Finite Group Databasebeta

← #391354

Subgroups — #391354

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23638C211yesmaximal, characteristic
2#20834C212yesmaximal
3#11329C212yesmaximal, complemented
4#11317C212yesmaximal, complemented
5#11291C212yesmaximal, complemented
6#11221C212yesmaximal, complemented
7#11201C212yesmaximal
8#11097C211yesmaximal, characteristic
9#10714C211yesmaximal, characteristic
10C24×C8C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C2×#424C2212yes—
14C2×#424C2212yes—
15C2×#407C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2280C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19#2077C412yescomplemented
20#2077C412yescomplemented
21#2076C411yescharacteristic, complemented
22#2076C411yescharacteristic, complemented
23#2075C411yescharacteristic, complemented
24#2075C411yescharacteristic, complemented
25#1850—24no—
26#1850—24no—
27C2×#341C2212yes—
28C2×#341C2211yescharacteristic
29#1735C2211yescharacteristic
30#1735C2212yes—
31C2×#339C2212yes—
32C2×#339C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35#1729C2212yes—
36#1729C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C24×C4C2311yescharacteristic
40C22×M16C2312yes—
41C22×M16C2312yes—
42C23×C8—42no—
43C23×C8—42no—
44C23×C8—41no—
45C23×C8—41no—
46C23×C8D412yes—
47C23×C8D412yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C2×#117C2312yes—
52C2×#117C2312yes—
53C22×C22⋊C4—21no—
54C22×C42C2312yes—
55C2×#108—42no—
56C2×#108—41no—
57#427—41no—
58#427—41no—
59C2×#107C2×C411yescharacteristic
60C2×#107C2×C411yescharacteristic
61C2×#106C2×C412yes—
62#424—42no—
63#424—42no—
64#424—42no—
65#424—42no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68#407—42no—
69#407—42no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74C2×C22⋊C8C2×C411yescharacteristic
75C2×C22⋊C8C2×C411yescharacteristic
76C2×C22⋊C8C2×C411yescharacteristic
77C2×C22⋊C8C2×C411yescharacteristic
78#381—22no—
79#380C2×C412yes—
80#380C2×C412yes—
81#380C2×C412yes—
82#380C2×C412yes—
83#379—22no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—41no—
87C4×C22⋊C4—41no—
88C2×#95—21no—
89#341—42no—
90#341—41no—
91#340C2×C411yescharacteristic
92#340C2×C411yescharacteristic
93#340C2×C412yes—
94#340C2×C412yes—
95#340C2×C412yes—
96#340C2×C412yes—
97#340C2×C412yes—
98#339—42no—
99#339—42no—
100#339—42no—
101#339—42no—
102#336—24no—
103#336—24no—
104#336—42no—
105#336—42no—
106C25#4111yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4C2411yescharacteristic
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—44no—
153C22×C8—44no—
154C22×C8—44no—
155C22×C8—48no—
156C22×C8—44no—
157C22×C8C22×C411yescharacteristic
158C22×C8C22×C411yescharacteristic
159C22×C8C22×C411yescharacteristic
160C22×C8C22×C411yescharacteristic
161C22×C8C2×D414yes—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191#108—44no—
192#107—41no—
193#107—41no—
194#106—42no—
195C4⋊C8—48no—
196C4⋊C8—48no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—44no—
202C22⋊C8—44no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—42no—
208C24—21no—
209C24—21no—
210C24—22no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24C4×D411yescharacteristic
216C24C4×D411yescharacteristic
217C24C2×#4111yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C4H(5,𝔽2)11yescharacteristic
265C22×C4H(5,𝔽2)12yes—
266C22×C4H(5,𝔽2)11yescharacteristic
267C22×C4H(5,𝔽2)12yes—
268C22×C42−1+412yes—
269C22×C42−1+412yes—
270C22×C42−1+411yescharacteristic
271M16—42no—
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276M16—42no—
277M16—42no—
278M16—42no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—48no—
291C2×C8—48no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—48no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—24no—
337C23—24no—
338C23—22no—
339C23—22no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—22no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23#44312yes—
356C23#44412yes—
357C23C2×C4×D411yescharacteristic
358C23#53811yescharacteristic
359C23#53912yes—
360C23#53912yes—
361C23#53911yescharacteristic
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
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—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#51811yescharacteristic
401C2×C4#52012yes—
402C2×C4#52012yes—
403C2×C4#52011yescharacteristic
404C2×C4#53411yescharacteristic
405C2×C4#53412yes—
406C2×C4#53612yes—
407C2×C4#53711yescharacteristic
408C2×C4#56011yescharacteristic
409C2×C4#56012yes—
410C2×C4H(3,𝔽4)11yescharacteristic
411C2×C4#56212yes—
412C2×C4#56212yes—
413C2×C4#56311yescharacteristic
414C2×C4#56311yescharacteristic
415C2×C4#56312yes—
416C2×C4#56312yes—
417C2×C4#56312yes—
418C8—41no—
419C8—41no—
420C8—41no—
421C8—41no—
422C8—44no—
423C8—82no—
424C8—82no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—24no—
434C22—24no—
435C22—22no—
436C22—22no—
437C22—48no—
438C22—44no—
439C22—44no—
440C22—41no—
441C22—41no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22#232212yes—
447C22#232511yescharacteristic
448C22#282712yes—
449C22#283411yescharacteristic
450C22#283511yescharacteristic
451C22C2×#44311yescharacteristic
452C22C2×#44411yescharacteristic
453C22#295512yes—
454C4—41no—
455C4—41no—
456C4—84no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—82no—
461C4#237611yescharacteristic
462C4#237911yescharacteristic
463C4#238312yes—
464C4#281211yescharacteristic
465C4#281412yes—
466C4#281711yescharacteristic
467C2—22no—
468C2—21no—
469C2—21no—
470C2—42no—
471C2—42no—
472C2#1462711yescharacteristic
473C2#2209512yes—
474C2#2209612yes—
475C2#2220411yescharacteristic
476C2#2220511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11329C2semidirect
4. #11317C2semidirect
5. #11291C2semidirect
6. #11221C2semidirect
19. #2077C4semidirect
20. #2077C4semidirect
21. #2076C4semidirect
22. #2076C4semidirect
23. #2075C4semidirect
24. #2075C4semidirect

Maximal subgroups

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

subgroupindex
#236382
#208342
#113292
#113172
#112912
#112212
#112012
#110972
#107142