Finite Group Databasebeta

← #201422

Subgroups — #201422

Order 512.

1320 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#21833C212yesmaximal, complemented
4#20632C211yesmaximal, characteristic
5#9474C218yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×#536—21no—
8C22×C4⋊Q8C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2989—24no—
11#2983—21no—
12#2975—21no—
13#2973C2214yes—
14#2965—24no—
15#2920—22no—
16C2×#431C2212yes—
17C2×#410—21no—
18#2857—22no—
19#2621C414yescomplemented
20#2547—21no—
21#2547C412yescomplemented
22#2546—21no—
23#2546C412yescomplemented
24#2486—22no—
25#2486C2214yes—
26#2310—21no—
27#2285C2212yes—
28#2285C2211yescharacteristic
29#1990—24no—
30#1988—24no—
31#1823C2218yes—
32#1799—22no—
33#1799C2214yes—
34#1740—21no—
35#1289C2218yes—
36C22×#41D411yescharacteristic
37C23×Q8—21no—
38#536—48no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C412yes—
48C2×#124—22no—
49C2×#123—24no—
50C2×C22⋊Q8—24no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D412yes—
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2312yes—
61C2×#117—21no—
62C2×#117—22no—
63C2×#117C2×C412yes—
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69#446—28no—
70#436—28no—
71#435—216no—
72#432—24no—
73#432—24no—
74#431—24no—
75#431—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79#423—24no—
80C2×C4⋊C8D414yes—
81#410—44no—
82C2×#99—24no—
83#407—24no—
84C2×C22⋊C8C2314yes—
85C4×M16—24no—
86C2×C4×C8D414yes—
87#398—24no—
88#398C2×C418yes—
89#397—24no—
90#397C2×C418yes—
91#391—22no—
92#391—22no—
93#391—24no—
94#389—24no—
95#389—24no—
96#384—21no—
97#384—22no—
98#382—22no—
99#382C2×C412yes—
100#381—24no—
101#380—24no—
102#380—24no—
103#380—21no—
104#380—22no—
105#380C2314yes—
106#380C2312yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111#342—24no—
112#340—216no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—24no—
116C2×#41—22no—
117C2×#41—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—44no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8C22⋊C411yescharacteristic
126C22×D4C22⋊C411yescharacteristic
127C22×D4C2×D412yes—
128C23×C4—22no—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C2×M16—28no—
136C22×C8C2×D418yes—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—48no—
142C4⋊Q8—41no—
143C4⋊Q8—42no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—42no—
148#124—48no—
149#123—416no—
150C22⋊Q8—416no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163#117—22no—
164#117—22no—
165#117—24no—
166#117—24no—
167#117—44no—
168#117—44no—
169#117—44no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C22×C412yes—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22×C412yes—
194C2×C22⋊C4C2×D412yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—42no—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212C4⋊C8—28no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215#99—48no—
216C22⋊C8—28no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219#97—28no—
220C4×C8—24no—
221C4×C8—24no—
222#95—22no—
223#95—28no—
224#95—28no—
225#95—24no—
226#95—42no—
227#95C22×C414yes—
228C24C2×C22⋊C412yes—
229C24C22×D411yescharacteristic
230#41—44no—
231#41—44no—
232#41—44no—
233#41—44no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×Q8—84no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12112yes—
284C22×C4#12112yes—
285C22×C4AGL(1,ℤ/8)12yes—
286C22×C4AGL(1,ℤ/8)14yes—
287C22×C4#13712yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D412yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4112yes—
292M16—88no—
293C2×C8—28no—
294C2×C8—48no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—28no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—416no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—416no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—44no—
326C42—84no—
327C42C4×D412yes—
328C42C4×D412yes—
329C42C4×D412yes—
330C42C4×D412yes—
331C42#12011yescharacteristic
332C42#12012yes—
333C42#12011yescharacteristic
334C42#12311yescharacteristic
335C42#12311yescharacteristic
336C42#12312yes—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—42no—
344C23C22×C22⋊C411yescharacteristic
345C23C2×#12112yes—
346C23C2×AGL(1,ℤ/8)12yes—
347C23C2×AGL(1,ℤ/8)11yescharacteristic
348C23C2×#13711yescharacteristic
349Q8—44no—
350Q8—84no—
351Q8—84no—
352Q8—88no—
353D4—44no—
354D4—44no—
355D4—44no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4#38611yescharacteristic
389C2×C4#38612yes—
390C2×C4#38611yescharacteristic
391C2×C4#38611yescharacteristic
392C2×C4#38612yes—
393C2×C4#38611yescharacteristic
394C2×C4#41712yes—
395C2×C4#41712yes—
396C2×C4#41812yes—
397C2×C4#41812yes—
398C2×C4#46814yes—
399C2×C4#46814yes—
400C2×C4#46914yes—
401C2×C4#46914yes—
402C2×C4C2×C4×D412yes—
403C2×C4C2×#12011yescharacteristic
404C2×C4C2×#12311yescharacteristic
405C8—48no—
406C22—21no—
407C22—22no—
408C22—21no—
409C22—22no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—44no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—82no—
418C22C2×#38611yescharacteristic
419C22C2×#41811yescharacteristic
420C22C2×#41711yescharacteristic
421C22C2×#46812yes—
422C22C2×#46912yes—
423C4—21no—
424C4—22no—
425C4—21no—
426C4—22no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—84no—
431C4—84no—
432C4—164no—
433C4#189512yes—
434C4#189512yes—
435C4#189514yes—
436C4#190912yes—
437C4#302812yes—
438C2—21no—
439C2—22no—
440C2—21no—
441C2—22no—
442C2—82no—
443C2#2096211yescharacteristic
444C2C2×#189511yescharacteristic
445C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21833C2semidirect
5. #9474C2semidirect
6. #7981C2semidirect
19. #2621C4semidirect
21. #2547C4semidirect
23. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#602772
#353162
#218332
#206322
#94742
#79812