Finite Group Databasebeta

← #274829

Subgroups — #274829

Order 512.

741 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32957C211yesmaximal, characteristic
2#27961C211yesmaximal, characteristic
3#22510C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#10623C212yesmaximal, complemented
6#10482C212yesmaximal, complemented
7#9224C212yesmaximal, complemented
8#9127C211yesmaximal, characteristic
9#8830C211yesmaximal, characteristic
10#8724C211yesmaximal, characteristic
11#8657C212yesmaximal, complemented
12C22×C4×Q8C2211yescharacteristic
13C2×#477C2211yescharacteristic
14C2×#477C2211yescharacteristic
15C2×#446C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103—21no—
18#2857C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2100—22no—
21#1990C2212yes—
22#1985—22no—
23#1984C2212yes—
24#1863—21no—
25#1863—21no—
26#1852—21no—
27#1834C2212yes—
28#1824—21no—
29#1824—21no—
30#1803—21no—
31#1803—21no—
32#1803C2212yes—
33#1803C2211yescharacteristic
34#1803C2212yes—
35#1803C2211yescharacteristic
36#1797—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1787—21no—
40#1749—21no—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#333C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8D411yescharacteristic
50C23×C8—21no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C22⋊Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68#477—24no—
69#477—42no—
70#446—24no—
71#446—42no—
72C2×#107—21no—
73C2×#107—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103—21no—
82C2×#103—21no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—21no—
86C2×#103—21no—
87C2×#103—44no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8D411yescharacteristic
91C2×C22⋊C8D411yescharacteristic
92C2×#97—22no—
93C2×C4×C8C2312yes—
94#391—21no—
95#389—21no—
96#389—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384C2311yescharacteristic
101#384C2312yes—
102#380—21no—
103#380—21no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—21no—
111#340—21no—
112#336—22no—
113#336—22no—
114#336—21no—
115#336—21no—
116#336D412yes—
117#334—42no—
118#333—42no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8C2×D411yescharacteristic
128C22×Q8C2×D411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C22⋊Q8—82no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—44no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—44no—
156C4×Q8#4114yes—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C22⋊C4—41no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—44no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196#107—42no—
197#107—42no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—44no—
205C4⋊C8—42no—
206#103—42no—
207#103—44no—
208#103—42no—
209#103—42no—
210#103—42no—
211#103—44no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—42no—
215C22⋊C8—42no—
216#97—44no—
217C4×C8—22no—
218C4×C8—22no—
219#95—41no—
220#95—41no—
221C24#12111yescharacteristic
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—28no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—81no—
241C2×Q8—81no—
242C2×Q8#12312yes—
243C2×Q8AGL(1,ℤ/8)12yes—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
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—42no—
265C22×C4—81no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#13512yes—
273C22×C4#13712yes—
274C22×C4#13712yes—
275C22×C4#13712yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—84no—
298C4⋊C4—21no—
299C4⋊C4—24no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C4⋊C4—81no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4#12312yes—
323C22⋊C4—82no—
324C42—21no—
325C42—21no—
326C42—24no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42C2×#4112yes—
333C42H(5,𝔽2)12yes—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23#46412yes—
339C23#47012yes—
340C23#47112yes—
341C23C2×#12111yescharacteristic
342Q8—22no—
343Q8—22no—
344Q8—22no—
345Q8—22no—
346Q8—48no—
347Q8—41no—
348Q8—41no—
349Q8—41no—
350Q8—41no—
351Q8—162no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—84no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—164no—
392C2×C4#46412yes—
393C2×C4#47012yes—
394C2×C4C2×#12311yescharacteristic
395C2×C4#53411yescharacteristic
396C2×C4#54612yes—
397C2×C4#55112yes—
398C2×C4C2×#13511yescharacteristic
399C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
400C2×C4C2×#13711yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4C2×#13711yescharacteristic
403C2×C4#57511yescharacteristic
404C2×C4#57812yes—
405C8—42no—
406C8—42no—
407C8—81no—
408C8—82no—
409C8—81no—
410C8—82no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22#170414yes—
418C22#241211yescharacteristic
419C22C2×#46411yescharacteristic
420C22#304411yescharacteristic
421C22C2×#47011yescharacteristic
422C22#305511yescharacteristic
423C22C2×#47111yescharacteristic
424C22#306811yescharacteristic
425C4—22no—
426C4—22no—
427C4—22no—
428C4—42no—
429C4—42no—
430C4—41no—
431C4—42no—
432C4—81no—
433C4—161no—
434C4—162no—
435C4—161no—
436C4#309211yescharacteristic
437C4#309511yescharacteristic
438C4#319311yescharacteristic
439C4#319611yescharacteristic
440C4#330012yes—
441C4#331212yes—
442C2—24no—
443C2C2×#170411yescharacteristic
444C2#1917411yescharacteristic
445C2#1944111yescharacteristic
446C2#1947411yescharacteristic
447C2#2452911yescharacteristic
448C2#2455611yescharacteristic
449C2#2457311yescharacteristic
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. #10623C2semidirect
6. #10482C2semidirect
7. #9224C2semidirect
11. #8657C2semidirect

Maximal subgroups

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

subgroupindex
#329572
#279612
#225102
C2×#18032
#106232
#104822
#92242
#91272
#88302
#87242
#86572