Finite Group Databasebeta

← #200590

Subgroups — #200590

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#20728C211yesmaximal, characteristic
4#17742C212yesmaximal, complemented
5#9402C214yesmaximal, complemented
6#9393C214yesmaximal, complemented
7#8094C212yesmaximal, complemented
8C2×#536—21no—
9C22×C4⋊Q8C2211yescharacteristic
10C2×C4×#41C2211yescharacteristic
11#2990—24no—
12#2983C2212yes—
13#2973—22no—
14#2972C2212yes—
15#2970—24no—
16#2920C2212yes—
17#2920C2212yes—
18C2×#432—22no—
19#2857—22no—
20#2621C414yes—
21#2547—21no—
22#2547C2212yes—
23#2546—21no—
24#2546C2212yes—
25#2486—22no—
26#2486C414yes—
27#2310—21no—
28#2285C2211yescharacteristic
29#1822—21no—
30#1801C2212yes—
31#1800C2212yes—
32#1785—21no—
33C4×#101—21no—
34C4×#100—21no—
35#1556—24no—
36#1549—24no—
37#1546C2218yes—
38#1303C2218yes—
39C22×#41D411yescharacteristic
40C23×Q8—21no—
41#536—48no—
42C2×C4⋊Q8—22no—
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×#124—22no—
52C2×#123—24no—
53C2×C22⋊Q8—24no—
54C4×#41—22no—
55C4×#41—22no—
56C4×#41—22no—
57C4×#41—22no—
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×D4C2×C412yes—
61C2×C4×D4D412yes—
62C2×C4×D4D411yescharacteristic
63C2×C4×D4C2311yescharacteristic
64C2×#117—21no—
65C2×#117C2×C412yes—
66C2×#117D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C412yes—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71#446—28no—
72#435—216no—
73C8×D4—28no—
74#432—24no—
75#432—24no—
76#432—22no—
77#432—24no—
78#432—22no—
79#432—22no—
80#432D414yes—
81#431—22no—
82#431—22no—
83#431—22no—
84#431C2314yes—
85#423D414yes—
86#423C2314yes—
87C2×C4⋊C8—24no—
88C2×#101—22no—
89C2×#100—22no—
90#407—22no—
91#407—22no—
92C2×C22⋊C8—24no—
93C4×M16—22no—
94C4×M16—22no—
95C2×#97—24no—
96#398—24no—
97#398C2×C418yes—
98#397—24no—
99#397C2×C418yes—
100#391—22no—
101#391—22no—
102#391—24no—
103#389—24no—
104#389—24no—
105#384—21no—
106#382C2×C412yes—
107#381—24no—
108#380—24no—
109#380—24no—
110#380—21no—
111#380C2×C412yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2312yes—
114#343—22no—
115#343—22no—
116#332—216no—
117C2×#41—24no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—42no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8C2×D411yescharacteristic
128C22×D4C22⋊C412yes—
129C22×D4C2×D411yescharacteristic
130C23×C4—22no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C2×M16—24no—
138C2×M16—24no—
139C22×C8—28no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—48no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151#124—48no—
152#123—416no—
153C22⋊Q8—416no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166#117—22no—
167#117—22no—
168#117—24no—
169#117—24no—
170#117—44no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22×C414yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22×C412yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C412yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8C2×D418yes—
213#101—44no—
214#100—44no—
215C22⋊C8—28no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220#97—28no—
221#97—24no—
222#97—42no—
223C4×C8—24no—
224C4×C8—42no—
225#95—22no—
226#95—28no—
227#95—28no—
228#95C22×C414yes—
229C24C2×C22⋊C411yescharacteristic
230C24C2×C22⋊C412yes—
231#41—48no—
232#41—48no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—88no—
244C2×Q8—88no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—42no—
250C2×D4AGL(1,ℤ/8)14yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C4×D412yes—
277C22×C4C4×D412yes—
278C22×C4#12012yes—
279C22×C4#12112yes—
280C22×C4#12112yes—
281C22×C4#12312yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—44no—
286M16—44no—
287M16—84no—
288C2×C8—28no—
289C2×C8—44no—
290C2×C8—88no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—416no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—44no—
309C22⋊C4—416no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—44no—
317C42C4×D412yes—
318C42C4×D412yes—
319C42#12011yescharacteristic
320C42#12011yescharacteristic
321C42#12112yes—
322C42#12112yes—
323C42#12311yescharacteristic
324C42#12311yescharacteristic
325C42C22×D412yes—
326C42C2×#4112yes—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—42no—
335C23#38612yes—
336C23#41814yes—
337C23C22×C22⋊C411yescharacteristic
338Q8—44no—
339Q8—84no—
340Q8—84no—
341Q8—88no—
342D4—28no—
343D4—48no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—88no—
376C2×C4#38611yescharacteristic
377C2×C4#38611yescharacteristic
378C2×C4#38611yescharacteristic
379C2×C4#38612yes—
380C2×C4#38611yescharacteristic
381C2×C4C2×C4×D412yes—
382C2×C4C2×#12011yescharacteristic
383C2×C4C2×#12112yes—
384C2×C4C2×#12311yescharacteristic
385C2×C4C2×AGL(1,ℤ/8)12yes—
386C2×C4#57512yes—
387C8—48no—
388C8—84no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—24no—
395C22—44no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—82no—
401C22C2×#38611yescharacteristic
402C22C2×#41811yescharacteristic
403C22#289611yescharacteristic
404C4—21no—
405C4—22no—
406C4—21no—
407C4—22no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—84no—
414C4—84no—
415C4#191512yes—
416C4#302212yes—
417C4#305514yes—
418C2—21no—
419C2—21no—
420C2—42no—
421C2—44no—
422C2#2095511yescharacteristic
423C2#2097311yescharacteristic
424C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #17742C2semidirect
5. #9402C2semidirect
6. #9393C2semidirect
7. #8094C2semidirect

Maximal subgroups

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

subgroupindex
#602822
#353162
#207282
#177422
#94022
#93932
#80942