Finite Group Databasebeta

← #6854248

Subgroups — #6854248

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic
3#33310C211yesmaximal, characteristic
4#32343C218yesmaximal
5#30307C212yesmaximal
6#29219C211yesmaximal, characteristic
7#29207C212yesmaximal
8#29191C212yesmaximal
9#28511C211yesmaximal, characteristic
10#25836C214yesmaximal
11#25454C214yesmaximal
12#22218C212yesmaximal
13C4×#487C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×#557C2212yes—
16C2×#498C2214yes—
17C2×#487C2211yescharacteristic
18C2×#477C2214yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2859C2212yes—
22#2859C2212yes—
23#2582—21no—
24#2568C2212yes—
25#2567C2212yes—
26#2565C2214yes—
27#2552—21no—
28#2550—22no—
29#2546—21no—
30#2546—21no—
31#2507C2214yes—
32#2506—22no—
33#2506—22no—
34#2451—22no—
35#2369C2212yes—
36#2365—21no—
37#2365—21no—
38#2355—21no—
39Q8×C4⋊C4—21no—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C42×Q8C2211yescharacteristic
45#2099C2214yes—
46#2095C2214yes—
47#2085C2218yes—
48#2066C2218yes—
49#2059C2218yes—
50#2059C2218yes—
51#1990C2211yescharacteristic
52#1990C2212yes—
53#1988C2214yes—
54#1984C2212yes—
55#1956C2211yescharacteristic
56#1956C2212yes—
57#1949C2218yes—
58#1939C2212yes—
59#1933C2214yes—
60#1927C2218yes—
61#1916C2214yes—
62#1865C2218yes—
63#1865C2218yes—
64#1845C2212yes—
65#1844C2212yes—
66#1844C2214yes—
67#1844C2212yes—
68#1834C2212yes—
69#1834C2211yescharacteristic
70C4×C4⋊C8C2212yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×#103C2214yes—
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75#557—48no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125—24no—
86C2×#125—22no—
87C2×#125C2314yes—
88C2×C4×Q8—22no—
89C2×C4×Q8—21no—
90C2×C4×Q8—22no—
91C2×C4×Q8D411yescharacteristic
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2314yes—
95C2×C4×Q8C2311yescharacteristic
96#498—48no—
97#487Q818yescomplemented
98#477—48no—
99C2×#107C2318yes—
100C2×#106C2318yes—
101C2×#106C2318yes—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2314yes—
104C2×#103C2314yes—
105C2×#103C2318yes—
106C2×#103C2314yes—
107C2×C4×C8C2312yes—
108C2×C4×C8C2312yes—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2312yes—
111C2×C4×C8C2311yescharacteristic
112#400—24no—
113#398—24no—
114#398—24no—
115#398—24no—
116#398—24no—
117#395—24no—
118#391—22no—
119#391—22no—
120#391—22no—
121#391—22no—
122#391—21no—
123#391—24no—
124#391C2312yes—
125#391C2314yes—
126#389—22no—
127#389—21no—
128#389—21no—
129#389—22no—
130#389—22no—
131#389—21no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—24no—
140#389C2314yes—
141#389C2312yes—
142#389C2314yes—
143#389C2314yes—
144#387—24no—
145#387—24no—
146#387—24no—
147#384—22no—
148#384—21no—
149#384—21no—
150#384—21no—
151#384—21no—
152#384C2311yescharacteristic
153#384C2311yescharacteristic
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#384C2312yes—
158#382—22no—
159#382C2314yes—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4D412yes—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C43C2312yes—
176C43C2311yescharacteristic
177#340C2314yes—
178#340C2314yes—
179#340C2318yes—
180#340C2318yes—
181#336C2314yes—
182#336C2314yes—
183#336C2314yes—
184#336C2312yes—
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8C2×D411yescharacteristic
189C22×Q8C2412yes—
190C22×C8C2414yes—
191C22×C8C2412yes—
192C22×C8C2412yes—
193C4⋊Q8—28no—
194C4⋊Q8—28no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—48no—
198C4⋊Q8—44no—
199C4⋊Q8—44no—
200C4⋊Q8—82no—
201C4⋊Q8—82no—
202C4⋊Q8C2×Q814yes—
203C4⋊Q8C2×Q814yes—
204#125—48no—
205#125—416no—
206#125—84no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—28no—
211C4×Q8—48no—
212C4×Q8—48no—
213C4×Q8—44no—
214C4×Q8—84no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2412yes—
264C2×C42C2412yes—
265C2×C42C2411yescharacteristic
266C2×C42C2411yescharacteristic
267C2×C42C2411yescharacteristic
268#107—48no—
269#106—48no—
270#106—48no—
271C4⋊C8—28no—
272C4⋊C8—24no—
273#103—28no—
274#103—48no—
275#103C2×Q8116yes—
276C4×C8—24no—
277C4×C8—24no—
278C4×C8—22no—
279C4×C8—24no—
280C4×C8C2×Q814yes—
281#95—24no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95C2×D412yes—
294#95C2×D412yes—
295#95C2×D412yes—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—42no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—42no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8C22×Q814yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4C22×D412yes—
319C22×C4C22×D412yes—
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2511yescharacteristic
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8C22×Q814yes—
337C2×C82−1+414yes—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4C22×Q814yes—
375C42—24no—
376C42—24no—
377C42—24no—
378C42—24no—
379C42—22no—
380C42—22no—
381C42—24no—
382C42—22no—
383C42—24no—
384C42—48no—
385C42—44no—
386C42—44no—
387C42—84no—
388C42#13514yes—
389C42#13714yes—
390C42C22×Q812yes—
391C42H(5,𝔽2)14yes—
392C422−1+412yes—
393C23C23×D411yescharacteristic
394Q8—28no—
395Q8—48no—
396Q8—44no—
397Q8—44no—
398Q8—88no—
399Q8—164no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4#53512yes—
437C2×C4#53512yes—
438C2×C4D4×Q812yes—
439C2×C4D4×Q812yes—
440C2×C4C2×#13512yes—
441C2×C4C2×#13512yes—
442C2×C4C2×#13512yes—
443C2×C4C2×#13712yes—
444C2×C4C2×#13712yes—
445C2×C4C2×#13712yes—
446C2×C4#57712yes—
447C2×C4#57712yes—
448C2×C4C23×Q811yescharacteristic
449C2×C4C2×H(5,𝔽2)12yes—
450C2×C4C2×2−1+412yes—
451C2×C4C2×2−1+411yescharacteristic
452C8—24no—
453C8—24no—
454C8—44no—
455C22C2×#53511yescharacteristic
456C22C2×D4×Q811yescharacteristic
457C22#349911yescharacteristic
458C22C22×#13511yescharacteristic
459C22C22×#13711yescharacteristic
460C22C2×#57711yescharacteristic
461C22C2×#57711yescharacteristic
462C4—22no—
463C4—22no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—44no—
470C4—44no—
471C4—82no—
472C4—84no—
473C4—82no—
474C4—84no—
475C4#321014yes—
476C4#338114yes—
477C4#338714yes—
478C4#350312yes—
479C4#355212yes—
480C2#3961611yescharacteristic
481C2C2×#321011yescharacteristic
482C2#6082011yescharacteristic
483C2C2×#338111yescharacteristic
484C2C2×#338711yescharacteristic
485C2#6247011yescharacteristic
486C2#6247811yescharacteristic
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. #487Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#365602
#333102
#323432
#303072
#292192
#292072
#291912
#285112
#258362
#254542
#222182
C4×#4872
#213392