Finite Group Databasebeta

← #6379362

Subgroups — #6379362

Order 512.

3010 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3504C211yesmaximal, characteristic
2#34217C212yesmaximal, complemented
3#22513C2116yesmaximal, complemented
4#21334C214yesmaximal, complemented
5#21332C212yesmaximal, complemented
6#20634C212yesmaximal
7#20632C211yesmaximal, characteristic
8#20393C212yesmaximal
9C4×#410C211yesmaximal, characteristic
10#3504—24no—
11#3504—24no—
12#3504—28no—
13C2×#557C414yescomplemented
14C2×#554C414yescomplemented
15C2×#554C2218yes—
16C2×#544C414yescomplemented
17C2×#544C2212yes—
18C2×#533C412yescomplemented
19C2×#533C2211yescharacteristic
20C22×#125C412yescomplemented
21C22×#125C2211yescharacteristic
22C2×#520C2212yes—
23C2×C4×#41C2211yescharacteristic
24C2×#410C2212yes—
25C2×#410C2211yescharacteristic
26C2×#410C2211yescharacteristic
27#2370—28no—
28#2345C22116yes—
29#2330—24no—
30#2316C2214yes—
31C4×#125—24no—
32#2285C2212yes—
33#2274C2214yes—
34C4×#117C2212yes—
35#1996C22116yes—
36#1996C22132yes—
37#1799C2218yes—
38#1799C22116yes—
39C4×#99C2218yes—
40C4×#99C22116yes—
41#1740C2214yes—
42#1740C2214yes—
43#1740C2212yes—
44#1740C2212yes—
45C22×#41C2311yescharacteristic
46#557—216no—
47#557—216no—
48#554—216no—
49#554—216no—
50#554—232no—
51#554—232no—
52#544—216no—
53#544—216no—
54#544—28no—
55#544—28no—
56#533—24no—
57#533—24no—
58#533—28no—
59#533—22no—
60#533—22no—
61#533—24no—
62C2×C4⋊Q8C2×C414yes—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2×C418yes—
65C2×C4⋊Q8C2×C414yes—
66C2×#125—24no—
67C2×#125—24no—
68C2×#125—28no—
69C2×#125C2×C411yescharacteristic
70C2×#125C2×C414yes—
71C2×#125C2×C418yes—
72C2×#125C2×C411yescharacteristic
73C2×#125C2×C418yes—
74C2×#125C2×C414yes—
75C2×#125C2×C412yes—
76C2×#125C2×C412yes—
77C2×#125C2×C414yes—
78C2×#125D414yes—
79C2×#125D414yes—
80C2×#125D418yes—
81C2×C22⋊Q8C2×C418yes—
82C2×C22⋊Q8C2×C418yes—
83C2×C22⋊Q8C2×C418yes—
84C2×C22⋊Q8C2314yes—
85C2×C22⋊Q8C2318yes—
86#520—28no—
87#520—28no—
88C4×#41—22no—
89C4×#41—22no—
90C4×#41—24no—
91C2×C4×Q8C2×C412yes—
92C2×C4×Q8C2×C414yes—
93C2×C4×Q8C2×C418yes—
94C2×C4×Q8C2×C414yes—
95C2×C4×D4C2×C412yes—
96C2×C4×D4C2×C414yes—
97C2×C4×D4C2314yes—
98C2×C4×D4C2318yes—
99C2×#117C2×C412yes—
100C2×#117C2×C414yes—
101C2×#117C2312yes—
102C2×#117C2314yes—
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2312yes—
106C22×C4⋊C4C2×C414yes—
107C22×C4⋊C4C2×C418yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C42C2×C412yes—
111C22×C42C2311yescharacteristic
112#410—28no—
113#410—24no—
114#410Q818yescomplemented
115C2×#99C2318yes—
116C2×#99C2314yes—
117C2×#99C2314yes—
118#389—28no—
119#389—28no—
120#389—216no—
121#387—216no—
122#387—216no—
123#384—22no—
124#382—22no—
125#380C23116yes—
126#380C2318yes—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—216no—
129C4×C4⋊C4—28no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C22⋊C4C2318yes—
133C4×C22⋊C4C23116yes—
134#376—22no—
135C43—22no—
136#342C2318yes—
137#342C23116yes—
138#342C23116yes—
139#342C2318yes—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—24no—
143C2×#41C2×Q814yes—
144C2×#41#4114yes—
145C22×Q8C22×C411yescharacteristic
146C22×Q8C22×C418yes—
147C22×Q8C22×C414yes—
148C22×D4C22×C411yescharacteristic
149C22×D4C2412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C418yes—
152C23×C4C22×C414yes—
153C23×C4C2414yes—
154C23×C4C2412yes—
155C23×C4C2411yescharacteristic
156C4⋊Q8—28no—
157C4⋊Q8—28no—
158C4⋊Q8—232no—
159C4⋊Q8—216no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162#125—28no—
163#125—28no—
164#125—216no—
165#125—216no—
166#125—232no—
167#125—216no—
168#125—21no—
169#125—21no—
170#125—22no—
171#125—21no—
172#125—21no—
173#125—22no—
174#125C22⋊C414yes—
175#125C22⋊C414yes—
176#125C22⋊C418yes—
177#125C22⋊C418yes—
178#125C22⋊C418yes—
179#125C22⋊C4116yes—
180#125C22⋊C4116yes—
181C22⋊Q8—28no—
182C22⋊Q8—28no—
183C22⋊Q8—232no—
184C22⋊Q8—216no—
185C22⋊Q8—216no—
186C22⋊Q8—232no—
187C22⋊Q8—232no—
188C4×Q8—216no—
189C4×Q8—216no—
190C4×Q8—216no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—28no—
194C4×Q8—28no—
195C4×D4—28no—
196C4×D4—28no—
197C4×D4—232no—
198C4×D4—216no—
199C4×D4—24no—
200C4×D4—24no—
201#117—22no—
202#117—22no—
203#117—24no—
204#117—24no—
205#117—24no—
206#117—28no—
207#117—28no—
208#117—24no—
209#117—24no—
210#117—24no—
211#117—28no—
212#117—24no—
213#117—28no—
214#117C2×Q818yes—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—216no—
219C2×C4⋊C4—216no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4C22×C418yes—
228C2×C4⋊C4C22×C418yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C418yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C418yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C22×C418yes—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×D4116yes—
242C2×C4⋊C4C2×D4116yes—
243C2×C22⋊C4C22×C418yes—
244C2×C22⋊C4C22×C414yes—
245C2×C22⋊C4C2414yes—
246C2×C22⋊C4C2418yes—
247C2×C22⋊C4C2414yes—
248C2×C22⋊C4C2412yes—
249C2×C22⋊C4C2414yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—216no—
258C2×C42C22×C412yes—
259C2×C42C22×C418yes—
260C2×C42C22×C414yes—
261C2×C42C22×C414yes—
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C412yes—
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D414yes—
268#99—216no—
269#99—28no—
270#99C2×Q8116yes—
271#95—28no—
272#95—216no—
273#95—28no—
274#95—216no—
275C24C23×C412yes—
276C24C2511yescharacteristic
277#41—28no—
278#41—28no—
279#41—28no—
280#41—28no—
281C2×Q8—22no—
282C2×Q8—216no—
283C2×Q8—28no—
284C2×Q8—22no—
285C2×Q8—88no—
286C2×Q8—84no—
287C2×Q8C4×Q814yes—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4C4×Q814yes—
293C2×D4C22×Q814yes—
294C2×D4C2×#4114yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—216no—
302C22×C4—28no—
303C22×C4—24no—
304C22×C4—82no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—88no—
308C22×C4C4×Q818yes—
309C22×C4C23×C412yes—
310C22×C4C23×C414yes—
311C22×C4C23×C414yes—
312C22×C4C23×C414yes—
313C22×C4C23×C412yes—
314C22×C4C23×C412yes—
315C22×C4C23×C418yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D414yes—
319C22×C4C22×D4116yes—
320C22×C4C22×D414yes—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×Q812yes—
324C22×C4C2×#4112yes—
325C22×C42−1+414yes—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—216no—
331C4⋊C4—216no—
332C4⋊C4—216no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—28no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—216no—
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—44no—
343C4⋊C4—48no—
344C4⋊C4—816no—
345C4⋊C4—88no—
346C4⋊C4C2×C22⋊C418yes—
347C4⋊C4C2×C22⋊C418yes—
348C4⋊C4C2×C22⋊C4116yes—
349C4⋊C4C2×C22⋊C4116yes—
350C4⋊C4C2×C22⋊C4116yes—
351C4⋊C4C2×C22⋊C4132yes—
352C22⋊C4—28no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—216no—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C22⋊C4—28no—
359C22⋊C4—24no—
360C22⋊C4—28no—
361C22⋊C4C22×Q818yes—
362C42—22no—
363C42—22no—
364C42—24no—
365C42—24no—
366C42—216no—
367C42—28no—
368C42—28no—
369C42—48no—
370C42—44no—
371C42—88no—
372C42—816no—
373C42C2×C22⋊C411yescharacteristic
374C42C2×C22⋊C411yescharacteristic
375C42C2×C22⋊C412yes—
376C42C2×C22⋊C412yes—
377C42C2×C22⋊C412yes—
378C42C2×C22⋊C414yes—
379C42C2×C22⋊C414yes—
380C23—22no—
381C23—22no—
382C23—42no—
383C23—44no—
384C23C2×C4×Q814yes—
385C23C24×C411yescharacteristic
386C23C23×D412yes—
387C23C23×Q811yescharacteristic
388C23C22×#4111yescharacteristic
389C23C2×2−1+412yes—
390Q8—28no—
391Q8—816no—
392Q8—88no—
393D4—28no—
394D4—28no—
395D4—28no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—28no—
404C2×C4—44no—
405C2×C4—416no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—816no—
422C2×C4C22×C22⋊C412yes—
423C2×C4C22×C22⋊C412yes—
424C2×C4C22×C22⋊C4116yes—
425C2×C4C22×C22⋊C414yes—
426C2×C4C22×C22⋊C412yes—
427C2×C4C22×C22⋊C411yescharacteristic
428C2×C4C22×C22⋊C411yescharacteristic
429C2×C4C2×C4×Q814yes—
430C2×C4C2×C4×Q812yes—
431C2×C4C2×C4×Q812yes—
432C2×C4#51912yes—
433C2×C4#51912yes—
434C2×C4#54814yes—
435C2×C4#54814yes—
436C2×C4D4×Q814yes—
437C2×C4D4×Q814yes—
438C22—24no—
439C22—22no—
440C22—24no—
441C22—41no—
442C22—41no—
443C22—42no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22C23×C22⋊C411yescharacteristic
448C22C22×C4×Q811yescharacteristic
449C22C2×#51911yescharacteristic
450C22C2×#54812yes—
451C22C2×D4×Q812yes—
452C4—24no—
453C4—22no—
454C4—22no—
455C4—216no—
456C4—24no—
457C4—48no—
458C4—44no—
459C4—44no—
460C4—44no—
461C4—88no—
462C4—84no—
463C4Q8×C22⋊C412yes—
464C4Q8×C22⋊C412yes—
465C4Q8×C22⋊C414yes—
466C2—22no—
467C2—22no—
468C2—21no—
469C2—21no—
470C2—44no—
471C2C2×Q8×C22⋊C411yescharacteristic
472C2#6007012yes—
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC22×#41C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#125 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34217C2semidirect
3. #22513C2semidirect
4. #21334C2semidirect
5. #21332C2semidirect
13. C2×#557C4semidirect
14. C2×#554C4semidirect
16. C2×#544C4semidirect
18. C2×#533C4semidirect
20. C22×#125C4semidirect
114. #410Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#35042
#342172
#225132
#213342
#213322
#206342
#206322
#203932
C4×#4102