Finite Group Databasebeta

← #2808636

Subgroups — #2808636

Order 512.

2370 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic
3#16553C214yesmaximal
4#16522C212yesmaximal
5#15854C212yesmaximal
6#15762C212yesmaximal
7#15762C214yesmaximal
8#15318C214yesmaximal
9#15104C212yesmaximal
10#14881C212yesmaximal
11#14552C214yesmaximal
12#14454C211yesmaximal, characteristic
13#14239C212yesmaximal
14C22×C4⋊Q8C2214yes—
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4⋊Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18#2567—28no—
19#2551—28no—
20C2×#400C2212yes—
21C2×#398C2214yes—
22C2×#395C2212yes—
23C2×#395C2212yes—
24C2×#395C2214yes—
25C2×#395C2212yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#395C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1446C2214yes—
39#1446C2212yes—
40#1446C2214yes—
41#1445C2214yes—
42#1445C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2212yes—
48#1444C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2212yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66#1434C2212yes—
67#1434C2212yes—
68#1434C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×Q8C2312yes—
72C23×Q8C2312yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—216no—
75C2×C4⋊Q8—42no—
76C2×C4⋊Q8—42no—
77C2×C4⋊Q8—48no—
78C2×C4⋊Q8—48no—
79C2×C4⋊Q8—48no—
80C2×C4⋊Q8—44no—
81C2×C4×Q8—24no—
82C2×C4×Q8—42no—
83C2×C4×Q8—42no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2314yes—
105C22×C42C2312yes—
106#400Q8116yescomplemented
107#398—48no—
108#395—216no—
109#395—28no—
110#395—28no—
111#395—44no—
112#395—48no—
113#395—44no—
114#395—48no—
115#391—24no—
116#391—24no—
117#391—44no—
118#389—216no—
119#389—216no—
120#384—24no—
121#384D418yes—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—28no—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—28no—
156C22×Q8—24no—
157C22×Q8—28no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—44no—
171C23×C4C2412yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2411yescharacteristic
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C4⋊Q8—416no—
185C4⋊Q8—464no—
186C4⋊Q8—432no—
187C4×Q8—416no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D4116yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×Q818yes—
233C2×C4⋊C4C2×Q818yes—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—28no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—42no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—44no—
248C2×C42C2×D414yes—
249#95—24no—
250#95—28no—
251#95—24no—
252#95—24no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—44no—
263#95—44no—
264#95—42no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—42no—
271#95—44no—
272#95—44no—
273#95C2×Q818yes—
274#95C2×Q8116yes—
275#95C2×Q8116yes—
276C24C2511yescharacteristic
277C2×Q8—28no—
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—416no—
282C2×Q8—416no—
283C2×Q8—48no—
284C2×Q8—48no—
285C2×Q8—44no—
286C2×Q8—48no—
287C2×Q8—44no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8—88no—
291C2×Q8—88no—
292C2×Q8—84no—
293C2×Q8—88no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8—88no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4C22×D414yes—
357C22×C4C22×D414yes—
358C22×C4C22×D412yes—
359C22×C4C22×D414yes—
360C22×C4C22×Q818yes—
361C22×C4C22×Q818yes—
362C22×C4C22×Q814yes—
363C22×C4C22×Q814yes—
364C22×C4C22×Q814yes—
365C22×C4H(5,𝔽2)18yes—
366C22×C4H(5,𝔽2)12yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C42−1+414yes—
369C22×C42−1+414yes—
370C22×C42−1+414yes—
371C4⋊C4—216no—
372C4⋊C4—216no—
373C4⋊C4—216no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—416no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—416no—
380C4⋊C4—416no—
381C4⋊C4—416no—
382C4⋊C4—44no—
383C4⋊C4—416no—
384C4⋊C4—88no—
385C4⋊C4—88no—
386C4⋊C4—84no—
387C42—28no—
388C42—28no—
389C42—48no—
390C42—416no—
391C42—48no—
392C42—44no—
393C23C23×D411yescharacteristic
394C23C23×Q812yes—
395C23C2×H(5,𝔽2)11yescharacteristic
396C23C2×H(5,𝔽2)11yescharacteristic
397C23C2×H(5,𝔽2)14yes—
398C23C2×2−1+412yes—
399C23C2×2−1+412yes—
400C23C2×2−1+412yes—
401Q8—416no—
402Q8—816no—
403Q8—816no—
404Q8—816no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4#53614yes—
477C2×C4D4×Q818yes—
478C2×C4Q8214yes—
479C22C2×#53611yescharacteristic
480C22C2×D4×Q812yes—
481C22C2×Q8211yescharacteristic
482C22#348411yescharacteristic
483C22#348412yes—
484C22#348611yescharacteristic
485C22#350312yes—
486C22#350314yes—
487C22#350314yes—
488C22#353212yes—
489C22#353712yes—
490C22#353812yes—
491C22#353812yes—
492C22#354411yescharacteristic
493C22#355014yes—
494C22#355214yes—
495C4—24no—
496C4—24no—
497C4—28no—
498C4—44no—
499C4—42no—
500C4—48no—
501C4—44no—
502C4—44no—
503C4—48no—
504C4—48no—
505C4—44no—
506C4—82no—
507C4—84no—
508C2#3670811yescharacteristic
509C2#3959212yes—
510C2#4493412yes—
511C2#4543512yes—
512C2#5107614yes—
513C2#5200411yescharacteristic
514C2#5859112yes—
515C2#5915111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
106. #400Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25512
#165532
#165222
#158542
#157622
#153182
#151042
#148812
#145522
#144542
#142392