Finite Group Databasebeta

← #6594495

Subgroups — #6594495

Order 512.

1481 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C212yesmaximal
2#33263C211yesmaximal, characteristic, complemented
3#33210C211yesmaximal, characteristic, complemented
4#33207C211yesmaximal, characteristic, complemented
5#29178C212yesmaximal, complemented
6#28004C214yesmaximal, complemented
7#27961C212yesmaximal, complemented
8#27877C212yesmaximal, complemented
9#27851C214yesmaximal, complemented
10#27666C214yesmaximal
11#22618C214yesmaximal, complemented
12#21727C211yesmaximal, characteristic, complemented
13#20824C211yesmaximal, characteristic, complemented
14C2×#1845C211yesmaximal, characteristic, complemented
15C4×#435C211yesmaximal, characteristic
16C2×#498C2211yescharacteristic
17C2×#435C2212yes—
18C22×#107C2214yes—
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2859C2212yes—
22#2857C2212yes—
23#2857C2212yes—
24#2570—22no—
25#2511—22no—
26#2507C2214yes—
27#2497C2214yes—
28#2487C2214yes—
29#2442—22no—
30#2441—22no—
31#2423—22no—
32#2417—22no—
33#2360—22no—
34#2352—22no—
35#2345C2214yes—
36C4×C4⋊Q8C2212yes—
37#2285C2212yes—
38C2×#384C2212yes—
39C42×D4C2211yescharacteristic
40#2100C2218yes—
41#2086C2218yes—
42#2075C2218yes—
43#1990C2212yes—
44#1990C2212yes—
45#1990C2212yes—
46#1944C2214yes—
47#1944C2214yes—
48#1943C2214yes—
49#1943C2214yes—
50#1941C2214yes—
51#1918C2218yes—
52#1918C2218yes—
53#1845C2211yescharacteristic
54#1845C2214yes—
55#1845C2211yescharacteristic
56#1845C2212yes—
57#1844C2211yescharacteristic
58#1834C2212yes—
59#1834C2212yes—
60#1824C2218yes—
61#1809C2214yes—
62#1809C2214yes—
63#1807C2214yes—
64#1776C2214yes—
65#1776C2214yes—
66#1771C2212yes—
67C4×C4⋊C8C2211yescharacteristic
68#1753C2212yes—
69#1752C2212yes—
70C4×C22⋊C8C2211yescharacteristic
71C4×C22⋊C8C2211yescharacteristic
72C4×#97C2211yescharacteristic
73C22×M16C2312yes—
74C23×C8C2312yes—
75C2×C4⋊Q8C2312yes—
76C2×#125—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×D4D412yes—
81C2×C4×D4C2311yescharacteristic
82C2×#117—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4C2314yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#498—24no—
89#435—28no—
90C2×#107—216no—
91C2×#107C2314yes—
92C2×#107C2314yes—
93C2×#107C2318yes—
94C2×#106C2314yes—
95C2×#106C2314yes—
96C2×C4⋊C8C2312yes—
97C2×C22⋊C8C2312yes—
98C2×C22⋊C8C2312yes—
99C4×M16—24no—
100C2×#97C2312yes—
101C2×#97C2312yes—
102C2×#97C2312yes—
103C2×C4×C8—24no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2312yes—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#400—24no—
110#400—24no—
111#399—24no—
112#398—24no—
113#398—24no—
114#396—24no—
115#391—22no—
116#391—22no—
117#391—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—24no—
124#389C2314yes—
125#389C2314yes—
126#389C2314yes—
127#389C2314yes—
128#389C2314yes—
129#388—24no—
130#387—24no—
131#387—24no—
132#387—24no—
133#387—24no—
134#386—24no—
135#385—24no—
136#384—22no—
137#384—22no—
138#384C2312yes—
139#384C2312yes—
140#384C2312yes—
141#384C2314yes—
142#382—22no—
143#382—22no—
144#380—22no—
145#380—22no—
146#380—22no—
147#380—24no—
148#380C2314yes—
149#380C2314yes—
150#380C2314yes—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4D412yes—
154C4×C4⋊C4C2314yes—
155C4×C4⋊C4C2314yes—
156C4×C4⋊C4C2311yescharacteristic
157C4×C22⋊C4—24no—
158C4×C22⋊C4D412yes—
159C4×C22⋊C4D412yes—
160C4×C22⋊C4C2311yescharacteristic
161C4×C22⋊C4C2311yescharacteristic
162C43C2311yescharacteristic
163#340C2318yes—
164#340C2318yes—
165#340C2318yes—
166#340C2318yes—
167#340C2318yes—
168#336C2312yes—
169#336C2312yes—
170#336C2312yes—
171C22×Q8—22no—
172C22×D4C2×D411yescharacteristic
173C23×C4—24no—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×M16—24no—
179C2×M16—24no—
180C2×M16—24no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8#4118yes—
184C22×C8C2412yes—
185C22×C8C2412yes—
186C22×C8C2412yes—
187C22×C8C2412yes—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190#125—84no—
191C22⋊Q8—48no—
192C22⋊Q8—84no—
193C4×Q8—84no—
194C4×D4—28no—
195C4×D4—24no—
196#117—84no—
197C2×C4⋊C4—216no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—84no—
214C2×C4⋊C4—84no—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C2414yes—
219C2×C4⋊C4C2414yes—
220C2×C4⋊C4C2414yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4C2×D412yes—
225C2×C22⋊C4C2×D412yes—
226C2×C22⋊C4C2×D411yescharacteristic
227C2×C22⋊C4C2×D411yescharacteristic
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2411yescharacteristic
239C2×C42C2411yescharacteristic
240C2×C42C2411yescharacteristic
241C2×C42C2411yescharacteristic
242C2×C42C2411yescharacteristic
243#107—216no—
244#107—28no—
245#107—28no—
246#106—28no—
247#106—28no—
248C4⋊C8—24no—
249C22⋊C8—24no—
250C22⋊C8—24no—
251#97—24no—
252#97—24no—
253#97—24no—
254C4×C8—24no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8#4114yes—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95C2×D412yes—
270#95C2×D412yes—
271#95C2×D412yes—
272C24C22×D411yescharacteristic
273C24C22×D411yescharacteristic
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×D4—22no—
278C2×D4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—48no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4#13718yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D412yes—
302C22×C4C22×D412yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C2×#4114yes—
308C22×C4C2511yescharacteristic
309M16—48no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—216no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8C2×#4114yes—
320C2×C8C2×#4114yes—
321C2×C8C2×#4114yes—
322C2×C82−1+414yes—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—24no—
330C4⋊C4—48no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—48no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C22⋊C4—22no—
346C22⋊C4—24no—
347C22⋊C4—22no—
348C22⋊C4—24no—
349C22⋊C4—48no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—21no—
357C42—21no—
358C42—22no—
359C42—22no—
360C42—21no—
361C42—21no—
362C42—84no—
363C42—84no—
364C42C2×#4112yes—
365C422−1+412yes—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—22no—
374C23#53212yes—
375C23C2×#13714yes—
376C23C23×D411yescharacteristic
377Q8—88no—
378D4—44no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—216no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—168no—
415C2×C4#53212yes—
416C2×C4#54714yes—
417C2×C4C2×#13714yes—
418C2×C4C2×#13714yes—
419C2×C4#57614yes—
420C2×C4C22×#4111yescharacteristic
421C2×C4C22×#4112yes—
422C2×C4C2×2−1+412yes—
423C2×C4C2×2−1+411yescharacteristic
424C8—24no—
425C8—42no—
426C8—42no—
427C8#55218yes—
428C22—24no—
429C22—22no—
430C22—24no—
431C22—22no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—42no—
436C22—41no—
437C22#318514yes—
438C22C2×#53211yescharacteristic
439C22C2×#54711yescharacteristic
440C22#349911yescharacteristic
441C22C22×#13712yes—
442C22C2×#57612yes—
443C4—24no—
444C4—22no—
445C4—42no—
446C4—41no—
447C4—41no—
448C4—42no—
449C4—41no—
450C4—84no—
451C4—84no—
452C4—84no—
453C4—84no—
454C4#334618yes—
455C4C2×#55212yes—
456C4#350912yes—
457C2—24no—
458C2—81no—
459C2#3618111yescharacteristic
460C2C2×#318511yescharacteristic
461C2#6071411yescharacteristic
462C2C2×#334612yes—
463C2#6248012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33263C2semidirect
3. #33210C2semidirect
4. #33207C2semidirect
5. #29178C2semidirect
6. #28004C2semidirect
7. #27961C2semidirect
8. #27877C2semidirect
9. #27851C2semidirect
11. #22618C2semidirect
12. #21727C2semidirect
13. #20824C2semidirect
14. C2×#1845C2semidirect

Maximal subgroups

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

subgroupindex
#361892
#332632
#332102
#332072
#291782
#280042
#279612
#278772
#278512
#276662
#226182
#217272
#208242
C2×#18452
C4×#4352