Finite Group Databasebeta

← #106556

Subgroups — #106556

Order 512.

824 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#20629C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#8128C212yesmaximal, complemented
5#7935C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2989—24no—
8#2983—21no—
9#2975C412yes—
10#2973—21no—
11#2973C412yes—
12#2965—24no—
13#2920—21no—
14#2920C2212yes—
15#2920C2212yes—
16C2×#431—21no—
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×#408C412yes—
20#1798—21no—
21#1798—21no—
22#1797—21no—
23#1797—21no—
24#1729C412yes—
25#1467—21no—
26#1466—21no—
27#1465C412yes—
28#1465C412yes—
29#1458—21no—
30#1458—21no—
31#1309—24no—
32#1283—24no—
33C22×#41C2×C411yescharacteristic
34C22×M16C2×C412yes—
35C23×C8C2×C412yes—
36C4×#41—22no—
37C4×#41—22no—
38C4×#41—22no—
39C4×#41—22no—
40C2×C4×Q8D411yescharacteristic
41C2×C4×Q8Q811yescharacteristic
42C2×C4×D4C2×C411yescharacteristic
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4Q811yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117D411yescharacteristic
50C2×#117D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42C2311yescharacteristic
54#446—28no—
55#436—28no—
56#435—28no—
57#435—28no—
58#432—22no—
59#432—22no—
60#432—22no—
61#432—22no—
62#432—22no—
63#432—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71#431—22no—
72#423—22no—
73#423—22no—
74#423—22no—
75#423—22no—
76#423D414yes—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84#408—44no—
85#407—22no—
86#407C2×C414yes—
87#407C2×C414yes—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C4×M16—22no—
92C4×M16—22no—
93C4×M16D414yes—
94C2×C4×C8—22no—
95#336—22no—
96#336—22no—
97#324—22no—
98#324—22no—
99#323—22no—
100#323—22no—
101C2×#41—22no—
102C2×#41—22no—
103C2×#41—22no—
104C2×#41—22no—
105C2×#41—41no—
106C2×#41—41no—
107C22×Q8C4⋊C411yescharacteristic
108C22×D4C4211yescharacteristic
109C22×D4C4211yescharacteristic
110C22×D4C4⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—42no—
124C2×M16—42no—
125C2×M16C22⋊C418yes—
126C22×C8—24no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—24no—
130C22×C8—42no—
131C22×C8—42no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4—22no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—22no—
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C4⋊C411yescharacteristic
157C2×C4⋊C4C4⋊C411yescharacteristic
158C2×C4⋊C4C4⋊C411yescharacteristic
159C2×C4⋊C4C4⋊C411yescharacteristic
160C2×C22⋊C4C4211yescharacteristic
161C2×C22⋊C4C4211yescharacteristic
162C2×C22⋊C4C22⋊C411yescharacteristic
163C2×C22⋊C4C22⋊C411yescharacteristic
164C2×C22⋊C4C4⋊C411yescharacteristic
165C2×C22⋊C4C4⋊C411yescharacteristic
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C4⋊C411yescharacteristic
179C2×C42C4⋊C411yescharacteristic
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q811yescharacteristic
186C2×C42#4112yes—
187C2×C42#4112yes—
188C4⋊C8—24no—
189C4⋊C8—24no—
190C4⋊C8—24no—
191C4⋊C8—24no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196C4⋊C8—42no—
197C4⋊C8—42no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—44no—
207C22⋊C8—44no—
208#97—24no—
209#97—24no—
210#97—42no—
211C4×C8—24no—
212C4×C8—24no—
213C4×C8—42no—
214C24#9511yescharacteristic
215C24#9511yescharacteristic
216C24C2×C4211yescharacteristic
217#41—44no—
218#41—44no—
219#41—44no—
220#41—44no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4#9511yescharacteristic
262C22×C4#9511yescharacteristic
263C22×C4#9511yescharacteristic
264C22×C4#9511yescharacteristic
265C22×C4#9511yescharacteristic
266C22×C4#9511yescharacteristic
267C22×C4#10012yes—
268C22×C4#10112yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C4⋊C411yescharacteristic
274C22×C4C2×C4⋊C411yescharacteristic
275C22×C4C2×C4⋊C411yescharacteristic
276C22×C4#11712yes—
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4C4×D412yes—
281M16—48no—
282M16—84no—
283M16—84no—
284M16—84no—
285C2×C8—28no—
286C2×C8—44no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—84no—
294C2×C8—84no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42#12012yes—
320C42#12011yescharacteristic
321C42#12011yescharacteristic
322C42#12112yes—
323C42#12112yes—
324C42C22⋊Q812yes—
325C42C22⋊Q812yes—
326C42#12312yes—
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23#34312yes—
343C23C2×#9511yescharacteristic
344C23C4×C22⋊C412yes—
345Q8—44no—
346D4—44no—
347D4—44no—
348D4—44no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4#34312yes—
393C2×C4#37911yescharacteristic
394C2×C4#38012yes—
395C2×C4#38111yescharacteristic
396C2×C4#38511yescharacteristic
397C2×C4#38511yescharacteristic
398C2×C4#38512yes—
399C2×C4#38612yes—
400C2×C4#38611yescharacteristic
401C2×C4#38611yescharacteristic
402C2×C4C2×#10011yescharacteristic
403C2×C4C2×#10111yescharacteristic
404C2×C4#41311yescharacteristic
405C2×C4#41311yescharacteristic
406C8—48no—
407C8—84no—
408C8—82no—
409C8—82no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—81no—
426C22—81no—
427C22C22⋊#9511yescharacteristic
428C22C2×#34311yescharacteristic
429C22#174211yescharacteristic
430C4—22no—
431C4—21no—
432C4—21no—
433C4—22no—
434C4—22no—
435C4—21no—
436C4—22no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4#179112yes—
443C4#190011yescharacteristic
444C4#190011yescharacteristic
445C4#190111yescharacteristic
446C4#190111yescharacteristic
447C2—21no—
448C2—21no—
449C2—42no—
450C2—42no—
451C2—81no—
452C2#1073911yescharacteristic
453C2#1074711yescharacteristic
454C2#1075711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC24C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8128C2semidirect
5. #7935C2semidirect

Maximal subgroups

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

subgroupindex
#602772
#206292
#173612
#81282
#79352