203C4⋊Q8#4114yes—
204#125—48no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—48no—
209C4×Q8—44no—
210C4×Q8—44no—
211C4×Q8—84no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2412yes—
259C2×C42C2412yes—
260C2×C42C2411yescharacteristic
261C2×C42C2411yescharacteristic
262C2×C42C2411yescharacteristic
263#107—28no—
264#107—28no—
265#106—48no—
266C4⋊C8—28no—
267C4⋊C8—24no—
268#103—216no—
269#103—216no—
270#103—48no—
271#103—48no—
272C4×C8—24no—
273C4×C8—24no—
274C4×C8—22no—
275C4×C8—24no—
276C4×C8—22no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—24no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95C2×D412yes—
292#95C2×D412yes—
293#95C2×D412yes—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—28no—
300C2×Q8—28no—
301C2×Q8—24no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—44no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—82no—
309C2×Q8—82no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4C22×D412yes—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D412yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2511yescharacteristic
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8H(5,𝔽2)14yes—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4C2×#4114yes—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—22no—
376C42—22no—
377C42—24no—
378C42—22no—
379C42—24no—
380C42—24no—
381C42—44no—
382C42—44no—
383C42—84no—
384C42—84no—
385C42#13514yes—
386C42#13714yes—
387C42C2×#4112yes—
388C422−1+414yes—
389C422−1+412yes—
390C23C23×D411yescharacteristic
391Q8—48no—
392Q8—44no—
393Q8—48no—
394Q8—48no—
395Q8—84no—
396Q8—164no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4#53612yes—
434C2×C4#53612yes—
435C2×C4#54812yes—
436C2×C4#54812yes—
437C2×C4C2×#13512yes—
438C2×C4C2×#13512yes—
439C2×C4C2×#13512yes—
440C2×C4C2×#13712yes—
441C2×C4C2×#13712yes—
442C2×C4C2×#13712yes—
443C2×C4#57712yes—
444C2×C4#57812yes—
445C2×C4C22×#4111yescharacteristic
446C2×C4C2×H(5,𝔽2)12yes—
447C2×C4C2×2−1+412yes—
448C2×C4C2×2−1+411yescharacteristic
449C8—24no—
450C8—44no—
451C8—42no—
452C22C2×#53611yescharacteristic
453C22C2×#54811yescharacteristic
454C22#349811yescharacteristic
455C22C22×#13511yescharacteristic
456C22C22×#13711yescharacteristic
457C22C2×#57711yescharacteristic
458C22C2×#57811yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—44no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—84no—
468C4—82no—
469C4—82no—
470C4—82no—
471C4—84no—
472C4#323114yes—
473C4#336014yes—
474C4#337414yes—
475C4#352212yes—
476C4#355012yes—
477C2#3916811yescharacteristic
478C2C2×#323111yescharacteristic
479C2#6087811yescharacteristic
480C2C2×#336011yescharacteristic
481C2C2×#337411yescharacteristic
482C2#6241811yescharacteristic
483C2#6242211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#365982
#365632
#332252
#308142
#302062
#292062
#287482
#286562
#285152
#264612
#263172
#222182
C4×#5002
#213392