Finite Group Databasebeta

← #238145

Subgroups — #238145

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5#10356C214yesmaximal, complemented
6#10353C214yesmaximal, complemented
7#8045C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103—22no—
15#2571C2212yes—
16#2570—22no—
17#2569—22no—
18#2552C2212yes—
19#2442—22no—
20#2402—22no—
21C4×#125—22no—
22#2296—21no—
23#2285—21no—
24#2285C2211yescharacteristic
25C4×#117C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990C2214yes—
28#1988C2214yes—
29#1987—22no—
30#1987—22no—
31#1981—22no—
32#1981C2212yes—
33#1981C2212yes—
34#1824—22no—
35#1736—21no—
36#1736—21no—
37#1296C2214yes—
38#1279C2212yes—
39#1279C2214yes—
40#1279C2212yes—
41C23×Q8—21no—
42C22×M16—21no—
43C23×C8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8Q814yes—
51C2×C4⋊Q8C2312yes—
52C2×#125—22no—
53C2×#117—21no—
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D412yes—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#431—22no—
62#431—22no—
63#423—22no—
64#423—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—48no—
75C2×C22⋊C8—21no—
76C2×C4×C8C2312yes—
77#400—24no—
78#400—24no—
79#398—24no—
80#398—24no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#387—24no—
86#387—24no—
87#385—24no—
88#384—22no—
89#384C2312yes—
90#382—22no—
91#382—22no—
92#382—21no—
93#380—24no—
94#380—21no—
95#380D412yes—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4C2311yescharacteristic
105C4×C4⋊C4C2312yes—
106C4×C4⋊C4C2311yescharacteristic
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109#376—21no—
110C2×#95—21no—
111C43—21no—
112#340—28no—
113#340—24no—
114#340—24no—
115C22×Q8—24no—
116C22×Q8—22no—
117C22×Q8—44no—
118C22×Q8—44no—
119C22×Q8—41no—
120C22×Q8—42no—
121C23×C4—22no—
122C23×C4—41no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—42no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—44no—
133C22×C8—42no—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8C2×Q814yes—
144#125—44no—
145#125—44no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—42no—
151#117—42no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D414yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q812yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194C2×C42C2411yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202#103—48no—
203#103—48no—
204C22⋊C8—42no—
205C4×C8—22no—
206C4×C8—22no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—21no—
217#95—22no—
218C24#12011yescharacteristic
219C2×Q8—24no—
220C2×Q8—416no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8—84no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—82no—
241C22×C4—84no—
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4C22⋊Q812yes—
246C22×C4C22⋊Q812yes—
247C22×C4C22⋊Q812yes—
248C22×C4#12412yes—
249C22×C4#12412yes—
250C22×C4#12412yes—
251C22×C4#13512yes—
252C22×C4#13512yes—
253C22×C4#13712yes—
254C22×C4#13712yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258M16—84no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—84no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C4⋊C4AGL(1,ℤ/8)14yes—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—44no—
294C42—41no—
295C42—41no—
296C42—42no—
297C42—41no—
298C42—41no—
299C42—42no—
300C42—44no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—82no—
305C42C22⋊Q812yes—
306C42C22⋊Q812yes—
307C42C22⋊Q812yes—
308C42#12412yes—
309C42#12412yes—
310C42#12412yes—
311C42C22×Q811yescharacteristic
312C42C2×#4111yescharacteristic
313C42C2×#4111yescharacteristic
314C42C2×#4111yescharacteristic
315C23—22no—
316C23—22no—
317C23#39312yes—
318C23#45212yes—
319C23#45212yes—
320C23C2×#12011yescharacteristic
321Q8—48no—
322Q8—88no—
323Q8—82no—
324Q8—82no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—88no—
350C2×C4—88no—
351C2×C4—84no—
352C2×C4#39312yes—
353C2×C4#39312yes—
354C2×C4#39312yes—
355C2×C4#45212yes—
356C2×C4#45212yes—
357C2×C4#47914yes—
358C2×C4#47914yes—
359C2×C4#48814yes—
360C2×C4#49114yes—
361C2×C4C2×C22⋊Q811yescharacteristic
362C2×C4C2×C22⋊Q811yescharacteristic
363C2×C4C2×C22⋊Q811yescharacteristic
364C2×C4C2×#12411yescharacteristic
365C2×C4C2×#12411yescharacteristic
366C2×C4C2×#12411yescharacteristic
367C2×C4C2×#13511yescharacteristic
368C2×C4C2×#13511yescharacteristic
369C2×C4C2×AGL(1,ℤ/8)12yes—
370C2×C4C2×#13711yescharacteristic
371C2×C4C2×#13711yescharacteristic
372C2×C4#57512yes—
373C8—82no—
374C8—82no—
375C8—82no—
376C22—24no—
377C22—24no—
378C22—22no—
379C22—22no—
380C22#202614yes—
381C22C2×#39311yescharacteristic
382C22C2×#45211yescharacteristic
383C22C2×#45211yescharacteristic
384C22#300612yes—
385C22#302212yes—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—44no—
391C4—82no—
392C4—84no—
393C4—82no—
394C4—82no—
395C4—84no—
396C4#202614yes—
397C4C2×#47911yescharacteristic
398C4C2×#47911yescharacteristic
399C4#307712yes—
400C4C2×#48811yescharacteristic
401C4C2×#49111yescharacteristic
402C4#313811yescharacteristic
403C4#314011yescharacteristic
404C2—24no—
405C2#1880312yes—
406C2#1880512yes—
407C2C2×#202611yescharacteristic
408C2#2345411yescharacteristic
409C2#2346411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10356C2semidirect
6. #10353C2semidirect

Maximal subgroups

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

subgroupindex
#349482
#218342
#213132
C2×#19812
#103562
#103532
#80452
#79012
#78912