Finite Group Databasebeta

← #118391

Subgroups — #118391

Order 512.

790 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21948C211yesmaximal, characteristic
2#17578C211yesmaximal, characteristic
3C22×#328C211yesmaximal, characteristic
4#17397C211yesmaximal, characteristic
5#8020C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7#7909C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2280C2211yescharacteristic
14C22×#95C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1810—22no—
18#1520—22no—
19#1511—22no—
20#1508—22no—
21C2×#328—22no—
22C2×#328—22no—
23C2×#328C414yescomplemented
24#1472—22no—
25#1472—22no—
26#1463—22no—
27#1294—22no—
28#1282—22no—
29#1280—22no—
30C23×Q8Q811yescharacteristic
31C24×C4—21no—
32C24×C4C2311yescharacteristic
33C22×M16—21no—
34C22×M16—21no—
35C22×M16—21no—
36C23×C8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C22⋊Q8D412yes—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4Q811yescharacteristic
52C22×C22⋊C4—21no—
53C2×#108—24no—
54#407—21no—
55#407—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407D412yes—
65#407D412yes—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8C2×C412yes—
75C2×C22⋊C8D412yes—
76C2×C22⋊C8D412yes—
77C2×C22⋊C8D412yes—
78#381—21no—
79#380—21no—
80#380D412yes—
81#379—21no—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95C2×C412yes—
90#341—24no—
91#328—28no—
92#328—28no—
93#324—28no—
94C25#4111yescharacteristic
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—82no—
101C22×Q8C4⋊C412yes—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—44no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—82no—
109C23×C4—82no—
110C23×C4—82no—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×Q811yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—82no—
132C2×M16—82no—
133C2×M16—82no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—82no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156C22⋊Q8—41no—
157#117—82no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C4⋊C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C42—41no—
181#108—84no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8C22⋊C414yes—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—41no—
202#95—41no—
203#95—41no—
204C24—22no—
205C24—22no—
206C24—42no—
207C24—42no—
208C24C4×D411yescharacteristic
209C24#12111yescharacteristic
210C24C22⋊Q811yescharacteristic
211C24#12311yescharacteristic
212C24#12411yescharacteristic
213C24#12411yescharacteristic
214C24#12611yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—816no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C4⋊C411yescharacteristic
262C22×C4#11711yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4C22⋊Q811yescharacteristic
279C22×C4C22⋊Q811yescharacteristic
280C22×C4C22⋊Q811yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4#12311yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12411yescharacteristic
286C22×C4#12611yescharacteristic
287C22×C4#12711yescharacteristic
288M16—44no—
289M16—44no—
290M16—82no—
291M16—82no—
292M16—82no—
293M16—82no—
294C2×C8—24no—
295C2×C8—44no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—82no—
300C2×C8—82no—
301C2×C8—82no—
302C2×C8—82no—
303C2×C8—82no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—82no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C42—82no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—22no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—48no—
345C23—48no—
346C23#38511yescharacteristic
347C23#38811yescharacteristic
348C23#39011yescharacteristic
349C23#39311yescharacteristic
350C23#39411yescharacteristic
351C23#39611yescharacteristic
352C23#39711yescharacteristic
353C23#47112yes—
354C23#47312yes—
355Q8—82no—
356Q8—82no—
357Q8—82no—
358Q8—82no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
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—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—88no—
396C2×C4—84no—
397C2×C4—88no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—84no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#38011yescharacteristic
406C2×C4#38511yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#38811yescharacteristic
409C2×C4#39211yescharacteristic
410C2×C4#39411yescharacteristic
411C2×C4#39611yescharacteristic
412C2×C4#39911yescharacteristic
413C8—44no—
414C8—82no—
415C8—82no—
416C8—82no—
417C22—28no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—44no—
439C22—44no—
440C22—42no—
441C22—42no—
442C22—44no—
443C22#143911yescharacteristic
444C22#186112yes—
445C22#197711yescharacteristic
446C22#197911yescharacteristic
447C22#203212yes—
448C22#205212yes—
449C22#206412yes—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—84no—
459C4—82no—
460C4—82no—
461C4—162no—
462C2—21no—
463C2—22no—
464C2—22no—
465C2—22no—
466C2—22no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2#1108911yescharacteristic
471C2#1111411yescharacteristic
472C2#1112811yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
23. C2×#328C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#175782
C22×#3282
#173972
#80202
#79252
#79092