Finite Group Databasebeta

← #414249

Subgroups — #414249

Order 512.

942 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23617C212yesmaximal
2C2×#1941C211yesmaximal, characteristic, complemented
3#20383C211yesmaximal, characteristic
4#11330C211yesmaximal, characteristic, complemented
5#11271C211yesmaximal, characteristic, complemented
6#11204C212yesmaximal, complemented
7#11106C211yesmaximal, characteristic
8#11106C211yesmaximal, characteristic
9#10896C212yesmaximal, complemented
10#10844C211yesmaximal, characteristic, complemented
11#10774C212yesmaximal
12C22×#131C2211yescharacteristic
13C2×#428C2212yes—
14C22×#106C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#423C2212yes—
17C2×#417C2212yes—
18C2×#417C2212yes—
19C2×#408C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2297C2211yescharacteristic
23C2×#389C2212yes—
24#2068—22no—
25#2067—22no—
26#2066—22no—
27#2065—22no—
28#1941—21no—
29#1941—21no—
30#1941—21no—
31#1941—21no—
32#1753—21no—
33#1753—21no—
34C8×C22⋊C4—21no—
35C8×C22⋊C4—21no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#341C2211yescharacteristic
41C2×#341C2212yes—
42C2×#341C2211yescharacteristic
43#1736C2212yes—
44C2×#340C2212yes—
45#1729C2211yescharacteristic
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×#117C2311yescharacteristic
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C42C2311yescharacteristic
65#428—24no—
66#428—42no—
67C2×#107—22no—
68C2×#106—21no—
69C2×#106—21no—
70C2×#106—24no—
71C2×#106—21no—
72C2×#106—21no—
73C2×#106—22no—
74C2×#106—42no—
75#423—24no—
76#423—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79#417—24no—
80#417—24no—
81#417—42no—
82#417—42no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#102—22no—
86C2×#102—22no—
87#408—22no—
88#408Q814yes—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91#405—42no—
92C2×#97—21no—
93C2×C4×C8—21no—
94C2×C4×C8—42no—
95C2×C4×C8D411yescharacteristic
96C2×C4×C8D411yescharacteristic
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99#391D412yes—
100#390D412yes—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#386—22no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C2×#95—22no—
109#343—24no—
110#343—42no—
111#343—42no—
112#343—42no—
113#341—24no—
114#341—22no—
115#341—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#336—21no—
121#336—21no—
122#336—21no—
123#336—21no—
124C2×#41—21no—
125C2×#41—21no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41C2×Q812yes—
131C2×#41C2×Q812yes—
132C22×Q8C2×D411yescharacteristic
133C22×D4—21no—
134C22×D4C2×D411yescharacteristic
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4#4112yes—
139C23×C4C2411yescharacteristic
140#131—48no—
141#131—48no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—24no—
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×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—42no—
161C2×M16—42no—
162C2×M16C2×Q812yes—
163C2×M16C2×Q812yes—
164C2×M16#4114yes—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—24no—
173C22×C8—22no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×Q812yes—
181C22×C8C2×Q812yes—
182C22×C8#4112yes—
183C22×C8#4112yes—
184#117—22no—
185#117—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—22no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—84no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211#107—44no—
212#106—24no—
213#106—24no—
214#106—44no—
215#106—42no—
216#106—42no—
217C4⋊C8—44no—
218C4⋊C8—44no—
219#103—44no—
220#103—44no—
221#102—44no—
222#102—44no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225#97—42no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8—41no—
229C4×C8—41no—
230C4×C8—42no—
231#95—22no—
232#95—22no—
233#95—42no—
234#95—42no—
235C24—21no—
236C24C2×#4111yescharacteristic
237#41—24no—
238#41—24no—
239#41—24no—
240#41—24no—
241#41—42no—
242#41—42no—
243#41—42no—
244#41—42no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8C4⋊Q812yes—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4C4⋊Q812yes—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4#12112yes—
280C22×C4#12112yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×Q811yescharacteristic
284C22×C4C22×Q811yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4112yes—
288C22×C4C2×#4112yes—
289C22×C4H(5,𝔽2)12yes—
290C22×C4H(5,𝔽2)12yes—
291M16—24no—
292M16—24no—
293M16—24no—
294M16—24no—
295M16—48no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—42no—
301M16—42no—
302M16—42no—
303M16—42no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—48no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8C22⋊Q812yes—
330C2×C8C22⋊Q812yes—
331C2×C8C22⋊Q812yes—
332C2×C8C22⋊Q812yes—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23C2×#12111yescharacteristic
364C23#53311yescharacteristic
365C23#53311yescharacteristic
366C23#55312yes—
367C23#55312yes—
368Q8—24no—
369Q8—42no—
370D4—24no—
371D4—44no—
372D4—44no—
373D4—42no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—84no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—162no—
400C2×C4—162no—
401C2×C4#47114yes—
402C2×C4C2×C22⋊Q811yescharacteristic
403C2×C4C2×C22⋊Q811yescharacteristic
404C2×C4#52511yescharacteristic
405C2×C4#52912yes—
406C2×C4C2×C4⋊Q811yescharacteristic
407C2×C4#53211yescharacteristic
408C2×C4#53211yescharacteristic
409C2×C4#53312yes—
410C2×C4#53411yescharacteristic
411C2×C4#53511yescharacteristic
412C2×C4#53812yes—
413C2×C4#54312yes—
414C2×C4#54612yes—
415C2×C4#54612yes—
416C2×C4#55312yes—
417C2×C4#55312yes—
418C2×C4#55412yes—
419C2×C4#55412yes—
420C8—22no—
421C8—22no—
422C8—22no—
423C8—22no—
424C8—42no—
425C8—42no—
426C8—44no—
427C8—82no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—82no—
440C22—82no—
441C22#253511yescharacteristic
442C22#254311yescharacteristic
443C22#258712yes—
444C22#260812yes—
445C22#285711yescharacteristic
446C22C2×#47112yes—
447C22#306712yes—
448C4—24no—
449C4—42no—
450C4—42no—
451C4—82no—
452C4—84no—
453C4—162no—
454C4#239611yescharacteristic
455C4#254912yes—
456C4#255912yes—
457C4#256611yescharacteristic
458C4#257911yescharacteristic
459C4#262611yescharacteristic
460C2—22no—
461C2—22no—
462C2—82no—
463C2#1516511yescharacteristic
464C2#2817512yes—
465C2#2859312yes—
466C2#3322011yescharacteristic
467C2#3322111yescharacteristic
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
2. C2×#1941C2semidirect
4. #11330C2semidirect
5. #11271C2semidirect
6. #11204C2semidirect
9. #10896C2semidirect
10. #10844C2semidirect

Maximal subgroups

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

subgroupindex
#236172
C2×#19412
#203832
#113302
#112712
#112042
#111062
#108962
#108442
#107742