Finite Group Databasebeta

← #223183

Subgroups — #223183

Order 512.

995 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#36180C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic
3#22650C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9139C212yesmaximal, complemented
7#9130C212yesmaximal, complemented
8#8723C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10C4×#332C211yesmaximal, characteristic, complemented
11#8043C212yesmaximal, complemented
12C2×#554—21no—
13C2×#436—21no—
14C2×#431C2212yes—
15C2×#423C2212yes—
16#2868C2211yescharacteristic
17#2857C2212yes—
18#2567—21no—
19#2552—21no—
20#2552C412yes—
21#2502—22no—
22#2440—22no—
23#2440C414yes—
24D4×C4⋊C4—21no—
25D4×C4⋊C4C412yes—
26C4×C4⋊Q8C2211yescharacteristic
27C2×#384C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1939—21no—
31#1923—22no—
32#1851—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1797C2212yes—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42C2×#332C2212yes—
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50C22×M16—22no—
51#554—44no—
52#554—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2311yescharacteristic
57C2×#125—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—24no—
60C2×C4×Q8—21no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2×C414yes—
68C22×C42C2312yes—
69#436—44no—
70#431—24no—
71#431—42no—
72#423—24no—
73#423—42no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#398—24no—
92#398—24no—
93#395—24no—
94#391—21no—
95#391—22no—
96#389—22no—
97#389—22no—
98#389—21no—
99#389—22no—
100#389C2×C412yes—
101#389C2×C414yes—
102#385—24no—
103#385—24no—
104#384—22no—
105#384—21no—
106#384—21no—
107#384—22no—
108#384C2×C412yes—
109#384C2×C414yes—
110#384C2311yescharacteristic
111#384C2312yes—
112#380—22no—
113#380C2×C414yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C22⋊C4C2×C412yes—
120C4×C22⋊C4D412yes—
121C4×C22⋊C4D412yes—
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—21no—
126#336—21no—
127#332—22no—
128#332—22no—
129#332—42no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×D4C22⋊C411yescharacteristic
133C23×C4—24no—
134C23×C4C22×C412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C2×M16—44no—
138C2×M16—44no—
139C2×M16—82no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—81no—
153C4⋊Q8—81no—
154C4⋊Q8—82no—
155#125—48no—
156C22⋊Q8—44no—
157C22⋊Q8—44no—
158C22⋊Q8—416no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×D4—24no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—48no—
165C4×D4D814yes—
166C4×D4SD1614yes—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—84no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C414yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2411yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—44no—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—81no—
224C4⋊C8—81no—
225C4⋊C8#4112yes—
226C4⋊C8#4112yes—
227C22⋊C8—44no—
228C22⋊C8—44no—
229C22⋊C8—82no—
230C4×C8—44no—
231C4×C8—42no—
232C4×C8—42no—
233#95—22no—
234#95—22no—
235#95—24no—
236#95—24no—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24C2×C22⋊C412yes—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—44no—
248C2×Q8—82no—
249C2×Q8—82no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4#10214yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—42no—
270C22×C4—48no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×D814yes—
281C22×C4C2×SD1614yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—88no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—41no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—48no—
323C4⋊C4—84no—
324C4⋊C4—84no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4#10114yes—
328C4⋊C4#10214yes—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—48no—
332C22⋊C4#10218yes—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—22no—
344C42—41no—
345C42—44no—
346C42C2×D812yes—
347C42C2×SD1612yes—
348C42#13712yes—
349C42#13712yes—
350C42C2×#4111yescharacteristic
351C42C2×#4111yescharacteristic
352C42H(5,𝔽2)11yescharacteristic
353C422−1+411yescharacteristic
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23C2×#10214yes—
361C23C22×C22⋊C411yescharacteristic
362Q8—81no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366Q8—164no—
367D4—28no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—168no—
404C2×C4C2×#10114yes—
405C2×C4C2×#10112yes—
406C2×C4C2×#10212yes—
407C2×C4C2×#10214yes—
408C2×C4C2×#10212yes—
409C2×C4#41912yes—
410C2×C4#41912yes—
411C2×C4C4×#4111yescharacteristic
412C2×C4#52011yescharacteristic
413C2×C4#54611yescharacteristic
414C2×C4#54611yescharacteristic
415C2×C4#54711yescharacteristic
416C2×C4#54711yescharacteristic
417C2×C4C22×D811yescharacteristic
418C2×C4C22×SD1611yescharacteristic
419C2×C4C2×#13711yescharacteristic
420C2×C4C2×#13711yescharacteristic
421C2×C4#57611yescharacteristic
422C2×C4#57711yescharacteristic
423C2×C4#57711yescharacteristic
424C2×C4#57811yescharacteristic
425C8—42no—
426C8—42no—
427C8—82no—
428C8—81no—
429C8—81no—
430C8—82no—
431C22—24no—
432C22—24no—
433C22—24no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22#233911yescharacteristic
439C22C22×#10111yescharacteristic
440C22#288811yescharacteristic
441C22C22×#10211yescharacteristic
442C22C2×#41911yescharacteristic
443C22#289711yescharacteristic
444C22#289811yescharacteristic
445C4—22no—
446C4—22no—
447C4—21no—
448C4—21no—
449C4—24no—
450C4—24no—
451C4—41no—
452C4—42no—
453C4—82no—
454C4—84no—
455C4—82no—
456C4#154614yes—
457C4#329411yescharacteristic
458C4#329811yescharacteristic
459C4#333111yescharacteristic
460C4#333311yescharacteristic
461C4#333411yescharacteristic
462C4#333511yescharacteristic
463C4#335611yescharacteristic
464C4#335711yescharacteristic
465C2—44no—
466C2C2×#154611yescharacteristic
467C2#1772011yescharacteristic
468C2#1776011yescharacteristic
469C2#1777711yescharacteristic
470C2#2248111yescharacteristic
471C2#2248511yescharacteristic
472C2#2250211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9898C2semidirect
5. #9896C2semidirect
6. #9139C2semidirect
7. #9130C2semidirect
8. #8723C2semidirect
9. #8586C2semidirect
10. C4×#332C2semidirect
11. #8043C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#227952
#226502
#98982
#98962
#91392
#91302
#87232
#85862
C4×#3322
#80432