Finite Group Databasebeta

← #414221

Subgroups — #414221

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23745C211yesmaximal, characteristic
2#23552C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#20435C211yesmaximal, characteristic
5#11382C211yesmaximal, characteristic
6#11354C211yesmaximal, characteristic
7#11249C212yesmaximal
8#11242C212yesmaximal
9#10896C212yesmaximal
10#10864C211yesmaximal, characteristic
11#10827C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#424C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2296C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2100—21no—
22#2100—21no—
23#2099—21no—
24#2099—21no—
25#2058—21no—
26#2058—21no—
27#2057—21no—
28#2057—21no—
29#1990—21no—
30#1990—21no—
31#1990—21no—
32#1990—21no—
33#1771—21no—
34#1771—21no—
35C8×C4⋊C4—21no—
36C8×C4⋊C4—21no—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#341C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47#1729C2211yescharacteristic
48#1728C2211yescharacteristic
49#1728C2211yescharacteristic
50#1728C2211yescharacteristic
51C23×Q8—21no—
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2312yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—42no—
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C42C2311yescharacteristic
72C2×#108—24no—
73C2×#108—24no—
74C2×#108—42no—
75#428—24no—
76#428—42no—
77C2×#107—22no—
78C2×#106—22no—
79#424—22no—
80#424Q814yescomplemented
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83#405—22no—
84#405—42no—
85#405Q814yescomplemented
86C2×#97—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—42no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95#389D411yescharacteristic
96#389D411yescharacteristic
97#389D411yescharacteristic
98#389D411yescharacteristic
99#385—21no—
100#385—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107#343—24no—
108#343—24no—
109#343—24no—
110#343—42no—
111#341—24no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#336—21no—
123#336—21no—
124#336—21no—
125#336—21no—
126#336—21no—
127#336—21no—
128#336—21no—
129#336—21no—
130C22×Q8—44no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8—82no—
134C22×Q8—82no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4#4111yescharacteristic
138C23×C4#4111yescharacteristic
139C23×C4C2411yescharacteristic
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16C2×Q812yes—
158C2×M16C2×Q812yes—
159C2×M16#4114yes—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×Q812yes—
173C22×C8C2×Q812yes—
174C22×C8#4112yes—
175C22×C8#4112yes—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178C4⋊Q8—44no—
179C4⋊Q8—44no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182#117—21no—
183#117—21no—
184#117—22no—
185#117—22no—
186#117C2×Q812yes—
187#117C2×Q812yes—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4#4114yes—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227#108—48no—
228#107—44no—
229#106—44no—
230C4⋊C8—24no—
231C4⋊C8—24no—
232#97—24no—
233#97—42no—
234C4×C8—24no—
235C4×C8—41no—
236C4×C8—41no—
237C4×C8—41no—
238C4×C8—41no—
239C4×C8—42no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244C24C2×#4111yescharacteristic
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—816no—
248C2×Q8—82no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×Q8—82no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×Q811yescharacteristic
279C22×C4C22×Q811yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C4H(5,𝔽2)12yes—
285C22×C42−1+412yes—
286M16—24no—
287M16—24no—
288M16—24no—
289M16—24no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
295M16—42no—
296M16—42no—
297M16—42no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8C22⋊Q812yes—
323C2×C8C22⋊Q812yes—
324C2×C8C22⋊Q812yes—
325C2×C8C22⋊Q812yes—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4C4⋊Q812yes—
349C4⋊C4C4⋊Q812yes—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—42no—
353C42—22no—
354C42—22no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—22no—
360C42—22no—
361C42—41no—
362C42—41no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23C2×#12111yescharacteristic
368C23#53311yescharacteristic
369C23#53311yescharacteristic
370C23#55112yes—
371C23#55612yes—
372Q8—88no—
373Q8—88no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—162no—
405C2×C4—162no—
406C2×C4#47112yes—
407C2×C4#47112yes—
408C2×C4#47212yes—
409C2×C4#47312yes—
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4#52511yescharacteristic
413C2×C4#52912yes—
414C2×C4C2×C4⋊Q811yescharacteristic
415C2×C4#53211yescharacteristic
416C2×C4#53211yescharacteristic
417C2×C4#53312yes—
418C2×C4#53411yescharacteristic
419C2×C4#53611yescharacteristic
420C2×C4#53912yes—
421C2×C4#54412yes—
422C2×C4#54612yes—
423C2×C4#54812yes—
424C2×C4#55112yes—
425C2×C4#55412yes—
426C2×C4#55612yes—
427C2×C4#55712yes—
428C8—22no—
429C8—22no—
430C8—22no—
431C8—22no—
432C8—44no—
433C8—42no—
434C8—42no—
435C8—82no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22#253411yescharacteristic
445C22#254511yescharacteristic
446C22#258812yes—
447C22#260512yes—
448C22#285711yescharacteristic
449C22C2×#47111yescharacteristic
450C22C2×#47111yescharacteristic
451C22C2×#47211yescharacteristic
452C22C2×#47311yescharacteristic
453C4—22no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—84no—
461C4—81no—
462C4—162no—
463C4#239611yescharacteristic
464C4#255012yes—
465C4#256012yes—
466C4#256411yescharacteristic
467C4#258111yescharacteristic
468C4#262711yescharacteristic
469C2—22no—
470C2—22no—
471C2#1516411yescharacteristic
472C2#2813411yescharacteristic
473C2#2813511yescharacteristic
474C2#2866911yescharacteristic
475C2#2867011yescharacteristic
476C2#3322011yescharacteristic
477C2#3322211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
80. #424Q8semidirect
85. #405Q8semidirect

Maximal subgroups

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

subgroupindex
#237452
#235522
C2×#19902
#204352
#113822
#113542
#112492
#112422
#108962
#108642
#108272