Finite Group Databasebeta

← #6854597

Subgroups — #6854597

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic
4#32132C218yesmaximal
5#30305C212yesmaximal
6#29219C212yesmaximal
7#29218C211yesmaximal, characteristic
8#29206C211yesmaximal, characteristic
9#29176C212yesmaximal
10#26442C214yesmaximal
11#26280C214yesmaximal
12#22246C212yesmaximal
13C4×#498C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×Q82—21no—
16C2×#498C2211yescharacteristic
17#2868C2211yescharacteristic
18#2868C2211yescharacteristic
19#2868C2212yes—
20#2859C2212yes—
21#2582—21no—
22#2567—22no—
23#2567—21no—
24#2567C2212yes—
25#2552—21no—
26#2551—21no—
27#2551—22no—
28#2551C2212yes—
29#2551C2214yes—
30#2550—22no—
31#2507C2214yes—
32#2451—22no—
33#2369C2212yes—
34#2355—21no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4—21no—
37Q8×C4⋊C4C2212yes—
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C42×Q8C2211yescharacteristic
43#2100C2214yes—
44#2099C2218yes—
45#2088C2218yes—
46#2084C2218yes—
47#2058C2214yes—
48#2058C2218yes—
49#1990C2214yes—
50#1990C2214yes—
51#1990C2212yes—
52#1990C2212yes—
53#1990C2211yescharacteristic
54#1990C2211yescharacteristic
55#1988C2218yes—
56#1987C2214yes—
57#1984C2212yes—
58#1948C2214yes—
59#1941C2214yes—
60#1939C2212yes—
61#1920C2218yes—
62#1920C2218yes—
63#1919C2218yes—
64#1916C2214yes—
65#1844C2212yes—
66#1844C2211yescharacteristic
67#1834C2212yes—
68#1834C2212yes—
69#1834C2212yes—
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—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2314yes—
88C2×C4⋊Q8C2312yes—
89C2×#125—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—22no—
95C2×C4×Q8D411yescharacteristic
96C2×C4×Q8D411yescharacteristic
97C2×C4×Q8C2311yescharacteristic
98#498Q818yescomplemented
99C2×#106C2314yes—
100C2×#106C2318yes—
101C2×#106C2314yes—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2314yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2312yes—
106C2×C4×C8C2312yes—
107C2×C4×C8C2312yes—
108C2×C4×C8C2311yescharacteristic
109#398—24no—
110#398—24no—
111#395—24no—
112#395—24no—
113#395—24no—
114#395—24no—
115#391—22no—
116#391—22no—
117#391—21no—
118#391—22no—
119#391—21no—
120#391—24no—
121#389—22no—
122#389—24no—
123#389—22no—
124#389—24no—
125#389—22no—
126#389—21no—
127#389—22no—
128#389—22no—
129#389—21no—
130#389—22no—
131#389—22no—
132#389—24no—
133#389C2314yes—
134#389C2312yes—
135#389C2314yes—
136#389C2312yes—
137#389C2314yes—
138#389C2314yes—
139#387—24no—
140#384—22no—
141#384—22no—
142#384—21no—
143#384—21no—
144#384—22no—
145#384—21no—
146#384—21no—
147#384C2312yes—
148#384C2311yescharacteristic
149#384C2312yes—
150#384C2311yescharacteristic
151#384C2314yes—
152#384C2312yes—
153#384C2312yes—
154#382—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4D412yes—
161C4×C4⋊C4D411yescharacteristic
162C4×C4⋊C4D411yescharacteristic
163C4×C4⋊C4D412yes—
164C4×C4⋊C4C2312yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2312yes—
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2311yescharacteristic
169C4×C4⋊C4C2312yes—
170C43C2312yes—
171C43C2311yescharacteristic
172#340C2318yes—
173#340C2318yes—
174#340C2318yes—
175#340C2314yes—
176#340C2318yes—
177#340C2318yes—
178#340C2314yes—
179#336C2312yes—
180#336C2314yes—
181#336C2314yes—
182#336C2314yes—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8C2×D411yescharacteristic
189C22×C8C2414yes—
190C22×C8C2412yes—
191C22×C8C2412yes—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—28no—
195C4⋊Q8—44no—
196C4⋊Q8—48no—
197C4⋊Q8—416no—
198C4⋊Q8—48no—
199C4⋊Q8—48no—
200C4⋊Q8—44no—
201C4⋊Q8—84no—
202C4⋊Q8—82no—
203C4⋊Q8—82no—
204C4⋊Q8—82no—
205C4⋊Q8C2×Q814yes—
206C4⋊Q8C2×Q814yes—
207#125—84no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—48no—
211C4×Q8—48no—
212C4×Q8—44no—
213C4×Q8—44no—
214C4×Q8—84no—
215C4×Q8SD1618yes—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2412yes—
266C2×C42C2412yes—
267C2×C42C2411yescharacteristic
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270#106—44no—
271#106—48no—
272#106C2×Q8116yes—
273C4⋊C8—24no—
274C4⋊C8—44no—
275C4×C8—24no—
276C4×C8—22no—
277C4×C8—24no—
278C4×C8—42no—
279C4×C8C2×Q814yes—
280#95—24no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95—24no—
286#95—22no—
287#95—24no—
288#95—22no—
289#95—24no—
290#95C2×D412yes—
291#95C2×D412yes—
292#95C2×D412yes—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—44no—
298C2×Q8—42no—
299C2×Q8—48no—
300C2×Q8—44no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—82no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—82no—
308C2×Q8—82no—
309C2×Q8—82no—
310C2×Q8C2×SD1614yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2511yescharacteristic
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—22no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8C22×Q814yes—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—28no—
349C4⋊C4—44no—
350C4⋊C4—48no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—82no—
367C4⋊C4—84no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4C2×SD1618yes—
373C4⋊C4C2×SD1614yes—
374C4⋊C4C22×Q814yes—
375C4⋊C4C22×Q814yes—
376C42—24no—
377C42—24no—
378C42—22no—
379C42—24no—
380C42—22no—
381C42—24no—
382C42—22no—
383C42—24no—
384C42—48no—
385C42—44no—
386C42—44no—
387C42—84no—
388C42C2×SD1618yes—
389C42C2×SD1614yes—
390C42#13714yes—
391C42C22×Q812yes—
392C422−1+414yes—
393C422−1+412yes—
394C23C23×D411yescharacteristic
395Q8—28no—
396Q8—48no—
397Q8—48no—
398Q8—168no—
399Q8—164no—
400Q8—164no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4#53612yes—
436C2×C4#53612yes—
437C2×C4D4×Q812yes—
438C2×C4D4×Q812yes—
439C2×C4C22×SD1614yes—
440C2×C4C22×SD1614yes—
441C2×C4C22×SD1612yes—
442C2×C4C22×SD1612yes—
443C2×C4C22×SD1612yes—
444C2×C4C2×#13712yes—
445C2×C4C2×#13712yes—
446C2×C4C2×#13712yes—
447C2×C4#57612yes—
448C2×C4#57712yes—
449C2×C4C23×Q811yescharacteristic
450C2×C4C2×2−1+412yes—
451C2×C4C2×2−1+411yescharacteristic
452C2×C4#58512yes—
453C8—24no—
454C8—42no—
455C8—44no—
456C22C2×#53611yescharacteristic
457C22C2×D4×Q811yescharacteristic
458C22#350011yescharacteristic
459C22C23×SD1611yescharacteristic
460C22C22×#13711yescharacteristic
461C22C2×#57611yescharacteristic
462C22C2×#57711yescharacteristic
463C4—24no—
464C4—22no—
465C4—22no—
466C4—24no—
467C4—22no—
468C4—42no—
469C4—44no—
470C4—44no—
471C4—42no—
472C4—84no—
473C4—82no—
474C4—82no—
475C4—84no—
476C4#322914yes—
477C4Q8×SD1614yes—
478C4#338514yes—
479C4#350212yes—
480C4#355512yes—
481C2#3961911yescharacteristic
482C2C2×#322911yescharacteristic
483C2#6087711yescharacteristic
484C2C2×Q8×SD1611yescharacteristic
485C2C2×#338511yescharacteristic
486C2#6252511yescharacteristic
487C2#6254011yescharacteristic
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
98. #498Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
Q8×C4⋊Q82
#332892
#321322
#303052
#292192
#292182
#292062
#291762
#264422
#262802
#222462
C4×#4982
#213392