Finite Group Databasebeta

← #6434118

Subgroups — #6434118

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#29918C218yesmaximal, complemented
4#29915C214yesmaximal, complemented
5#29909C214yesmaximal, complemented
6#23745C211yesmaximal, characteristic, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23732C211yesmaximal, characteristic, complemented
9C2×#2095C211yesmaximal, characteristic, complemented
10#23660C212yesmaximal, complemented
11#23633C212yesmaximal, complemented
12#20913C212yesmaximal
13#20847C212yesmaximal
14#20684C211yesmaximal, characteristic
15C2×#536C2211yescharacteristic
16C2×#520C2212yes—
17C2×C4×#41C2211yescharacteristic
18C2×#428C2212yes—
19C22×#107C2212yes—
20C2×#419C2212yes—
21C22×#103C2212yes—
22C2×#408C2212yes—
23#2855C2214yes—
24#2625C2214yes—
25#2447—22no—
26#2446—22no—
27#2419—24no—
28Q8×C4⋊C4—22no—
29D4×C4⋊C4—24no—
30D4×C4⋊C4—22no—
31#2317C2212yes—
32#2317C2214yes—
33#2316C2211yescharacteristic
34#2311C2211yescharacteristic
35#2310C2211yescharacteristic
36#2296C2211yescharacteristic
37C2×#391C2211yescharacteristic
38#2285C2211yescharacteristic
39#2285C2212yes—
40C2×#384C2211yescharacteristic
41C2×#384C2212yes—
42#2101C2214yes—
43#2100C2212yes—
44#2100C2214yes—
45#2099C2212yes—
46#2099C2214yes—
47#2097C2214yes—
48#2096C2212yes—
49#2095C2214yes—
50#2095C2214yes—
51#2095C2212yes—
52#2085C2218yes—
53#2083C2218yes—
54#2074C2218yes—
55#2072C2218yes—
56#1873C2218yes—
57#1872C2218yes—
58#1854C2218yes—
59#1852C2218yes—
60#1818C2214yes—
61#1817C2214yes—
62#1736C2212yes—
63#1736C2212yes—
64#1735C2212yes—
65#1735C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C22×#41C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C22×M16C2312yes—
72C23×C8C2312yes—
73#536—28no—
74C2×C4⋊Q8—21no—
75C2×#124—21no—
76C2×#123—22no—
77C2×C22⋊Q8C2314yes—
78#520—44no—
79#520—44no—
80C4×#41—28no—
81C2×C4×Q8—22no—
82C2×C4×Q8D412yes—
83C2×C4×D4—24no—
84C2×C4×D4—22no—
85C2×C4×D4D414yes—
86C2×C4×D4D412yes—
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2312yes—
90C2×#117C2314yes—
91C2×#117C2311yescharacteristic
92C2×#117C2312yes—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100#428—44no—
101C2×#107—44no—
102C2×#107C2314yes—
103C2×#107C2318yes—
104C2×#106C2314yes—
105#419—44no—
106C2×#103—44no—
107C2×#103C2314yes—
108C2×#103C2318yes—
109C2×#103C2314yes—
110#408—28no—
111C2×C22⋊C8C2318yes—
112#400—24no—
113#398—24no—
114#397—24no—
115#397—24no—
116#391—21no—
117#391—22no—
118#391—22no—
119#391C2314yes—
120#391C2312yes—
121#391C2312yes—
122#390—21no—
123#389—22no—
124#389—24no—
125#389—28no—
126#389C2314yes—
127#389C2312yes—
128#389C2312yes—
129#387—24no—
130#387—24no—
131#386—22no—
132#385—24no—
133#385—22no—
134#385—24no—
135#385—28no—
136#384—22no—
137#384—24no—
138#384C2314yes—
139#384C2312yes—
140#384C2312yes—
141#384C2311yescharacteristic
142#384C2311yescharacteristic
143#382C2314yes—
144#382C2312yes—
145#381—24no—
146#381—22no—
147#380—22no—
148#380—24no—
149#380—22no—
150#380C2314yes—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—22no—
154C2×#95—22no—
155#340C2314yes—
156#340C2318yes—
157#340C2314yes—
158#340C2314yes—
159#340C2318yes—
160#340C2314yes—
161#336C2318yes—
162#336C2312yes—
163#336C2312yes—
164#336C2312yes—
165#336C2312yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41#4118yes—
170C22×Q8—21no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8C2×D411yescharacteristic
176C22×Q8C2412yes—
177C22×D4C2×D412yes—
178C22×D4C2×D411yescharacteristic
179C23×C4—21no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4C2×D412yes—
184C23×C4C2×D414yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—24no—
190C22×C8—24no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8C2414yes—
194C22×C8C2414yes—
195C4⋊Q8—44no—
196#124—44no—
197#123—48no—
198C22⋊Q8—216no—
199C4×Q8—28no—
200C4×Q8—48no—
201C4×D4—216no—
202C4×D4—28no—
203C4×D4—416no—
204C4×D4—48no—
205#117—28no—
206#117—24no—
207#117—24no—
208#117—24no—
209#117—24no—
210#117—22no—
211#117—22no—
212#117—44no—
213#117—44no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—84no—
232C2×C4⋊C4—88no—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4C2×D412yes—
248C2×C22⋊C4C2×D414yes—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C2×D412yes—
256C2×C42C2×D414yes—
257C2×C42C2412yes—
258C2×C42C2411yescharacteristic
259C2×C42C2411yescharacteristic
260#107—48no—
261#107—44no—
262#106—44no—
263#103—48no—
264#103—44no—
265#103—44no—
266C22⋊C8—216no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277C24C22×D412yes—
278C24C22×D411yescharacteristic
279#41—216no—
280#41—216no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8#12314yes—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4#12318yes—
298C2×D4#12314yes—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—28no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—48no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—84no—
322C22×C4—88no—
323C22×C4—82no—
324C22×C4C22×D412yes—
325C22×C4C22×D414yes—
326C22×C4C22×D412yes—
327C22×C4C22×D412yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4118yes—
330C22×C4C2×#4114yes—
331C22×C4H(5,𝔽2)14yes—
332C22×C42−1+412yes—
333C22×C42−1+412yes—
334C22×C4C2511yescharacteristic
335M16—48no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—48no—
343C4⋊C4—28no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—82no—
364C4⋊C4—84no—
365C4⋊C4—88no—
366C4⋊C4—84no—
367C22⋊C4—28no—
368C22⋊C4—24no—
369C22⋊C4—48no—
370C22⋊C4—42no—
371C22⋊C4—48no—
372C22⋊C4—44no—
373C22⋊C4—42no—
374C22⋊C4—84no—
375C42—28no—
376C42—24no—
377C42—22no—
378C42—24no—
379C42—22no—
380C42—48no—
381C42—42no—
382C42—44no—
383C42—42no—
384C23—22no—
385C23—22no—
386C23—24no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23C2×#12314yes—
391C23C2×#12312yes—
392C23#53512yes—
393C23#53612yes—
394C23C23×D411yescharacteristic
395Q8—28no—
396Q8—44no—
397Q8—44no—
398Q8—84no—
399D4—216no—
400D4—28no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—48no—
414C2×C4—416no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—84no—
433C2×C4—162no—
434C2×C4—164no—
435C2×C4—168no—
436C2×C4C2×#12314yes—
437C2×C4C2×#12312yes—
438C2×C4#53512yes—
439C2×C4#53612yes—
440C2×C4#53918yes—
441C2×C4#53918yes—
442C2×C4#57814yes—
443C2×C4C22×#4112yes—
444C2×C4C2×H(5,𝔽2)12yes—
445C2×C4C2×2−1+411yescharacteristic
446C2×C4C2×2−1+411yescharacteristic
447C8—44no—
448C8—44no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22—24no—
453C22—24no—
454C22—28no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22#262314yes—
459C22C22×#12311yescharacteristic
460C22C2×#53511yescharacteristic
461C22C2×#53611yescharacteristic
462C22C2×#57812yes—
463C22#358512yes—
464C4—24no—
465C4—24no—
466C4—42no—
467C4—44no—
468C4—82no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—84no—
473C4—82no—
474C4—82no—
475C4#262314yes—
476C4C2×#53912yes—
477C4C2×#53912yes—
478C2—24no—
479C2—44no—
480C2C2×#262311yescharacteristic
481C2#6052612yes—
482C2#6082111yescharacteristic
483C2#6082711yescharacteristic
484C2#6089312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29918C2semidirect
4. #29915C2semidirect
5. #29909C2semidirect
6. #23745C2semidirect
7. #23744C2semidirect
8. #23732C2semidirect
9. C2×#2095C2semidirect
10. #23660C2semidirect
11. #23633C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#341832
#299182
#299152
#299092
#237452
#237442
#237322
C2×#20952
#236602
#236332
#209132
#208472
#206842