Finite Group Databasebeta

← #192361

Subgroups — #192361

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21394C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic
5#13854C211yesmaximal, characteristic
6#8872C214yesmaximal
7#8867C214yesmaximal
8#8022C211yesmaximal, characteristic
9#7912C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2370—22no—
14#2369—21no—
15#2365—21no—
16#2355—21no—
17#2355—22no—
18Q8×C4⋊C4—21no—
19#2329C2214yes—
20#2317—21no—
21C4×C22⋊Q8—22no—
22#2285C2211yescharacteristic
23#2274—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1988—21no—
28#1981—21no—
29#1948—21no—
30#1947—21no—
31#1872C2212yes—
32#1872C2212yes—
33#1853C2212yes—
34#1852C2212yes—
35#1776—22no—
36#1763—22no—
37#1506C418yescomplemented
38C2×#328C2214yes—
39#1442—21no—
40#1442—21no—
41#1442C2212yes—
42#1442C2212yes—
43C4×#95—22no—
44#1294C2214yes—
45#1280C2214yes—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533Q814yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8D412yes—
58C2×#117—21no—
59C2×#117—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68#432—22no—
69#432—22no—
70#432—22no—
71#432C2×C414yes—
72C2×#107—22no—
73C2×#106—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×#103—22no—
77C2×#103—22no—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×#97—21no—
81C2×#97—21no—
82#391—22no—
83#391—24no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—24no—
88#387—24no—
89#387—24no—
90#387—24no—
91#385—24no—
92#384—22no—
93#384D411yescharacteristic
94#384D411yescharacteristic
95#382—22no—
96#382—21no—
97#380—21no—
98#380C2312yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—24no—
116#376—22no—
117#376C2×C414yes—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95C2312yes—
125C2×#95C2312yes—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#328C2×C4116yes—
131C22×Q8—21no—
132C22×Q8C2×D412yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8C4⋊C414yes—
146C4⋊Q8C4⋊C414yes—
147#125—24no—
148#125—24no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8C22×C414yes—
154C22⋊Q8C2×Q814yes—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—84no—
159C4×Q8C4⋊C418yes—
160#117—24no—
161#117—24no—
162#117—44no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
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—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q812yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#107—44no—
213#106—44no—
214C4⋊C8—24no—
215C4⋊C8—42no—
216#103—44no—
217#103—44no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8C22×C414yes—
222#97—24no—
223#97—42no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—41no—
238#95—41no—
239#95—44no—
240#95C22×C414yes—
241C24C22×D411yescharacteristic
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8C2×C4⋊C414yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4C22⋊Q812yes—
292C22×C4C22⋊Q812yes—
293C22×C4#13714yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×Q811yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—28no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C4⋊C4C2×C4⋊C414yes—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—44no—
337C42—28no—
338C42—48no—
339C42—44no—
340C42—44no—
341C42—82no—
342C42—82no—
343C42—84no—
344C42—84no—
345C42—82no—
346C42C2×C4⋊C412yes—
347C42C2×C4⋊C412yes—
348C42C4×D412yes—
349C42C4×D412yes—
350C42C22⋊Q812yes—
351C42C22⋊Q812yes—
352C42#12312yes—
353C42#12312yes—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23C2×C4×D411yescharacteristic
359C23C2×#12011yescharacteristic
360C23C2×C22⋊Q811yescharacteristic
361C23C2×#13712yes—
362C23#57512yes—
363Q8—44no—
364Q8—48no—
365Q8—82no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4#38512yes—
424C2×C4#38512yes—
425C2×C4#38512yes—
426C2×C4#38512yes—
427C2×C4#45412yes—
428C2×C4#45512yes—
429C2×C4C22×C4⋊C411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C8—44no—
434C8—82no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22C2×#38511yescharacteristic
444C22C4×#13712yes—
445C22#300411yescharacteristic
446C22#300511yescharacteristic
447C22C2×#45511yescharacteristic
448C22C2×#45411yescharacteristic
449C22#302511yescharacteristic
450C22#302511yescharacteristic
451C22#307812yes—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—44no—
460C4—44no—
461C4—44no—
462C4—84no—
463C4—82no—
464C4—82no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4#186014yes—
469C2—22no—
470C2—22no—
471C2#1818111yescharacteristic
472C2#1818211yescharacteristic
473C2#1818811yescharacteristic
474C2#1819111yescharacteristic
475C2C2×#186011yescharacteristic
476C2#2087411yescharacteristic
477C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #1506C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#213942
#213652
C2×#15062
#138542
#88722
#88672
#80222
#79122