Finite Group Databasebeta

← #6805743

Subgroups — #6805743

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic
3#33225C211yesmaximal, characteristic
4#30808C218yesmaximal
5#30205C212yesmaximal
6#29206C211yesmaximal, characteristic
7#28748C211yesmaximal, characteristic
8#28653C212yesmaximal
9#28512C212yesmaximal
10#26462C214yesmaximal
11#26320C214yesmaximal
12#22218C212yesmaximal
13C4×#498C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×Q82—21no—
16C2×#498C2211yescharacteristic
17C2×#487C2212yes—
18C2×#487C2214yes—
19C2×#475C2218yes—
20#2868C2211yescharacteristic
21#2868C2212yes—
22#2868C2211yescharacteristic
23#2862C2212yes—
24#2582—21no—
25#2571—21no—
26#2570—22no—
27#2568C2212yes—
28#2567—22no—
29#2567—21no—
30#2551—21no—
31#2551—22no—
32#2551C2214yes—
33#2551C2212yes—
34#2511—22no—
35#2446C2214yes—
36#2365C2212yes—
37#2355—21no—
38#2355—21no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C42×Q8C2211yescharacteristic
46#2096C2218yes—
47#2072C2218yes—
48#2066C2218yes—
49#2058C2214yes—
50#2043C2214yes—
51#2024C2218yes—
52#1990C2212yes—
53#1990C2211yescharacteristic
54#1990C2211yescharacteristic
55#1984C2212yes—
56#1981C2214yes—
57#1956C2212yes—
58#1956C2214yes—
59#1956C2212yes—
60#1956C2212yes—
61#1948C2214yes—
62#1941C2214yes—
63#1939C2212yes—
64#1927C2218yes—
65#1927C2218yes—
66#1916C2214yes—
67#1865C2218yes—
68#1844C2212yes—
69#1844C2211yescharacteristic
70C4×#106C2214yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×C4⋊C8C2212yes—
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75Q82—48no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2314yes—
88C2×C4⋊Q8C2312yes—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8D411yescharacteristic
95C2×C4×Q8C2314yes—
96C2×C4×Q8C2311yescharacteristic
97#498—24no—
98#487—28no—
99#487—48no—
100#475—416no—
101C2×#106C2314yes—
102C2×#106C2314yes—
103C2×#106C2318yes—
104C2×C4⋊C8C2314yes—
105C2×C4⋊C8C2312yes—
106C2×#103C2318yes—
107C2×#103C2318yes—
108C2×#103C2318yes—
109C2×#103C2318yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2312yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115#400—24no—
116#400—24no—
117#398—24no—
118#395—24no—
119#395—24no—
120#395—24no—
121#391—22no—
122#391—22no—
123#391—21no—
124#391—21no—
125#391—22no—
126#391—22no—
127#391C2314yes—
128#391C2314yes—
129#391C2314yes—
130#389—22no—
131#389—24no—
132#389—22no—
133#389—24no—
134#389—22no—
135#389—21no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—21no—
140#389C2312yes—
141#389C2314yes—
142#389C2312yes—
143#387—24no—
144#387—24no—
145#387—24no—
146#384—22no—
147#384—21no—
148#384—21no—
149#384—21no—
150#384—21no—
151#384—22no—
152#384—21no—
153#384—21no—
154#384C2311yescharacteristic
155#384C2311yescharacteristic
156#384C2312yes—
157#384C2314yes—
158#384C2312yes—
159#382—22no—
160#382—22no—
161#382—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4D412yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2311yescharacteristic
176C4×C4⋊C4C2312yes—
177C43C2312yes—
178C43C2311yescharacteristic
179#340C2318yes—
180#340C2314yes—
181#340C2314yes—
182#336C2312yes—
183#336C2314yes—
184#336C2314yes—
185#336C2314yes—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8C2×D411yescharacteristic
191C22×Q8C2414yes—
192C22×C8C2412yes—
193C22×C8C2414yes—
194C22×C8C2412yes—
195C4⋊Q8—22no—
196C4⋊Q8—24no—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—28no—
200C4⋊Q8—24no—
201C4⋊Q8—48no—
202C4⋊Q8—44no—
203C4⋊Q8—44no—
204C4⋊Q8—416no—
205C4⋊Q8—48no—
206C4⋊Q8—48no—
207C4⋊Q8—44no—
208C4⋊Q8—82no—
209C4⋊Q8—84no—
210C4⋊Q8#4114yes—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—48no—
214C4×Q8—48no—
215C4×Q8—48no—
216C4×Q8—44no—
217C4×Q8—44no—
218C4×Q8SD1618yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2412yes—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D412yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2412yes—
267C2×C42C2412yes—
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271#106—28no—
272#106—28no—
273#106—48no—
274C4⋊C8—28no—
275C4⋊C8—24no—
276#103—216no—
277#103—216no—
278#103—48no—
279#103—48no—
280C4×C8—24no—
281C4×C8—24no—
282C4×C8—22no—
283C4×C8—24no—
284C4×C8—22no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95C2×D412yes—
298#95C2×D412yes—
299#95C2×D412yes—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—28no—
305C2×Q8—28no—
306C2×Q8—24no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—48no—
310C2×Q8—44no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—84no—
315C2×Q8—82no—
316C2×Q8—84no—
317C2×Q8—82no—
318C2×Q8C2×SD1614yes—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2511yescharacteristic
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—24no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8H(5,𝔽2)14yes—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—28no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—84no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—84no—
377C4⋊C4—82no—
378C4⋊C4—84no—
379C4⋊C4C2×SD1618yes—
380C4⋊C4C2×SD1614yes—
381C4⋊C4C2×#4114yes—
382C42—24no—
383C42—22no—
384C42—24no—
385C42—22no—
386C42—24no—
387C42—22no—
388C42—24no—
389C42—24no—
390C42—48no—
391C42—44no—
392C42—44no—
393C42—84no—
394C42C2×SD1618yes—
395C42C2×SD1614yes—
396C42#13714yes—
397C42C2×#4112yes—
398C422−1+414yes—
399C422−1+412yes—
400C23C23×D411yescharacteristic
401Q8—28no—
402Q8—48no—
403Q8—48no—
404Q8—48no—
405Q8—84no—
406Q8—168no—
407Q8—164no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4#53612yes—
443C2×C4#53612yes—
444C2×C4#54812yes—
445C2×C4#54812yes—
446C2×C4C22×SD1614yes—
447C2×C4C22×SD1614yes—
448C2×C4C22×SD1612yes—
449C2×C4C22×SD1612yes—
450C2×C4C22×SD1612yes—
451C2×C4C2×#13712yes—
452C2×C4C2×#13712yes—
453C2×C4C2×#13712yes—
454C2×C4#57712yes—
455C2×C4#57812yes—
456C2×C4C22×#4111yescharacteristic
457C2×C4C2×H(5,𝔽2)12yes—
458C2×C4C2×2−1+412yes—
459C2×C4C2×2−1+411yescharacteristic
460C8—24no—
461C8—44no—
462C8—42no—
463C22C2×#53611yescharacteristic
464C22C2×#54811yescharacteristic
465C22#349811yescharacteristic
466C22C23×SD1611yescharacteristic
467C22C22×#13711yescharacteristic
468C22C2×#57711yescharacteristic
469C22C2×#57811yescharacteristic
470C4—24no—
471C4—24no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—44no—
476C4—42no—
477C4—42no—
478C4—84no—
479C4—82no—
480C4—84no—
481C4—82no—
482C4—82no—
483C4#322914yes—
484C4#335914yes—
485C4#336914yes—
486C4#352212yes—
487C4#355012yes—
488C2#3916811yescharacteristic
489C2C2×#322911yescharacteristic
490C2#6087811yescharacteristic
491C2C2×#335911yescharacteristic
492C2C2×#336911yescharacteristic
493C2#6241311yescharacteristic
494C2#6242411yescharacteristic
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
Q8×C4⋊Q82
#332252
#308082
#302052
#292062
#287482
#286532
#285122
#264622
#263202
#222182
C4×#4982
#213392