Finite Group Databasebeta

← #115912

Subgroups — #115912

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21930C211yesmaximal, characteristic
2C2×#1520C212yesmaximal
3#14159C211yesmaximal, characteristic
4#10846C211yesmaximal, characteristic
5C2×#1294C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380C2212yes—
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C2211yescharacteristic
11#1850—24no—
12#1797—22no—
13C2×#342—21no—
14C2×#342—21no—
15#1520—24no—
16#1520—24no—
17C2×#328—24no—
18#1446C412yescomplemented
19#1445C412yescomplemented
20#1436—22no—
21C22⋊#95—21no—
22C22⋊#95—21no—
23#1294—24no—
24C23×Q8C2×C411yescharacteristic
25C24×C4—21no—
26C24×C4C2311yescharacteristic
27C23×C8—22no—
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8—21no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8D412yes—
35C2×C22⋊Q8Q812yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C22⋊C4—21no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4D411yescharacteristic
42C22×C22⋊C4D411yescharacteristic
43C22×C42D412yes—
44C2×C4⋊C8—24no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8D414yes—
51#380—22no—
52#380—22no—
53#380—22no—
54#380D414yes—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95C2×C412yes—
66C2×#95C2×C412yes—
67#342—42no—
68#342—42no—
69#336—24no—
70#328—216no—
71C25C2×D411yescharacteristic
72C22×Q8—21no—
73C22×Q8—22no—
74C22×Q8—22no—
75C22×Q8—82no—
76C22×Q8C4⋊C412yes—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—44no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—82no—
87C23×C4—82no—
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C412yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×Q811yescharacteristic
96C23×C4#4112yes—
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4112yes—
100C22×C8—24no—
101C22×C8—42no—
102C22×C8—42no—
103C22×C8—84no—
104C22⋊Q8—24no—
105C22⋊Q8—24no—
106C22⋊Q8—24no—
107C22⋊Q8—24no—
108C22⋊Q8—42no—
109C22⋊Q8—42no—
110C22⋊Q8—42no—
111C22⋊Q8—42no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—84no—
124C2×C4⋊C4C4⋊C412yes—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4SD1612yes—
142C2×C22⋊C4Q1612yes—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—82no—
146C4⋊C8—84no—
147C22⋊C8—24no—
148C22⋊C8—24no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—44no—
155#95—44no—
156#95—41no—
157#95—42no—
158#95—41no—
159#95—82no—
160#95—82no—
161C24—42no—
162C24—42no—
163C24C2×C22⋊C411yescharacteristic
164C24#12112yes—
165C24#12111yescharacteristic
166C24#12112yes—
167C24C22⋊Q811yescharacteristic
168C24C2×SD1611yescharacteristic
169C24C2×Q1611yescharacteristic
170C24#13512yes—
171C24AGL(1,ℤ/8)11yescharacteristic
172C24#13711yescharacteristic
173C24#13712yes—
174C2×Q8—24no—
175C2×Q8—42no—
176C2×Q8—84no—
177C2×Q8—82no—
178C2×Q8—82no—
179C2×Q8—84no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—48no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—81no—
213C22×C4—82no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4—82no—
217C22×C4—81no—
218C22×C4—88no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4—84no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4#10314yes—
229C22×C4C2×C4⋊C411yescharacteristic
230C22×C4#11711yescharacteristic
231C22×C4C4×D412yes—
232C22×C4C4×D412yes—
233C22×C4#12011yescharacteristic
234C22×C4#12012yes—
235C22×C4#12112yes—
236C22×C4#12111yescharacteristic
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q812yes—
239C22×C4C22⋊Q811yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12312yes—
242C22×C4#12312yes—
243C22×C4#12412yes—
244C22×C4#12411yescharacteristic
245C22×C4#12612yes—
246C22×C4#12711yescharacteristic
247C2×C8—48no—
248C2×C8—44no—
249C2×C8—84no—
250C2×C8—84no—
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C42—44no—
270C42—82no—
271C23—22no—
272C23—24no—
273C23—22no—
274C23—24no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23—44no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—41no—
284C23—42no—
285C23—42no—
286C23—44no—
287C23#38011yescharacteristic
288C23#38611yescharacteristic
289C23#39011yescharacteristic
290C23#39412yes—
291C23#39612yes—
292C23C2×#10311yescharacteristic
293C23#41611yescharacteristic
294C23#41711yescharacteristic
295C23#41911yescharacteristic
296C23#46011yescharacteristic
297C23#46112yes—
298C23#46211yescharacteristic
299C23#46212yes—
300C23#46312yes—
301C23#46412yes—
302C23#46412yes—
303C23#46512yes—
304C23#46712yes—
305C23#46812yes—
306C23#47012yes—
307C23#47511yescharacteristic
308C23#47711yescharacteristic
309C23#47912yes—
310Q8—84no—
311Q8—84no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—88no—
354C2×C4—88no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—164no—
363C2×C4—164no—
364C2×C4#38512yes—
365C2×C4#38812yes—
366C2×C4#39211yescharacteristic
367C2×C4#39311yescharacteristic
368C2×C4#39711yescharacteristic
369C2×C4#39911yescharacteristic
370C8—84no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—24no—
382C22—24no—
383C22—22no—
384C22—22no—
385C22—24no—
386C22—24no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—42no—
391C22—41no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22#143911yescharacteristic
398C22#180311yescharacteristic
399C22#180511yescharacteristic
400C22#188412yes—
401C22#189211yescharacteristic
402C22#189411yescharacteristic
403C22#191312yes—
404C22#198111yescharacteristic
405C22#198311yescharacteristic
406C22#205512yes—
407C22#205612yes—
408C22#206612yes—
409C22#206812yes—
410C4—21no—
411C4—22no—
412C4—22no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—44no—
417C4—84no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—164no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—21no—
426C2—21no—
427C2—22no—
428C2—22no—
429C2—21no—
430C2#938312yes—
431C2#938512yes—
432C2#1104611yescharacteristic
433C2#1105611yescharacteristic
434C2#1107711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1446C4semidirect
19. #1445C4semidirect

Maximal subgroups

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

subgroupindex
#219302
C2×#15202
#141592
#108462
C2×#12942