Finite Group Databasebeta

← #6854476

Subgroups — #6854476

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic
4#31521C218yesmaximal
5#30307C212yesmaximal
6#29191C212yesmaximal
7#29144C212yesmaximal
8#29141C211yesmaximal, characteristic
9#28512C211yesmaximal, characteristic
10#24818C214yesmaximal
11#24332C214yesmaximal
12#22270C212yesmaximal
13C4×#494C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×#557C2212yes—
16C2×#494C2211yescharacteristic
17C2×#475C2214yes—
18C2×C4×Q16C2214yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2859C2212yes—
22#2859C2212yes—
23#2582C2212yes—
24#2568C2212yes—
25#2567—21no—
26#2565C2214yes—
27#2552—21no—
28#2550—22no—
29#2546—21no—
30#2546—21no—
31#2507—22no—
32#2506—22no—
33#2506—22no—
34#2451C2214yes—
35#2369—21no—
36#2365—21no—
37#2365—21no—
38#2355—21no—
39Q8×C4⋊C4C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C42×Q8C2211yescharacteristic
45#2096C2214yes—
46#2085C2218yes—
47#2066C2218yes—
48#2059C2218yes—
49#2026C2218yes—
50#2024C2214yes—
51#1990C2214yes—
52#1990C2214yes—
53#1984C2212yes—
54#1969C2214yes—
55#1956C2211yescharacteristic
56#1956C2212yes—
57#1956C2212yes—
58#1956C2211yescharacteristic
59#1939C2212yes—
60#1916C2214yes—
61#1878C2218yes—
62#1878C2218yes—
63#1865C2218yes—
64#1845C2211yescharacteristic
65#1844C2212yes—
66#1834C2212yes—
67#1834C2212yes—
68#1834C2212yes—
69#1805C2218yes—
70#1803C2214yes—
71C4×C4⋊C8C2212yes—
72C4×C4⋊C8C2211yescharacteristic
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75C22×Q16C2314yes—
76#557—48no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#125—24no—
87C2×#125—22no—
88C2×#125C2314yes—
89C2×C4×Q8—22no—
90C2×C4×Q8D411yescharacteristic
91C2×C4×Q8D411yescharacteristic
92C2×C4×Q8C2312yes—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2314yes—
95C2×C4×Q8C2314yes—
96C2×C4×Q8C2311yescharacteristic
97#494Q818yescomplemented
98#475—48no—
99C4×Q16—48no—
100C2×#107C2314yes—
101C2×#106C2318yes—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2314yes—
104C2×#103C2318yes—
105C2×#103C2318yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2312yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2312yes—
112C2×C4×C8C2311yescharacteristic
113#400—24no—
114#398—24no—
115#398—24no—
116#398—24no—
117#398—24no—
118#395—24no—
119#391—22no—
120#391—22no—
121#391—22no—
122#391—24no—
123#391C2312yes—
124#391C2314yes—
125#391C2314yes—
126#391C2312yes—
127#389—22no—
128#389—22no—
129#389—21no—
130#389—21no—
131#389—22no—
132#389—22no—
133#389—21no—
134#389—21no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—24no—
143#389C2314yes—
144#389C2314yes—
145#387—24no—
146#387—24no—
147#387—24no—
148#384—22no—
149#384—21no—
150#384—21no—
151#384—21no—
152#384—21no—
153#384C2312yes—
154#384C2311yescharacteristic
155#384C2312yes—
156#384C2311yescharacteristic
157#384C2312yes—
158#384C2312yes—
159#382—22no—
160#382C2314yes—
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⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2311yescharacteristic
176C43C2312yes—
177C43C2311yescharacteristic
178#340C2318yes—
179#340C2318yes—
180#340C2318yes—
181#336C2314yes—
182#336C2314yes—
183#336C2312yes—
184#336C2314yes—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8C2×D411yescharacteristic
188C22×Q8C2412yes—
189C22×Q8C2412yes—
190C2×Q16—44no—
191C2×Q16—44no—
192C2×Q16C2×Q8116yes—
193C22×C8C2414yes—
194C22×C8C2412yes—
195C22×C8C2412yes—
196C4⋊Q8—28no—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—44no—
200C4⋊Q8—48no—
201C4⋊Q8—44no—
202C4⋊Q8—44no—
203C4⋊Q8—44no—
204C4⋊Q8—82no—
205C4⋊Q8C2×Q814yes—
206C4⋊Q8C2×Q814yes—
207#125—48no—
208#125—416no—
209#125—84no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—28no—
214C4×Q8—28no—
215C4×Q8—48no—
216C4×Q8—48no—
217C4×Q8—48no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2412yes—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2412yes—
267C2×C42C2412yes—
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271#107—44no—
272#106—48no—
273C4⋊C8—24no—
274C4⋊C8—44no—
275#103—48no—
276#103—44no—
277#103—44no—
278#103—48no—
279C4×C8—24no—
280C4×C8—22no—
281C4×C8—24no—
282C4×C8—42no—
283C4×C8C2×Q814yes—
284#95—24no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95C2×D412yes—
297#95C2×D412yes—
298#95C2×D412yes—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8C22×Q814yes—
315C2×Q8C22×Q814yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4C22×D412yes—
323C22×C4C22×D412yes—
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2511yescharacteristic
333Q16—416no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8C22×Q814yes—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—22no—
383C42—22no—
384C42—24no—
385C42—24no—
386C42—22no—
387C42—24no—
388C42—48no—
389C42—44no—
390C42—44no—
391C42—84no—
392C42#13514yes—
393C42#13714yes—
394C42C22×Q812yes—
395C42H(5,𝔽2)14yes—
396C422−1+412yes—
397C23C23×D411yescharacteristic
398Q8—28no—
399Q8—28no—
400Q8—44no—
401Q8—44no—
402Q8—44no—
403Q8—88no—
404Q8—164no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4#53512yes—
442C2×C4#53512yes—
443C2×C4D4×Q812yes—
444C2×C4D4×Q812yes—
445C2×C4C2×#13512yes—
446C2×C4C2×#13512yes—
447C2×C4C2×#13512yes—
448C2×C4C2×#13712yes—
449C2×C4C2×#13712yes—
450C2×C4C2×#13712yes—
451C2×C4#57712yes—
452C2×C4#57812yes—
453C2×C4C23×Q811yescharacteristic
454C2×C4C2×H(5,𝔽2)12yes—
455C2×C4C2×2−1+411yescharacteristic
456C2×C4#58512yes—
457C8—24no—
458C8—42no—
459C8—44no—
460C22C2×#53511yescharacteristic
461C22C2×D4×Q811yescharacteristic
462C22#350011yescharacteristic
463C22C22×#13511yescharacteristic
464C22C22×#13711yescharacteristic
465C22C2×#57711yescharacteristic
466C22C2×#57811yescharacteristic
467C4—22no—
468C4—22no—
469C4—22no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—44no—
475C4—44no—
476C4—84no—
477C4—84no—
478C4—82no—
479C4—82no—
480C4#321014yes—
481C4#338314yes—
482C4#338714yes—
483C4#350112yes—
484C4#354712yes—
485C2#3961811yescharacteristic
486C2C2×#321011yescharacteristic
487C2#6082411yescharacteristic
488C2C2×#338311yescharacteristic
489C2C2×#338711yescharacteristic
490C2#6253111yescharacteristic
491C2#6253611yescharacteristic
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

Complements

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

NH ≅ G/Ntype
97. #494Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#365602
#332892
#315212
#303072
#291912
#291442
#291412
#285122
#248182
#243322
#222702
C4×#4942
#213392