Finite Group Databasebeta

← #190268

Subgroups — #190268

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34195C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#8769C212yesmaximal, complemented
5#8768C212yesmaximal, complemented
6#8765C212yesmaximal, complemented
7#8764C212yesmaximal, complemented
8#8096C212yesmaximal, complemented
9#7958C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11#2920C2212yes—
12C2×#431C2212yes—
13#2356C2211yescharacteristic
14#2356C2211yescharacteristic
15#2353C412yes—
16#2353C412yes—
17#2349C2211yescharacteristic
18#2349C2211yescharacteristic
19#2347—21no—
20#2343—22no—
21#2339—21no—
22#2320—21no—
23#2318—22no—
24#2283C412yes—
25#2283C2211yescharacteristic
26#2283C2211yescharacteristic
27C2×#383C412yes—
28#2275—21no—
29#1822—21no—
30#1801C2212yes—
31#1800C2212yes—
32#1785—21no—
33C4×#101—21no—
34C4×#100—21no—
35#1461C2212yes—
36#1460C2212yes—
37#1459—22no—
38#1458—22no—
39#1303C2214yes—
40#1303C2214yes—
41#1286C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8C2311yescharacteristic
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D412yes—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×#117D411yescharacteristic
57C2×#117D412yes—
58C22×C42C2×C412yes—
59C22×C42C2311yescharacteristic
60#432—22no—
61#432—22no—
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431C2314yes—
68#423C2314yes—
69C2×C4⋊C8—22no—
70C2×#101—22no—
71C2×#100—22no—
72#407—22no—
73C2×C22⋊C8—22no—
74C4×M16—22no—
75C2×C4×C8—22no—
76#391—21no—
77#391—21no—
78#389—21no—
79#389—21no—
80#388—22no—
81#388—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#386—22no—
87#386—22no—
88#385—22no—
89#385—22no—
90#383—22no—
91#383—22no—
92#383C2×C412yes—
93#383C2×C412yes—
94#383C2×C411yescharacteristic
95#383C2×C412yes—
96#383C2×C411yescharacteristic
97#382—22no—
98#382—21no—
99#382C2311yescharacteristic
100#382C2311yescharacteristic
101#381—22no—
102#381—21no—
103#381C2×C411yescharacteristic
104#381C2×C411yescharacteristic
105#380—21no—
106#380—21no—
107#380C2×C412yes—
108#380C2×C412yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2311yescharacteristic
114C4×C4⋊C4C2311yescharacteristic
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4C2×C412yes—
120#376—21no—
121#376—21no—
122#376C2×C412yes—
123#376C2×C412yes—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128#343—22no—
129#343—22no—
130#324—24no—
131#323—24no—
132C2×#41—24no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—42no—
136C22×Q8C2×D411yescharacteristic
137C22×D4C22⋊C412yes—
138C22×D4C2×D411yescharacteristic
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C22×C8—42no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—44no—
162C4×D4—44no—
163#117—22no—
164#117—22no—
165#117—24no—
166#117—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C2×D412yes—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42C22⋊C412yes—
195C2×C42C22⋊C412yes—
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C2×C42#4112yes—
204C2×C42C2411yescharacteristic
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—24no—
208C4⋊C8—42no—
209#101—44no—
210#100—44no—
211C22⋊C8—24no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215#97—42no—
216C4×C8—24no—
217C4×C8—42no—
218C4×C8—42no—
219#95—21no—
220#95—21no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95C22×C411yescharacteristic
238#95C22×C411yescharacteristic
239#95C22×C412yes—
240#95C22×C412yes—
241#95C22×C411yescharacteristic
242#95C22×C411yescharacteristic
243C24C2×C22⋊C411yescharacteristic
244C24C2×C22⋊C412yes—
245#41—48no—
246#41—48no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4#13514yes—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4#11712yes—
289C22×C4#11712yes—
290C22×C4#12312yes—
291C22×C4#12312yes—
292C22×C4#12312yes—
293C22×C4#12312yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297M16—44no—
298M16—44no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—84no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—82no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—42no—
327C42—42no—
328C42—82no—
329C42—82no—
330C42—82no—
331C42#11711yescharacteristic
332C42#11711yescharacteristic
333C42#11711yescharacteristic
334C42#11711yescharacteristic
335C42#12312yes—
336C42#12312yes—
337C42#12312yes—
338C42#12312yes—
339C42C2×#4112yes—
340C42C2×#4112yes—
341C23—22no—
342C23—22no—
343C23—21no—
344C23—21no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23#38112yes—
350C23#41614yes—
351C23C22×C22⋊C411yescharacteristic
352Q8—44no—
353D4—28no—
354D4—48no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—164no—
399C2×C4—164no—
400C2×C4#38111yescharacteristic
401C2×C4#38111yescharacteristic
402C2×C4#38111yescharacteristic
403C2×C4#38111yescharacteristic
404C2×C4#38112yes—
405C2×C4C2×#11711yescharacteristic
406C2×C4C2×#11711yescharacteristic
407C2×C4C2×#12312yes—
408C2×C4C2×#12312yes—
409C2×C4C2×#13512yes—
410C2×C4#57512yes—
411C8—84no—
412C8—84no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—24no—
417C22—22no—
418C22—22no—
419C22—44no—
420C22—44no—
421C22—42no—
422C22—42no—
423C22—81no—
424C22—81no—
425C22C2×#38111yescharacteristic
426C22C2×#41611yescharacteristic
427C22#289611yescharacteristic
428C4—21no—
429C4—21no—
430C4—22no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—42no—
437C4—42no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4—82no—
442C4—82no—
443C4—81no—
444C4—81no—
445C4#309114yes—
446C2—21no—
447C2—21no—
448C2—42no—
449C2—44no—
450C2#2069911yescharacteristic
451C2#2070811yescharacteristic
452C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8769C2semidirect
5. #8768C2semidirect
6. #8765C2semidirect
7. #8764C2semidirect
8. #8096C2semidirect
9. #7958C2semidirect

Maximal subgroups

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

subgroupindex
#341952
#207282
#173492
#87692
#87682
#87652
#87642
#80962
#79582