Finite Group Databasebeta

← #414062

Subgroups — #414062

Order 512.

964 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23734C211yesmaximal, characteristic
2#23453C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic, complemented
4#20384C211yesmaximal, characteristic
5#11383C211yesmaximal, characteristic, complemented
6#11354C211yesmaximal, characteristic, complemented
7#11249C212yesmaximal, complemented
8#11105C212yesmaximal
9#10896C212yesmaximal, complemented
10#10865C211yesmaximal, characteristic, complemented
11#10774C212yesmaximal
12C22×#131C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#108C2212yes—
15C2×#428C2212yes—
16C2×#417C2212yes—
17C2×#417C2212yes—
18C2×#408C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C22×#97C2211yescharacteristic
21#2297C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2096—21no—
24#2095—21no—
25#2094—21no—
26#2093—21no—
27#2026—21no—
28#2026—21no—
29#2025—21no—
30#2025—21no—
31#1992—22no—
32#1992—22no—
33#1753—22no—
34#1752—22no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2212yes—
40#1736C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1729C2211yescharacteristic
44#1728C2211yescharacteristic
45#1728C2211yescharacteristic
46#1728C2211yescharacteristic
47C22×#41C2311yescharacteristic
48C2×#131—22no—
49C2×#131—42no—
50C2×#131—42no—
51C2×#131Q814yes—
52C22×M16—21no—
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2312yes—
57C23×C8—21no—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×#125—24no—
61C2×#125—22no—
62C2×#125—21no—
63C2×#125—21no—
64C2×#125—42no—
65C2×#117C2311yescharacteristic
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C42C2311yescharacteristic
71C2×#108—24no—
72C2×#108—24no—
73C2×#108—42no—
74#428—24no—
75#428—42no—
76C2×#107—22no—
77C2×#106—22no—
78#417—24no—
79#417—24no—
80#417—42no—
81#417—42no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#102—22no—
85C2×#102—22no—
86#408—22no—
87#408Q814yes—
88C2×C22⋊C8—22no—
89#405—42no—
90C2×#97—21no—
91C2×#97—42no—
92C2×#97D412yes—
93C2×#97D412yes—
94C2×C4×C8—21no—
95#391D411yescharacteristic
96#391D411yescharacteristic
97#390D411yescharacteristic
98#390D411yescharacteristic
99#386—22no—
100#384—21no—
101#384—21no—
102#384—22no—
103C4×C22⋊C4—22no—
104#343—24no—
105#343—42no—
106#343—42no—
107#343—42no—
108#341—24no—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—22no—
113#340—22no—
114#336—22no—
115#336—21no—
116#336—21no—
117#336—22no—
118#336—21no—
119#336—21no—
120C2×#41—21no—
121C2×#41—21no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41C2×Q812yes—
127C2×#41C2×Q812yes—
128C22×Q8C2×D411yescharacteristic
129C22×D4—21no—
130C22×D4C2×D411yescharacteristic
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4#4112yes—
135C23×C4C2411yescharacteristic
136#131—48no—
137#131—48no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—22no—
141C2×M16—24no—
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—42no—
157C2×M16—42no—
158C2×M16C2×Q812yes—
159C2×M16C2×Q812yes—
160C2×M16#4114yes—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—24no—
169C22×C8—22no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D412yes—
175C22×C8C2×Q812yes—
176C22×C8C2×Q812yes—
177C22×C8#4112yes—
178C22×C8#4112yes—
179#125—28no—
180#125—44no—
181#125—44no—
182#125—44no—
183#117—22no—
184#117—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4#4114yes—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215#108—48no—
216#107—44no—
217#106—44no—
218#103—44no—
219#103—44no—
220#102—44no—
221#102—44no—
222C22⋊C8—28no—
223#97—42no—
224#97—42no—
225#97—42no—
226C4×C8—42no—
227#95—22no—
228#95—22no—
229C24—21no—
230C24C2×#4111yescharacteristic
231#41—24no—
232#41—24no—
233#41—24no—
234#41—24no—
235#41—42no—
236#41—42no—
237#41—42no—
238#41—42no—
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8C4⋊Q812yes—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4C4⋊Q812yes—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4#12112yes—
274C22×C4#12112yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×Q811yescharacteristic
278C22×C4C22×Q811yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4112yes—
282C22×C4C2×#4112yes—
283C22×C4H(5,𝔽2)12yes—
284C22×C4H(5,𝔽2)12yes—
285M16—24no—
286M16—24no—
287M16—24no—
288M16—24no—
289M16—48no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
295M16—42no—
296M16—42no—
297M16—42no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—48no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8C22⋊Q814yes—
322C2×C8C22⋊Q812yes—
323C2×C8C22⋊Q812yes—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—48no—
329C4⋊C4—48no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C22⋊C4—42no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23C2×#12111yescharacteristic
355C23#53311yescharacteristic
356C23#53311yescharacteristic
357C23#55112yes—
358C23#55912yes—
359Q8—24no—
360Q8—42no—
361D4—24no—
362D4—44no—
363D4—44no—
364D4—42no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—84no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4—162no—
392C2×C4—162no—
393C2×C4C2×C22⋊Q811yescharacteristic
394C2×C4C2×C22⋊Q811yescharacteristic
395C2×C4#52511yescharacteristic
396C2×C4#52912yes—
397C2×C4C2×C4⋊Q811yescharacteristic
398C2×C4#53211yescharacteristic
399C2×C4#53211yescharacteristic
400C2×C4#53312yes—
401C2×C4#53411yescharacteristic
402C2×C4#53511yescharacteristic
403C2×C4#53812yes—
404C2×C4#54312yes—
405C2×C4#54612yes—
406C2×C4#54612yes—
407C2×C4#55112yes—
408C2×C4#55412yes—
409C2×C4#55412yes—
410C2×C4#55912yes—
411C8—24no—
412C8—22no—
413C8—22no—
414C8—42no—
415C8—42no—
416C8—44no—
417C8—82no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—42no—
425C22—44no—
426C22—81no—
427C22—81no—
428C22—82no—
429C22#253411yescharacteristic
430C22#254411yescharacteristic
431C22#258912yes—
432C22#260712yes—
433C22#285711yescharacteristic
434C22#306812yes—
435C22#306912yes—
436C4—42no—
437C4—42no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4—84no—
442C4—162no—
443C4#239611yescharacteristic
444C4#254912yes—
445C4#255712yes—
446C4#256411yescharacteristic
447C4#258011yescharacteristic
448C4#262611yescharacteristic
449C2—22no—
450C2—22no—
451C2—82no—
452C2#1516111yescharacteristic
453C2#2813912yes—
454C2#2863411yescharacteristic
455C2#2863511yescharacteristic
456C2#3322312yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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
3. C2×#1992C2semidirect
5. #11383C2semidirect
6. #11354C2semidirect
7. #11249C2semidirect
9. #10896C2semidirect
10. #10865C2semidirect

Maximal subgroups

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

subgroupindex
#237342
#234532
C2×#19922
#203842
#113832
#113542
#112492
#111052
#108962
#108652
#107742