Finite Group Databasebeta

← #403295

Subgroups — #403295

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic
2C2×#1918C211yesmaximal, characteristic
3#15143C211yesmaximal, characteristic
4#13821C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11322C211yesmaximal, characteristic
7#11227C211yesmaximal, characteristic
8#11227C212yesmaximal
9#11207C212yesmaximal
10#10907C212yesmaximal
11#10849C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C2×#395—21no—
17C2×#395—21no—
18C2×#389C2211yescharacteristic
19C2×#389C2212yes—
20C2×#384C2212yes—
21C2×C4×C4⋊C4—21no—
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1988—24no—
25#1918—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1444—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C42—21no—
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65C22×C42C2312yes—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107Q814yescomplemented
69C2×C4⋊C8—22no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8—42no—
74#395—44no—
75#395—44no—
76#389—24no—
77#389—22no—
78#389—22no—
79#389Q818yescomplemented
80#384—24no—
81#384—42no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—42no—
85C4×C4⋊C4—44no—
86C4×C4⋊C4D414yes—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95D411yescharacteristic
96C2×#95D412yes—
97C2×#95D411yescharacteristic
98#340—22no—
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—22no—
108#336—24no—
109C22×Q8—44no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—82no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2411yescharacteristic
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8C2×Q812yes—
142C4⋊Q8—28no—
143C4⋊Q8—48no—
144C4⋊Q8—44no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×Q814yes—
186C2×C4⋊C4C2×Q812yes—
187C2×C4⋊C4C2×Q812yes—
188C2×C4⋊C4C2×Q814yes—
189C2×C4⋊C4#4114yes—
190C2×C4⋊C4#4112yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42—82no—
203C2×C42C2×D412yes—
204C2×C42C2×Q814yes—
205C2×C42#4112yes—
206C2×C42#4114yes—
207#107—28no—
208C4⋊C8—44no—
209C4⋊C8—48no—
210#95—24no—
211#95—28no—
212#95—22no—
213#95—24no—
214#95—44no—
215#95—44no—
216#95—42no—
217#95—42no—
218#95—44no—
219#95—81no—
220#95D814yes—
221C24C22×D411yescharacteristic
222C2×Q8—48no—
223C2×Q8—88no—
224C2×Q8—88no—
225C2×Q8—84no—
226C2×Q8—84no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4C22⋊Q814yes—
284C22×C4C22⋊Q814yes—
285C22×C4C2×D812yes—
286C22×C4C2×D814yes—
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4#13714yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C22×Q812yes—
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4111yescharacteristic
296C22×C4H(5,𝔽2)12yes—
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C42−1+411yescharacteristic
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—41no—
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—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—416no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—88no—
339C4⋊C4—88no—
340C4⋊C4—84no—
341C4⋊C4C22⋊Q814yes—
342C4⋊C4C4⋊Q814yes—
343C42—22no—
344C42—24no—
345C42—22no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—24no—
350C42—48no—
351C42—88no—
352C23C2×#12111yescharacteristic
353C23C2×C22⋊Q812yes—
354C23#54612yes—
355C23#54611yescharacteristic
356C23D4×Q811yescharacteristic
357C23C22×D811yescharacteristic
358C23C2×AGL(1,ℤ/8)11yescharacteristic
359C23C2×#13712yes—
360C23#57712yes—
361C23#57711yescharacteristic
362C23#57811yescharacteristic
363Q8—816no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4#45914yes—
425C2×C4#47418yes—
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4#52511yescharacteristic
428C2×C4#52912yes—
429C2×C4C2×C4⋊Q811yescharacteristic
430C2×C4#53211yescharacteristic
431C2×C4#53312yes—
432C2×C4#53312yes—
433C2×C4#53511yescharacteristic
434C2×C4#53711yescharacteristic
435C2×C4#53812yes—
436C2×C4#54212yes—
437C2×C4#55112yes—
438C2×C4#55211yescharacteristic
439C2×C4#55412yes—
440C2×C4#55412yes—
441C2×C4#55411yescharacteristic
442C2×C4#55712yes—
443C2×C4Q8212yes—
444C8—42no—
445C8—82no—
446C8—81no—
447C22#241111yescharacteristic
448C22#241711yescharacteristic
449C22#245012yes—
450C22#245112yes—
451C22#285711yescharacteristic
452C22C2×#45911yescharacteristic
453C22#303911yescharacteristic
454C22#304311yescharacteristic
455C22#304611yescharacteristic
456C22C2×#47412yes—
457C22#307912yes—
458C22#308212yes—
459C22#308412yes—
460C22#329411yescharacteristic
461C22#329412yes—
462C22#331012yes—
463C22#331612yes—
464C22#332012yes—
465C22#332812yes—
466C22#333011yescharacteristic
467C22Q8×D811yescharacteristic
468C22#338411yescharacteristic
469C22#338712yes—
470C4—28no—
471C4—24no—
472C4—44no—
473C4—44no—
474C4—42no—
475C4—44no—
476C4—84no—
477C4—84no—
478C4—82no—
479C4—161no—
480C4#239611yescharacteristic
481C4#252012yes—
482C4#256711yescharacteristic
483C4#258411yescharacteristic
484C4#258612yes—
485C4#266211yescharacteristic
486C2#1487211yescharacteristic
487C2#2447611yescharacteristic
488C2#2449611yescharacteristic
489C2#2481111yescharacteristic
490C2#2483111yescharacteristic
491C2#2640812yes—
492C2#2642912yes—
493C2#2644012yes—
494C2#2645812yes—
495C2#3319511yescharacteristic
496C2#3320411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
68. C2×#107Q8semidirect
79. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19182
#151432
#138212
#113842
#113222
#112272
#112072
#109072
#108492