Finite Group Databasebeta

← #192281

Subgroups — #192281

Order 512.

876 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#21379C211yesmaximal, characteristic
3#17821C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic
5#8865C214yesmaximal, complemented
6#8864C214yesmaximal, complemented
7#8088C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2989—24no—
10#2983C2212yes—
11#2975—21no—
12#2973—21no—
13#2973C2212yes—
14#2965—24no—
15#2920—21no—
16#2920C2212yes—
17#2920C2212yes—
18C2×#432C2211yescharacteristic
19C2×#431—21no—
20C22×#97C2211yescharacteristic
21#1985C2212yes—
22#1955C2212yes—
23#1869—21no—
24#1869—21no—
25#1749—21no—
26#1749—21no—
27#1573C2218yes—
28#1572—24no—
29#1569—24no—
30#1465C412yes—
31#1465C412yes—
32#1464C412yes—
33#1464C412yes—
34#1461C2212yes—
35#1460C2212yes—
36#1302C2218yes—
37C22×#41Q811yescharacteristic
38C23×C8—21no—
39C4×#41—22no—
40C4×#41—22no—
41C4×#41—22no—
42C4×#41—22no—
43C2×C4×Q8Q811yescharacteristic
44C2×C4×Q8C2311yescharacteristic
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4C2×C411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4Q811yescharacteristic
50C2×C4×D4C2311yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117Q811yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57#446—28no—
58#436—28no—
59#435—28no—
60#435—28no—
61#432—22no—
62#432—22no—
63#432—22no—
64#432—22no—
65#432—22no—
66#432—22no—
67#432C2×C414yes—
68#432C2314yes—
69#431—22no—
70#431—22no—
71#431—22no—
72#431—22no—
73#431—22no—
74#431—22no—
75#431D414yes—
76#431C2314yes—
77#423—22no—
78#423—22no—
79#423D414yes—
80C2×C4⋊C8—22no—
81C2×C4⋊C8C2312yes—
82#407—22no—
83#407—22no—
84#407C2×C414yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C4×M16—22no—
90C4×M16—22no—
91C4×M16—22no—
92C2×#97—22no—
93C2×#97—22no—
94C2×#97—21no—
95C2×#97—21no—
96C2×#97C2312yes—
97C2×C4×C8—22no—
98#336—22no—
99#336—22no—
100#335—28no—
101#334—28no—
102#324C2×C414yes—
103#324C2×C414yes—
104#323C2×C414yes—
105#323C2×C414yes—
106C2×#41—22no—
107C2×#41—22no—
108C2×#41—42no—
109C2×#41—42no—
110C22×Q8C2×Q811yescharacteristic
111C22×D4C4⋊C411yescharacteristic
112C22×D4C4⋊C411yescharacteristic
113C22×D4C2×Q811yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×Q811yescharacteristic
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C22×C8—22no—
127C22×C8—24no—
128C22×C8—22no—
129C22×C8—44no—
130C22×C8—42no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×D4—22no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4—22no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153C2×C4⋊C4C4⋊C411yescharacteristic
154C2×C4⋊C4C4⋊C411yescharacteristic
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C2×Q811yescharacteristic
158C2×C4⋊C4C2×Q811yescharacteristic
159C2×C22⋊C4C4⋊C411yescharacteristic
160C2×C22⋊C4C4⋊C411yescharacteristic
161C2×C22⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C4⋊C411yescharacteristic
163C2×C22⋊C4C22×C411yescharacteristic
164C2×C22⋊C4C22×C411yescharacteristic
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42C4⋊C411yescharacteristic
174C2×C42C4⋊C411yescharacteristic
175C2×C42C4⋊C411yescharacteristic
176C2×C42C4⋊C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×Q811yescharacteristic
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42C2411yescharacteristic
187C4⋊C8—24no—
188C4⋊C8—24no—
189C4⋊C8—24no—
190C4⋊C8—24no—
191C4⋊C8—24no—
192C4⋊C8—24no—
193C4⋊C8C2×D418yes—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—24no—
199C22⋊C8—44no—
200C22⋊C8—44no—
201C22⋊C8—42no—
202C22⋊C8C22×C418yes—
203#97—24no—
204#97—24no—
205#97—24no—
206#97—24no—
207#97—41no—
208#97—41no—
209#97—41no—
210#97—41no—
211C4×C8—24no—
212C4×C8—24no—
213C4×C8—24no—
214C24C2×C4⋊C411yescharacteristic
215C24C2×C4⋊C411yescharacteristic
216C24C2×C4⋊C411yescharacteristic
217#41—48no—
218#41—48no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4C2×C4⋊C411yescharacteristic
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4C2×C4⋊C411yescharacteristic
260C22×C4C2×C4⋊C411yescharacteristic
261C22×C4C2×C4⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4C2×C4⋊C411yescharacteristic
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4#12012yes—
267C22×C4C22⋊Q812yes—
268C22×C4C22⋊Q812yes—
269C22×C4#12312yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×Q811yescharacteristic
273M16—84no—
274M16—84no—
275M16—84no—
276C2×C8—24no—
277C2×C8—28no—
278C2×C8—28no—
279C2×C8—42no—
280C2×C8—48no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—84no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42C4×D412yes—
311C42C4×D412yes—
312C42#12011yescharacteristic
313C42#12011yescharacteristic
314C42C22⋊Q812yes—
315C42C22⋊Q812yes—
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42C22×D412yes—
319C42C2×#4112yes—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23#38512yes—
334C23#43012yes—
335C23C22×C4⋊C411yescharacteristic
336Q8—44no—
337D4—44no—
338D4—44no—
339D4—44no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4#38511yescharacteristic
380C2×C4#38511yescharacteristic
381C2×C4#38511yescharacteristic
382C2×C4#38511yescharacteristic
383C2×C4#38512yes—
384C2×C4#43012yes—
385C2×C4C2×C4×D412yes—
386C2×C4C2×#12011yescharacteristic
387C2×C4C2×C22⋊Q812yes—
388C2×C4C2×#12311yescharacteristic
389C8—44no—
390C8—48no—
391C8—48no—
392C8—82no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22C2×#38511yescharacteristic
411C22C2×#43011yescharacteristic
412C22C2×#43011yescharacteristic
413C4—21no—
414C4—21no—
415C4—22no—
416C4—22no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4#301812yes—
428C2—21no—
429C2—21no—
430C2—42no—
431C2—42no—
432C2—81no—
433C2#2085111yescharacteristic
434C2#2085511yescharacteristic
435C2#2089211yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC22×#41Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17821C2semidirect
5. #8865C2semidirect
6. #8864C2semidirect
7. #8088C2semidirect

Maximal subgroups

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

subgroupindex
#602772
#213792
#178212
#173662
#88652
#88642
#80882