Finite Group Databasebeta

← #329426

Subgroups — #329426

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33207C211yesmaximal, characteristic
2#32957C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4C2×#1984C211yesmaximal, characteristic
5#10620C212yesmaximal, complemented
6#10483C212yesmaximal, complemented
7#10478C212yesmaximal, complemented
8#10273C212yesmaximal, complemented
9#9129C211yesmaximal, characteristic
10#9127C211yesmaximal, characteristic
11#8596C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#498C2211yescharacteristic
14C2×#477C2212yes—
15C2×#462C2212yes—
16C2×#451—21no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#1990C2211yescharacteristic
22#1990C2211yescharacteristic
23#1984—21no—
24#1984—21no—
25#1984C2212yes—
26#1984C2212yes—
27#1941—21no—
28#1941—21no—
29#1878—22no—
30#1872—21no—
31#1863—22no—
32#1852—21no—
33#1844C2211yescharacteristic
34#1807—21no—
35#1807—21no—
36#1803C2212yes—
37#1803C2212yes—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1787—21no—
41C2×#336—21no—
42C2×#336—21no—
43C2×#335C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8D411yescharacteristic
50C22×Q16—22no—
51C23×C8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×C4×Q8D412yes—
63C2×C4×Q8C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68#498—42no—
69#477—42no—
70#477—42no—
71#462—42no—
72#462—42no—
73#451—44no—
74C2×#107—22no—
75C2×#106—21no—
76C2×#106—21no—
77C2×#106—21no—
78C2×#106—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C4×C8—42no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—21no—
96#391—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#380—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#340—21no—
108#340—21no—
109#340—21no—
110#340—21no—
111#340—22no—
112#340—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—21no—
118#336—21no—
119#335—42no—
120#333—42no—
121#332—42no—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—42no—
129C22×Q8C2×D412yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C2×Q16—44no—
136C2×Q16—44no—
137C2×Q16—44no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8Q1614yescomplemented
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C22⋊Q8—42no—
159C22⋊Q8—42no—
160C22⋊Q8—82no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—44no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42C2411yescharacteristic
200#107—44no—
201#106—42no—
202#106—42no—
203#106—42no—
204#106—42no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—42no—
212#103—44no—
213#103—44no—
214#103—44no—
215#103—44no—
216C22⋊C8—42no—
217C22⋊C8—42no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—41no—
221C4×C8—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226C24#12111yescharacteristic
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—48no—
234C2×Q8—48no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—81no—
243C2×Q8—81no—
244C2×Q8#13514yes—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4C2×Q1612yes—
270C22×C4AGL(1,ℤ/8)12yes—
271C22×C4#13712yes—
272C22×C4#13712yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4111yescharacteristic
276Q16—88no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—84no—
298C2×C8C2×Q1612yes—
299C2×C8C2×Q1612yes—
300C2×C8#13712yes—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—81no—
321C4⋊C4—81no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C42—21no—
327C42—22no—
328C42—21no—
329C42—22no—
330C42—44no—
331C42—42no—
332C42—42no—
333C42H(5,𝔽2)12yes—
334C42H(5,𝔽2)12yes—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#46212yes—
340C23#46812yes—
341C23#47112yes—
342C23C2×#12111yescharacteristic
343Q8—24no—
344Q8—24no—
345Q8—48no—
346Q8—42no—
347Q8—42no—
348Q8—82no—
349Q8—82no—
350Q8—162no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—24no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4#46212yes—
388C2×C4#46812yes—
389C2×C4#53412yes—
390C2×C4#53411yescharacteristic
391C2×C4#53511yescharacteristic
392C2×C4#53812yes—
393C2×C4C22×Q1611yescharacteristic
394C2×C4C2×#13512yes—
395C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
396C2×C4C2×#13711yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4#57511yescharacteristic
400C8—22no—
401C8—22no—
402C8—22no—
403C8—41no—
404C8—82no—
405C8—82no—
406C8—82no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22#169214yes—
414C22#261711yescharacteristic
415C22C2×#46211yescharacteristic
416C22#303711yescharacteristic
417C22#305011yescharacteristic
418C22C2×#46811yescharacteristic
419C22C2×#47111yescharacteristic
420C22#306711yescharacteristic
421C4—22no—
422C4—22no—
423C4—22no—
424C4—44no—
425C4—41no—
426C4—82no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4#318812yes—
432C4#318911yescharacteristic
433C4#319012yes—
434C4#319511yescharacteristic
435C4#320911yescharacteristic
436C4#321511yescharacteristic
437C2—24no—
438C2#1911811yescharacteristic
439C2C2×#169211yescharacteristic
440C2#1926311yescharacteristic
441C2#1928511yescharacteristic
442C2#2971511yescharacteristic
443C2#2973511yescharacteristic
444C2#2975011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10620C2semidirect
6. #10483C2semidirect
7. #10478C2semidirect
8. #10273C2semidirect
149. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332072
#329572
#329062
C2×#19842
#106202
#104832
#104782
#102732
#91292
#91272
#85962