Finite Group Databasebeta

← #274119

Subgroups — #274119

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33147C211yesmaximal, characteristic
2#25902C211yesmaximal, characteristic
3#22625C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5#10601C212yesmaximal, complemented
6#10541C212yesmaximal, complemented
7#9737C212yesmaximal, complemented
8#9567C212yesmaximal, complemented
9#9129C211yesmaximal, characteristic
10#8829C211yesmaximal, characteristic
11#8342C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#487C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15#2857C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#2024—22no—
18#1990C2212yes—
19#1988C2211yescharacteristic
20#1988C2211yescharacteristic
21#1987—21no—
22#1987—21no—
23#1987C2212yes—
24#1987C2212yes—
25#1948C2211yescharacteristic
26#1948C2211yescharacteristic
27#1939C2212yes—
28#1919—21no—
29#1919—21no—
30#1916—22no—
31#1872—21no—
32#1872—21no—
33#1853—21no—
34#1824—21no—
35#1797—21no—
36#1797C2211yescharacteristic
37#1797C2211yescharacteristic
38C8×C4⋊C4C2212yes—
39C4×C22⋊C8—21no—
40C4×C22⋊C8C2211yescharacteristic
41C2×#340—21no—
42C2×#340—21no—
43C2×#335C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8—21no—
50C23×C8—21no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2312yes—
58C2×C22⋊Q8—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65#487—24no—
66#487—42no—
67C2×#106—21no—
68C2×#106—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×#103—22no—
77C2×#103—22no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8D411yescharacteristic
81C2×C22⋊C8D411yescharacteristic
82C2×C4×C8—22no—
83C2×C4×C8C2312yes—
84#391—21no—
85#389—21no—
86#389—21no—
87#384C2311yescharacteristic
88#380—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D411yescharacteristic
93C4×C4⋊C4D411yescharacteristic
94C4×C4⋊C4C2311yescharacteristic
95C4×C4⋊C4C2311yescharacteristic
96C2×#95—21no—
97#340—22no—
98#340—22no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—21no—
103#340—21no—
104#340—21no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#336—22no—
110#336—22no—
111#336—21no—
112#336—21no—
113#336D412yes—
114#335—42no—
115#333—24no—
116#333—42no—
117#332—42no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—44no—
121C22×Q8—41no—
122C22×Q8—82no—
123C22×Q8—82no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—42no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C4⋊Q8—24no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147C4⋊Q8—42no—
148C4⋊Q8#4114yes—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4Q1614yes—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C22⋊C4—41no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—82no—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42C2411yescharacteristic
186#106—42no—
187#106—42no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—42no—
195C4⋊C8—82no—
196#103—44no—
197#103—44no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—82no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—41no—
210#95—41no—
211C24#12111yescharacteristic
212C2×Q8—24no—
213C2×Q8—44no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—816no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4C2×Q1612yes—
253C22×C4#13512yes—
254C22×C4#13512yes—
255C22×C4#13712yes—
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—84no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—81no—
299C4⋊C4—82no—
300C4⋊C4C2×Q1612yes—
301C4⋊C4C2×Q1612yes—
302C4⋊C4#13712yes—
303C22⋊C4—82no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42—42no—
309C42—42no—
310C42—81no—
311C42—81no—
312C42C2×#4112yes—
313C42H(5,𝔽2)12yes—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23#46412yes—
319C23#46712yes—
320C23#47112yes—
321C23C2×#12111yescharacteristic
322Q8—84no—
323Q8—84no—
324Q8—88no—
325Q8—82no—
326Q8—82no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—88no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4#46412yes—
368C2×C4#46712yes—
369C2×C4#52511yescharacteristic
370C2×C4#53511yescharacteristic
371C2×C4#54612yes—
372C2×C4#55112yes—
373C2×C4C22×Q1611yescharacteristic
374C2×C4C2×#13511yescharacteristic
375C2×C4C2×#13511yescharacteristic
376C2×C4C2×#13711yescharacteristic
377C2×C4C2×#13711yescharacteristic
378C2×C4#57511yescharacteristic
379C2×C4#57712yes—
380C8—42no—
381C8—42no—
382C8—81no—
383C8—81no—
384C8—82no—
385C8—82no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22#168414yes—
393C22#241111yescharacteristic
394C22C2×#46411yescharacteristic
395C22#304211yescharacteristic
396C22C2×#46711yescharacteristic
397C22#305611yescharacteristic
398C22C2×#47111yescharacteristic
399C22#306811yescharacteristic
400C4—22no—
401C4—22no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—41no—
407C4—81no—
408C4—84no—
409C4—161no—
410C4—161no—
411C4#310411yescharacteristic
412C4#310811yescharacteristic
413C4#320911yescharacteristic
414C4#321111yescharacteristic
415C4#330112yes—
416C4#331612yes—
417C2—24no—
418C2C2×#168411yescharacteristic
419C2#1917611yescharacteristic
420C2#1939311yescharacteristic
421C2#1946311yescharacteristic
422C2#2448311yescharacteristic
423C2#2450211yescharacteristic
424C2#2452111yescharacteristic
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. #10601C2semidirect
6. #10541C2semidirect
7. #9737C2semidirect
8. #9567C2semidirect

Maximal subgroups

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

subgroupindex
#331472
#259022
#226252
C2×#19872
#106012
#105412
#97372
#95672
#91292
#88292
#83422