Finite Group Databasebeta

← #6854601

Subgroups — #6854601

Order 512.

1494 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#36560C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic
4#32130C218yesmaximal
5#30304C212yesmaximal
6#29219C211yesmaximal, characteristic
7#29219C212yesmaximal
8#29208C211yesmaximal, characteristic
9#29178C212yesmaximal
10#26440C214yesmaximal
11#26276C214yesmaximal
12#22247C212yesmaximal
13C4×#500C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×Q82—21no—
16C2×#557—21no—
17C2×#500C2211yescharacteristic
18#2868C2211yescharacteristic
19#2868C2211yescharacteristic
20#2868C2212yes—
21#2859C2212yes—
22#2568—21no—
23#2567—22no—
24#2567—21no—
25#2565—22no—
26#2551—21no—
27#2551—22no—
28#2551C2212yes—
29#2551C2214yes—
30#2546—21no—
31#2546C2212yes—
32#2506—22no—
33#2506C2214yes—
34#2365—21no—
35#2365C2212yes—
36Q8×C4⋊C4—21no—
37Q8×C4⋊C4C2212yes—
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C42×Q8C2211yescharacteristic
43#2099C2214yes—
44#2099C2218yes—
45#2086C2218yes—
46#2083C2218yes—
47#2057C2214yes—
48#2057C2218yes—
49#1990C2214yes—
50#1990C2214yes—
51#1990C2212yes—
52#1990C2211yescharacteristic
53#1990C2212yes—
54#1990C2211yescharacteristic
55#1988C2218yes—
56#1988C2214yes—
57#1984C2212yes—
58#1947C2214yes—
59#1942C2214yes—
60#1939C2212yes—
61#1918C2218yes—
62#1918C2218yes—
63#1918C2218yes—
64#1916C2214yes—
65#1845C2212yes—
66#1845C2211yescharacteristic
67#1834C2212yes—
68#1834C2212yes—
69#1834C2212yes—
70C4×#107C2214yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×C4⋊C8C2212yes—
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75Q82—48no—
76#557—48no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2314yes—
90C2×C4⋊Q8C2312yes—
91C2×#125—24no—
92C2×#125—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8D411yescharacteristic
100C2×C4×Q8D411yescharacteristic
101C2×C4×Q8C2311yescharacteristic
102#500Q818yescomplemented
103C2×#107C2314yes—
104C2×#107C2318yes—
105C2×#107C2314yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2314yes—
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#395—24no—
117#395—24no—
118#395—24no—
119#391—22no—
120#391—22no—
121#391—21no—
122#391—22no—
123#391—22no—
124#391—21no—
125#389—22no—
126#389—24no—
127#389—22no—
128#389—24no—
129#389—21no—
130#389—22no—
131#389—22no—
132#389—21no—
133#389—22no—
134#389—22no—
135#389C2312yes—
136#389C2314yes—
137#389C2314yes—
138#389C2312yes—
139#389C2314yes—
140#389C2314yes—
141#387—24no—
142#387—24no—
143#384—21no—
144#384—22no—
145#384—21no—
146#384—22no—
147#384—21no—
148#384—21no—
149#384C2312yes—
150#384C2311yescharacteristic
151#384C2312yes—
152#384C2311yescharacteristic
153#384C2314yes—
154#384C2312yes—
155#384C2312yes—
156#382—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4D412yes—
162C4×C4⋊C4D411yescharacteristic
163C4×C4⋊C4D411yescharacteristic
164C4×C4⋊C4D412yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2312yes—
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2312yes—
169C4×C4⋊C4C2311yescharacteristic
170C4×C4⋊C4C2312yes—
171C43C2312yes—
172C43C2311yescharacteristic
173#340C2318yes—
174#340C2318yes—
175#340C2318yes—
176#340C2314yes—
177#340C2318yes—
178#340C2318yes—
179#340C2314yes—
180#336C2312yes—
181#336C2314yes—
182#336C2314yes—
183#336C2314yes—
184C22×Q8—21no—
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8C2×D411yescharacteristic
190C22×C8C2414yes—
191C22×C8C2412yes—
192C22×C8C2412yes—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—28no—
196C4⋊Q8—44no—
197C4⋊Q8—48no—
198C4⋊Q8—48no—
199C4⋊Q8—48no—
200C4⋊Q8—416no—
201C4⋊Q8—44no—
202C4⋊Q8—48no—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205C4⋊Q8—82no—
206C4⋊Q8—84no—
207C4⋊Q8C2×Q814yes—
208C4⋊Q8C2×Q814yes—
209#125—416no—
210#125—48no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—48no—
214C4×Q8—48no—
215C4×Q8—48no—
216C4×Q8—44no—
217C4×Q8—48no—
218C4×Q8—44no—
219C4×Q8D818yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2412yes—
270C2×C42C2412yes—
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274#107—44no—
275#107—48no—
276#107C2×Q8116yes—
277C4⋊C8—24no—
278C4⋊C8—44no—
279C4×C8—24no—
280C4×C8—22no—
281C4×C8—24no—
282C4×C8—42no—
283C4×C8C2×Q814yes—
284#95—22no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—22no—
289#95—24no—
290#95—24no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95C2×D412yes—
295#95C2×D412yes—
296#95C2×D412yes—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—44no—
305C2×Q8—48no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—82no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—82no—
314C2×Q8C2×D814yes—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2511yescharacteristic
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8C22×Q814yes—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—44no—
359C4⋊C4—48no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—48no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—42no—
372C4⋊C4—82no—
373C4⋊C4—84no—
374C4⋊C4—82no—
375C4⋊C4—84no—
376C4⋊C4C2×D814yes—
377C4⋊C4C2×D818yes—
378C4⋊C4C22×Q814yes—
379C4⋊C4C22×Q814yes—
380C42—24no—
381C42—22no—
382C42—24no—
383C42—22no—
384C42—24no—
385C42—24no—
386C42—22no—
387C42—24no—
388C42—48no—
389C42—44no—
390C42—48no—
391C42—44no—
392C42C2×D818yes—
393C42C2×D814yes—
394C42#13714yes—
395C42C22×Q812yes—
396C422−1+414yes—
397C422−1+412yes—
398C23C23×D411yescharacteristic
399Q8—28no—
400Q8—48no—
401Q8—48no—
402Q8—88no—
403Q8—168no—
404Q8—164no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4#53612yes—
440C2×C4#53612yes—
441C2×C4D4×Q812yes—
442C2×C4D4×Q812yes—
443C2×C4C22×D814yes—
444C2×C4C22×D814yes—
445C2×C4C22×D812yes—
446C2×C4C22×D812yes—
447C2×C4C22×D812yes—
448C2×C4C2×#13712yes—
449C2×C4C2×#13712yes—
450C2×C4C2×#13712yes—
451C2×C4#57712yes—
452C2×C4#57812yes—
453C2×C4C23×Q811yescharacteristic
454C2×C4C2×2−1+412yes—
455C2×C4C2×2−1+411yescharacteristic
456C2×C4#58512yes—
457C8—24no—
458C8—42no—
459C8—44no—
460C22C2×#53611yescharacteristic
461C22C2×D4×Q811yescharacteristic
462C22#350011yescharacteristic
463C22C23×D811yescharacteristic
464C22C22×#13711yescharacteristic
465C22C2×#57711yescharacteristic
466C22C2×#57811yescharacteristic
467C4—24no—
468C4—22no—
469C4—22no—
470C4—24no—
471C4—22no—
472C4—42no—
473C4—44no—
474C4—44no—
475C4—42no—
476C4—82no—
477C4—84no—
478C4—82no—
479C4—84no—
480C4#322614yes—
481C4Q8×D814yes—
482C4#338714yes—
483C4#350212yes—
484C4#355512yes—
485C2#3961911yescharacteristic
486C2C2×#322611yescharacteristic
487C2#6087811yescharacteristic
488C2C2×Q8×D811yescharacteristic
489C2C2×#338711yescharacteristic
490C2#6252911yescharacteristic
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
102. #500Q8semidirect

Maximal subgroups

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

subgroupindex
#365602
Q8×C4⋊Q82
#332892
#321302
#303042
#292192
#292082
#291782
#264402
#262762
#222472
C4×#5002
#213392