Finite Group Databasebeta

← #467325

Subgroups — #467325

Order 512.

797 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2058C211yesmaximal, characteristic
2#21313C211yesmaximal, characteristic
3#21034C211yesmaximal, characteristic
4#20754C211yesmaximal, characteristic
5#11382C211yesmaximal, characteristic
6#11366C212yesmaximal
7#11303C212yesmaximal
8#11269C211yesmaximal, characteristic
9#11238C212yesmaximal
10#11216C212yesmaximal
11#11204C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#428C2211yescharacteristic
14C22×#106C2212yes—
15C2×#424C2212yes—
16C2×#423C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C4×#117C2211yescharacteristic
19#2058—22no—
20#2058—22no—
21#1988—21no—
22#1988—21no—
23#1987—21no—
24#1987—21no—
25#1921—21no—
26#1920—21no—
27#1919—21no—
28#1918—21no—
29#1832—22no—
30#1831—22no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#341C2212yes—
36C2×#341C2212yes—
37#1736C2211yescharacteristic
38#1736C2212yes—
39#1736C2211yescharacteristic
40#1736C2211yescharacteristic
41#1736C2212yes—
42C2×#340C2212yes—
43#1729C2212yes—
44#1728C2211yescharacteristic
45#1728C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C23×Q8—21no—
48C22×M16C2312yes—
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—42no—
59C2×#117C2311yescharacteristic
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C42C2311yescharacteristic
67#428—22no—
68#428Q814yescomplemented
69C2×#107—21no—
70C2×#106—21no—
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—42no—
74#424—24no—
75#424—42no—
76#423—42no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84C4×C4⋊C4D411yescharacteristic
85C4×C4⋊C4D411yescharacteristic
86C4×C4⋊C4D411yescharacteristic
87C4×C4⋊C4D411yescharacteristic
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90#376—21no—
91C2×#95—21no—
92C43—21no—
93#343—24no—
94#343—42no—
95#343—42no—
96#343—42no—
97#341—42no—
98#341—42no—
99#340—24no—
100#340—21no—
101#340—21no—
102#340—21no—
103#340—21no—
104#340—21no—
105#340—21no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#336—24no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336D412yes—
116#336D412yes—
117C22×Q8—44no—
118C22×Q8—41no—
119C22×Q8—41no—
120C22×Q8—82no—
121C22×Q8—82no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4#4111yescharacteristic
126C23×C4#4111yescharacteristic
127C23×C4C2411yescharacteristic
128C2×M16—21no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—21no—
146C2×M16—22no—
147C2×M16C2×Q812yes—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8C2×D414yes—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165#117—21no—
166#117—21no—
167#117—22no—
168#117—22no—
169#117C2×Q812yes—
170#117#4112yes—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×Q812yes—
196C2×C4⋊C4#4114yes—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211#107—24no—
212#106—24no—
213#106—48no—
214C4⋊C8—48no—
215C4⋊C8—42no—
216C4⋊C8—42no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—41no—
222#95—41no—
223C24C2×#4111yescharacteristic
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—816no—
230C2×Q8—82no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×Q811yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4112yes—
264C22×C4H(5,𝔽2)12yes—
265C22×C4H(5,𝔽2)11yescharacteristic
266C22×C42−1+412yes—
267C22×C42−1+412yes—
268C22×C42−1+411yescharacteristic
269M16—24no—
270M16—42no—
271M16—42no—
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276M16—42no—
277M16—42no—
278M16—42no—
279M16—42no—
280M16—42no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4C22⋊Q812yes—
322C4⋊C4C22⋊Q812yes—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—42no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—81no—
338C42—81no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23C2×#12111yescharacteristic
344C23#54112yes—
345C23#55112yes—
346C23#55211yescharacteristic
347C23#55411yescharacteristic
348Q8—88no—
349Q8—88no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—162no—
388C2×C4#47112yes—
389C2×C4#47112yes—
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4#52511yescharacteristic
392C2×C4#53511yescharacteristic
393C2×C4#53611yescharacteristic
394C2×C4#53611yescharacteristic
395C2×C4#53612yes—
396C2×C4#53711yescharacteristic
397C2×C4#54112yes—
398C2×C4#54212yes—
399C2×C4#54212yes—
400C2×C4#54412yes—
401C2×C4#54611yescharacteristic
402C2×C4#54612yes—
403C2×C4D4×Q811yescharacteristic
404C2×C4#55612yes—
405C2×C4#55712yes—
406C2×C4#56212yes—
407C2×C4#56312yes—
408C8—42no—
409C8—44no—
410C8—82no—
411C8—81no—
412C8—82no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22#241111yescharacteristic
422C22#241711yescharacteristic
423C22#262111yescharacteristic
424C22#275512yes—
425C22#276212yes—
426C22C2×#47111yescharacteristic
427C22C2×#47111yescharacteristic
428C22#306811yescharacteristic
429C22#306911yescharacteristic
430C4—22no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—41no—
436C4—82no—
437C4—81no—
438C4—81no—
439C4—84no—
440C4—161no—
441C4#258111yescharacteristic
442C4#262811yescharacteristic
443C4#263711yescharacteristic
444C4#270012yes—
445C4#271312yes—
446C4#282011yescharacteristic
447C2—22no—
448C2—22no—
449C2#1656511yescharacteristic
450C2#2452111yescharacteristic
451C2#2452311yescharacteristic
452C2#2485611yescharacteristic
453C2#2485811yescharacteristic
454C2#2988711yescharacteristic
455C2#2988911yescharacteristic
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
68. #428Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
#213132
#210342
#207542
#113822
#113662
#113032
#112692
#112382
#112162
#112042