Finite Group Databasebeta

← #415139

Subgroups — #415139

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic
4C2×#1843C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic
6#13783C211yesmaximal, characteristic
7#11232C212yesmaximal
8#11231C212yesmaximal
9#11228C212yesmaximal
10#11227C212yesmaximal
11#10850C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1988—24no—
25#1987—24no—
26#1949—24no—
27#1843—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36#1446—21no—
37#1446—21no—
38#1445—22no—
39#1436—22no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—21no—
45C23×Q8—21no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8Q814yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C42—21no—
64C22×C42—21no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#107—22no—
68C2×#107—42no—
69C2×#106—22no—
70C2×#106—42no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8D414yes—
83#389—24no—
84#389Q818yescomplemented
85#382—22no—
86#382—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4D414yes—
91#376—22no—
92#376—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95D411yescharacteristic
105C2×#95D411yescharacteristic
106C2×#95D412yes—
107#340—24no—
108#340—24no—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115C22×Q8—24no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—82no—
119C22×Q8—82no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C4⋊Q8—28no—
149C4⋊Q8—28no—
150C4⋊Q8—44no—
151C4⋊Q8—44no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×Q814yes—
184C2×C4⋊C4C2×Q814yes—
185C2×C4⋊C4C2×Q814yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—42no—
192C2×C42—82no—
193C2×C42—81no—
194C2×C42—81no—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×Q814yes—
198C2×C42C2×Q812yes—
199C2×C42#4114yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203#107—44no—
204#106—44no—
205C4⋊C8—28no—
206C4⋊C8—48no—
207C4⋊C8—48no—
208C4⋊C8—44no—
209#95—24no—
210#95—22no—
211#95—22no—
212#95—24no—
213#95—28no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95—82no—
218#95—82no—
219#95—82no—
220#95—82no—
221#95—82no—
222#95—82no—
223C24C22×D411yescharacteristic
224C2×Q8—48no—
225C2×Q8—88no—
226C2×Q8—84no—
227C2×Q8—84no—
228C2×Q8—88no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12112yes—
284C22×C4#12112yes—
285C22×C4C22⋊Q814yes—
286C22×C4C22⋊Q814yes—
287C22×C4C22⋊Q812yes—
288C22×C4C22⋊Q812yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×Q811yescharacteristic
292C22×C4C22×Q812yes—
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4H(5,𝔽2)12yes—
298C22×C42−1+412yes—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—48no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—88no—
337C4⋊C4C4⋊Q814yes—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—84no—
343C42—84no—
344C42#12114yes—
345C42C22⋊Q814yes—
346C42C22⋊Q814yes—
347C42#12614yes—
348C23C2×#12111yescharacteristic
349C23C2×#12111yescharacteristic
350C23C2×C22⋊Q811yescharacteristic
351C23C2×C22⋊Q812yes—
352C23#53712yes—
353C23#57612yes—
354C23#57711yescharacteristic
355C23#57712yes—
356C23#57711yescharacteristic
357C23#57811yescharacteristic
358C23#57811yescharacteristic
359Q8—816no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4#39614yes—
419C2×C4#39614yes—
420C2×C4C2×#12111yescharacteristic
421C2×C4C2×C22⋊Q811yescharacteristic
422C2×C4C2×C22⋊Q811yescharacteristic
423C2×C4C2×#12611yescharacteristic
424C2×C4C2×C4⋊Q811yescharacteristic
425C2×C4#53211yescharacteristic
426C2×C4#53312yes—
427C2×C4#53511yescharacteristic
428C2×C4#53611yescharacteristic
429C2×C4#54212yes—
430C2×C4#54612yes—
431C2×C4D4×Q812yes—
432C2×C4#55212yes—
433C2×C4#55412yes—
434C2×C4#55712yes—
435C2×C4#55712yes—
436C2×C4Q8212yes—
437C2×C4#55912yes—
438C8—42no—
439C8—42no—
440C8—82no—
441C22C2×#39611yescharacteristic
442C22#241712yes—
443C22#258212yes—
444C22#262011yescharacteristic
445C22#285711yescharacteristic
446C22#304111yescharacteristic
447C22#304311yescharacteristic
448C22#304611yescharacteristic
449C22#304611yescharacteristic
450C22#305711yescharacteristic
451C22#305911yescharacteristic
452C22#306212yes—
453C22#308112yes—
454C22#308112yes—
455C22#308212yes—
456C22#308211yescharacteristic
457C22#308211yescharacteristic
458C22#308312yes—
459C22#308412yes—
460C22#325712yes—
461C22#325812yes—
462C22#326212yes—
463C22#326312yes—
464C4—24no—
465C4—24no—
466C4—44no—
467C4—44no—
468C4—44no—
469C4—84no—
470C4—84no—
471C4—82no—
472C4—161no—
473C4—161no—
474C4#245112yes—
475C4#251112yes—
476C4#254411yescharacteristic
477C4#254611yescharacteristic
478C4#256511yescharacteristic
479C4#256711yescharacteristic
480C2#1518411yescharacteristic
481C2#2361311yescharacteristic
482C2#2361411yescharacteristic
483C2#2484912yes—
484C2#2485112yes—
485C2#2929212yes—
486C2#2929312yes—
487C2#2983911yescharacteristic
488C2#2984111yescharacteristic
489C2#3320311yescharacteristic
490C2#3320411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
84. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19872
C2×#19492
C2×#18432
#142822
#137832
#112322
#112312
#112282
#112272
#108502