Finite Group Databasebeta

← #106609

Subgroups — #106609

Order 512.

897 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3C4×#408C211yesmaximal, characteristic
4#8073C212yesmaximal, complemented
5#7898C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2990—24no—
8#2983C412yescomplemented
9#2973—21no—
10#2973C412yescomplemented
11#2972—21no—
12#2970—24no—
13#2920—21no—
14C2×#432—21no—
15C2×#432C2212yes—
16C2×#408—21no—
17C2×C4×M16C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2283C2212yes—
20#1820—21no—
21#1799C412yes—
22#1799C412yes—
23#1782—21no—
24C4×#99—21no—
25C4×#99—21no—
26#1750C412yescomplemented
27#1750C412yescomplemented
28C4×C22⋊C8—21no—
29C4×C22⋊C8—21no—
30#1729—21no—
31#1300—24no—
32#1279—24no—
33C22×#41D411yescharacteristic
34C22×M16—21no—
35C23×C8—21no—
36C4×#41—22no—
37C4×#41—22no—
38C4×#41—22no—
39C4×#41—22no—
40C2×C4×Q8C2×C411yescharacteristic
41C2×C4×Q8Q811yescharacteristic
42C2×C4×D4C2×C411yescharacteristic
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4D411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4Q811yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117C2×C411yescharacteristic
50C2×#117D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42C2311yescharacteristic
54#446—28no—
55#435—28no—
56#435—28no—
57C8×D4—28no—
58#432—22no—
59#432—22no—
60#432—22no—
61#432—22no—
62#432—22no—
63#432—22no—
64#432C2×C414yes—
65#432D414yes—
66#431—22no—
67#431—22no—
68#423—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8D414yes—
71C2×#99—22no—
72C2×#99—22no—
73#408—28no—
74#407—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C4×M16—22no—
80C4×M16—22no—
81C4×M16—22no—
82C2×#97—22no—
83C2×#97—22no—
84C2×#97C2×C412yes—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2×C412yes—
90#383—22no—
91#382—22no—
92#381—22no—
93#380C2×C414yes—
94C4×C4⋊C4—22no—
95C4×C22⋊C4—22no—
96#376—22no—
97#342—22no—
98#342—22no—
99#336—22no—
100#336—22no—
101C2×#41—24no—
102C2×#41—22no—
103C2×#41—22no—
104C2×#41—42no—
105C22×Q8C4⋊C411yescharacteristic
106C22×D4C22⋊C411yescharacteristic
107C22×D4C22⋊C411yescharacteristic
108C22×D4C4⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—42no—
121C2×M16—42no—
122C22×C8—24no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—24no—
126C22×C8—24no—
127C22×C8—42no—
128C22×C8—42no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—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—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4C4211yescharacteristic
153C2×C4⋊C4C4211yescharacteristic
154C2×C4⋊C4C4⋊C411yescharacteristic
155C2×C4⋊C4C4⋊C411yescharacteristic
156C2×C4⋊C4C4⋊C411yescharacteristic
157C2×C4⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4—24no—
159C2×C22⋊C4C4211yescharacteristic
160C2×C22⋊C4C4211yescharacteristic
161C2×C22⋊C4C22⋊C411yescharacteristic
162C2×C22⋊C4C22⋊C411yescharacteristic
163C2×C22⋊C4C4⋊C411yescharacteristic
164C2×C22⋊C4C4⋊C411yescharacteristic
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—42no—
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C4⋊C411yescharacteristic
179C2×C42C4⋊C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q811yescharacteristic
186C2×C42#4112yes—
187C2×C42#4112yes—
188C4⋊C8—24no—
189C4⋊C8—24no—
190C4⋊C8—24no—
191C4⋊C8C22⋊C418yes—
192#99—44no—
193#99—44no—
194C22⋊C8—28no—
195C22⋊C8—28no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—24no—
199C22⋊C8—24no—
200#97—24no—
201#97—24no—
202#97—24no—
203#97—24no—
204C4×C8—24no—
205C4×C8—24no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210#95—24no—
211#95—42no—
212#95—42no—
213#95—42no—
214C24#9511yescharacteristic
215C24#9511yescharacteristic
216C24C2×C22⋊C411yescharacteristic
217#41—48no—
218#41—48no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4#10414yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
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—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4#9511yescharacteristic
257C22×C4#9511yescharacteristic
258C22×C4#9511yescharacteristic
259C22×C4#9511yescharacteristic
260C22×C4#9511yescharacteristic
261C22×C4#9511yescharacteristic
262C22×C4C2×C4211yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C4⋊C411yescharacteristic
266C22×C4C2×C4⋊C411yescharacteristic
267C22×C4C2×C4⋊C411yescharacteristic
268C22×C4C4×D412yes—
269C22×C4C4×D412yes—
270C22×C4#12012yes—
271C22×C4#12112yes—
272C22×C4#12112yes—
273C22×C4#12312yes—
274M16—48no—
275M16—84no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—28no—
279C2×C8—48no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—84no—
285C2×C8—84no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—48no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—84no—
314C42C2×C22⋊C412yes—
315C42#11712yes—
316C42C4×D412yes—
317C42C4×D412yes—
318C42#12011yescharacteristic
319C42#12011yescharacteristic
320C42C22⋊Q812yes—
321C42C22⋊Q812yes—
322C42#12311yescharacteristic
323C42#12311yescharacteristic
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23#33912yes—
336C23C2×#9511yescharacteristic
337C23#38612yes—
338C23C2×#10412yes—
339C23#42112yes—
340Q8—44no—
341D4—28no—
342D4—44no—
343D4—44no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4#33912yes—
387C2×C4C4×C22⋊C412yes—
388C2×C4#37911yescharacteristic
389C2×C4#38012yes—
390C2×C4#38111yescharacteristic
391C2×C4#38511yescharacteristic
392C2×C4#38512yes—
393C2×C4#38511yescharacteristic
394C2×C4#38611yescharacteristic
395C2×C4#38611yescharacteristic
396C8—44no—
397C8—44no—
398C8—48no—
399C22—21no—
400C22—22no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—81no—
414C22—81no—
415C22C22⋊#9511yescharacteristic
416C22C2×#33911yescharacteristic
417C22#173311yescharacteristic
418C22#187114yes—
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—21no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—164no—
433C4#179412yes—
434C4#186812yes—
435C2—24no—
436C2—21no—
437C2—21no—
438C2—42no—
439C2—81no—
440C2#1073211yescharacteristic
441C2#1074811yescharacteristic
442C2#1075211yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ 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. #8073C2semidirect
5. #7898C2semidirect
8. #2983C4semidirect
10. #2973C4semidirect
26. #1750C4semidirect
27. #1750C4semidirect

Maximal subgroups

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

subgroupindex
#602822
#207232
C4×#4082
#80732
#78982