Finite Group Databasebeta

← #200517

Subgroups — #200517

Order 512.

1370 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3#20726C211yesmaximal, characteristic
4#17746C212yesmaximal, complemented
5#9399C214yesmaximal, complemented
6#9391C214yesmaximal, complemented
7#8098C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10#2975C2212yes—
11#2974—24no—
12#2973—22no—
13#2972C2212yes—
14D4×M16—24no—
15#2920C2212yes—
16#2920C2212yes—
17C2×#407—22no—
18#2563—22no—
19#2563—21no—
20#2563—21no—
21#2563C414yes—
22#2563C2212yes—
23#2563C2212yes—
24C2×#393C414yes—
25C2×#390—21no—
26C2×#382—21no—
27C2×#380—22no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1821—21no—
30#1800C2212yes—
31#1800C2212yes—
32#1783—21no—
33C4×#100—21no—
34C4×#100—21no—
35#1558—24no—
36#1548—24no—
37#1548C2218yescomplemented
38#1303C2218yescomplemented
39C23×Q8C2×C412yes—
40C23×D4D411yescharacteristic
41C24×C4D412yes—
42C22×M16—24no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2312yes—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C412yes—
56C2×C4×D4D412yes—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×C4×D4C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C22⋊C4—24no—
66C22×C22⋊C4C2×C412yes—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#436—28no—
70#435—216no—
71C8×D4—28no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432D414yes—
76#431—22no—
77#431—22no—
78#431—22no—
79#431C2314yes—
80#423D414yes—
81#423C2314yes—
82C2×#100—22no—
83C2×#100—22no—
84#407—24no—
85#407—24no—
86#407—22no—
87#407—24no—
88#407—22no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C4×M16—22no—
92C4×M16—22no—
93#393—24no—
94#393—24no—
95#393C2×C418yes—
96#393C2×C418yes—
97#390—24no—
98#390—22no—
99#390—22no—
100#382—24no—
101#382—22no—
102#382—22no—
103#380—24no—
104#380—24no—
105#380—24no—
106#380—24no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2312yes—
111C2×#95—24no—
112C2×#95C2×C412yes—
113#343—22no—
114#343—22no—
115#333—216no—
116C25C22⋊C412yes—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—84no—
121C22×Q8C22×C414yes—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—42no—
126C22×D4C22⋊C412yes—
127C22×D4C2×D411yescharacteristic
128C22×D4C2×D411yescharacteristic
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—48no—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C2×M16—24no—
145C2×M16—28no—
146C2×M16—24no—
147C2×M16—88no—
148C22×C8—28no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—88no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22×C412yes—
195C2×C22⋊C4C22×C412yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—44no—
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42C2411yescharacteristic
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8C2×D418yes—
211#100—44no—
212#100—44no—
213C22⋊C8—28no—
214C22⋊C8—28no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218C22⋊C8—42no—
219#97—24no—
220#97—42no—
221C4×C8—24no—
222C4×C8—42no—
223#95—28no—
224#95—28no—
225#95—22no—
226#95C22×C414yes—
227C24—24no—
228C24—22no—
229C24—22no—
230C24—42no—
231C24—44no—
232C24—44no—
233C24C2×C22⋊C411yescharacteristic
234C24C2×C22⋊C412yes—
235C24C2×C22⋊C412yes—
236C2×Q8—28no—
237C2×Q8—44no—
238C2×Q8—88no—
239C2×Q8—84no—
240C2×Q8—84no—
241C2×Q8—88no—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—48no—
250C2×D4—48no—
251C2×D4—42no—
252C2×D4#13714yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
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—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—48no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—84no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C4×D412yes—
290C22×C4C4×D412yes—
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4#12311yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300M16—48no—
301M16—44no—
302M16—44no—
303M16—88no—
304C2×C8—28no—
305C2×C8—44no—
306C2×C8—88no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—88no—
323C22⋊C4—88no—
324C42—44no—
325C42—44no—
326C42C4×D412yes—
327C42C4×D412yes—
328C42#12012yes—
329C42#12112yes—
330C42#12112yes—
331C42#12312yes—
332C42C22×D412yes—
333C42C2×#4112yes—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—416no—
350C23—44no—
351C23—44no—
352C23—82no—
353C23—82no—
354C23—88no—
355C23#38611yescharacteristic
356C23#38611yescharacteristic
357C23#41914yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—88no—
360Q8—88no—
361D4—28no—
362D4—48no—
363D4—44no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—88no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4#38612yes—
400C2×C4#38611yescharacteristic
401C2×C4#38612yes—
402C2×C4#38611yescharacteristic
403C2×C4#45514yes—
404C2×C4C2×C4×D412yes—
405C2×C4C2×#12011yescharacteristic
406C2×C4C2×#12112yes—
407C2×C4C2×#12311yescharacteristic
408C2×C4C2×#13712yes—
409C2×C4#57512yes—
410C8—48no—
411C8—84no—
412C22—24no—
413C22—24no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—24no—
421C22—44no—
422C22—44no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—82no—
429C22—88no—
430C22—88no—
431C22—82no—
432C22#188512yes—
433C22#191412yes—
434C22C2×#38611yescharacteristic
435C22C2×#41911yescharacteristic
436C22#289611yescharacteristic
437C4—22no—
438C4—22no—
439C4—21no—
440C4—21no—
441C4—22no—
442C4—42no—
443C4—42no—
444C4—84no—
445C4—84no—
446C4#188512yes—
447C4C2×#45512yes—
448C4#305614yes—
449C2—21no—
450C2—21no—
451C2—22no—
452C2—22no—
453C2—44no—
454C2—82no—
455C2C2×#188511yescharacteristic
456C2#2097211yescharacteristic
457C2C2×#191411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17746C2semidirect
5. #9399C2semidirect
6. #9391C2semidirect
7. #8098C2semidirect
37. #1548C22semidirect
38. #1303C22semidirect

Maximal subgroups

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

subgroupindex
#602302
C2×#25632
#207262
#177462
#93992
#93912
#80982