Finite Group Databasebeta

← #201207

Subgroups — #201207

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#17746C212yesmaximal
4#17365C211yesmaximal, characteristic
5#9452C218yesmaximal, complemented
6#7950C212yesmaximal
7C22×C4⋊Q8C412yes—
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2984—24no—
11#2983—21no—
12#2983C2214yes—
13#2982—21no—
14Q8×M16—24no—
15#2920—22no—
16#2920C2212yes—
17#2920C2212yes—
18#2546—21no—
19#2546—21no—
20#2546—21no—
21#2546—21no—
22#2546C414yes—
23#2546C2214yes—
24C2×#398—22no—
25C2×#391C412yes—
26C2×#389—22no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1558C2218yes—
29#1548—24no—
30#1548—24no—
31#1467C2214yes—
32#1463—22no—
33#1461—21no—
34#1461—21no—
35#1285C2218yes—
36C23×Q8C2×C412yes—
37C23×Q8C2×C411yescharacteristic
38C2×C4⋊Q8—22no—
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⋊Q8—48no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C418yes—
49C2×C4⋊Q8C2312yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4D412yes—
66C22×C42C2×C412yes—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69#446—28no—
70#446—216no—
71C8×Q8—28no—
72#432—24no—
73#432—24no—
74#432—22no—
75#432—22no—
76#432C2314yes—
77#432C2314yes—
78#431—24no—
79#431—24no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431D414yes—
84#423—24no—
85#423—22no—
86#423D414yes—
87#407—22no—
88#407—22no—
89C4×M16—24no—
90C4×M16—22no—
91C4×M16—22no—
92#398—24no—
93#398—24no—
94#398—24no—
95#398—24no—
96#391—22no—
97#391—22no—
98#391C2×C418yes—
99#389—24no—
100#389—24no—
101#389—24no—
102#389—24no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2312yes—
107C2×#95—24no—
108C2×#95—21no—
109#333—216no—
110#324—24no—
111#324—24no—
112C22×Q8—24no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—42no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22×C414yes—
124C22×Q8C22×C412yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—24no—
133C2×M16—42no—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—48no—
139C4⋊Q8—48no—
140C4⋊Q8—416no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—24no—
156C4×Q8—24no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—416no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—44no—
190C2×C42C22⋊C412yes—
191C2×C42C22⋊C412yes—
192C2×C42C22×C414yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—28no—
204C4⋊C8—28no—
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—42no—
208C4⋊C8C2×D418yes—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—42no—
213#97—28no—
214#97—24no—
215#97—24no—
216C4×C8—28no—
217C4×C8—24no—
218C4×C8—42no—
219#95—22no—
220#95—28no—
221#95—28no—
222#95—22no—
223C24C2×C22⋊C411yescharacteristic
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—24no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—28no—
232C2×Q8—42no—
233C2×Q8—48no—
234C2×Q8—48no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—88no—
241C2×Q8—88no—
242C2×Q8#10014yes—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12112yes—
275C22×C4#12112yes—
276C22×C4#12311yescharacteristic
277C22×C4#12311yescharacteristic
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281M16—48no—
282M16—84no—
283C2×C8—28no—
284C2×C8—44no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—416no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—416no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—44no—
311C42—44no—
312C42C4×D412yes—
313C42C4×D412yes—
314C42#12012yes—
315C42#12112yes—
316C42#12112yes—
317C42#12312yes—
318C42C22×D412yes—
319C42C2×#4112yes—
320C23—21no—
321C23—21no—
322C23—22no—
323C23—22no—
324C23#38611yescharacteristic
325C23#38611yescharacteristic
326C23C22×C22⋊C411yescharacteristic
327Q8—28no—
328Q8—48no—
329Q8—44no—
330Q8—88no—
331Q8—88no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—28no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—416no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—82no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4#38611yescharacteristic
368C2×C4#38612yes—
369C2×C4#38611yescharacteristic
370C2×C4#38612yes—
371C2×C4C2×#10014yes—
372C2×C4C2×#10012yes—
373C2×C4#41312yes—
374C2×C4#45614yes—
375C2×C4C2×C4×D412yes—
376C2×C4C2×#12011yescharacteristic
377C2×C4C2×#12112yes—
378C2×C4C2×#12311yescharacteristic
379C8—48no—
380C8—84no—
381C22—24no—
382C22—24no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22#188612yes—
390C22#190912yes—
391C22C2×#38611yescharacteristic
392C22C22×#10011yescharacteristic
393C22C2×#41311yescharacteristic
394C4—22no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—24no—
400C4—48no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—84no—
405C4#188612yes—
406C4#190014yes—
407C4C2×#45612yes—
408C2—21no—
409C2—21no—
410C2—22no—
411C2—22no—
412C2C2×#188611yescharacteristic
413C2C2×#190011yescharacteristic
414C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
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
5. #9452C2semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#25462
#177462
#173652
#94522
#79502