Finite Group Databasebeta

← #192188

Subgroups — #192188

Order 512.

925 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#21374C211yesmaximal, characteristic
3#21370C212yesmaximal, complemented
4#17349C211yesmaximal, characteristic
5#8860C214yesmaximal, complemented
6#8859C214yesmaximal, complemented
7#8054C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2989—24no—
10#2983—21no—
11#2975C2212yes—
12#2973—21no—
13#2973C2212yes—
14#2965—24no—
15#2920—21no—
16#2920C2212yes—
17C2×#431—21no—
18C2×#431C2211yescharacteristic
19C2×#431C2212yes—
20C2×C4×M16C2211yescharacteristic
21#1985—21no—
22#1955—21no—
23#1950—24no—
24#1870C2212yes—
25#1870C2212yes—
26#1855C2218yes—
27C4×#108—24no—
28C4×C22⋊C8—21no—
29C4×C22⋊C8—21no—
30#1461C412yes—
31#1460C412yes—
32#1459C2212yes—
33#1459C2212yes—
34#1458C412yes—
35#1458C412yes—
36#1298C2218yes—
37C22×#41Q811yescharacteristic
38C22×M16C2312yes—
39C4×#41—22no—
40C4×#41—22no—
41C4×#41—22no—
42C4×#41—22no—
43C2×C4×Q8C2×C411yescharacteristic
44C2×C4×Q8D411yescharacteristic
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4Q811yescharacteristic
48C2×C4×D4Q811yescharacteristic
49C2×C4×D4C2311yescharacteristic
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#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71#431—22no—
72#431—22no—
73#431—22no—
74#431—22no—
75#431C2×C414yes—
76#431C2×C414yes—
77C2×#108—28no—
78#423—22no—
79#423D414yes—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83#407—22no—
84#407C2314yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2314yes—
90C4×M16—22no—
91C4×M16—22no—
92C4×M16—22no—
93C4×M16D414yes—
94C2×#97—21no—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99#341—28no—
100#336—22no—
101#336—22no—
102#324C2×C414yes—
103#324C2×C414yes—
104#323C2×C414yes—
105#323C2×C414yes—
106C2×#41—24no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—42no—
110C22×Q8C4⋊C411yescharacteristic
111C22×D4C4⋊C411yescharacteristic
112C22×D4C2×Q811yescharacteristic
113C22×D4C2×Q811yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×Q811yescharacteristic
121C23×C4C2×Q811yescharacteristic
122C23×C4C2411yescharacteristic
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—44no—
126C2×M16C2×D418yes—
127C22×C8—22no—
128C22×C8—24no—
129C22×C8—24no—
130C22×C8—22no—
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⋊C4C4⋊C411yescharacteristic
156C2×C4⋊C4C4⋊C411yescharacteristic
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C22⋊C4C4⋊C411yescharacteristic
160C2×C22⋊C4C4⋊C411yescharacteristic
161C2×C22⋊C4C22×C411yescharacteristic
162C2×C22⋊C4C22×C411yescharacteristic
163C2×C22⋊C4C2×Q811yescharacteristic
164C2×C22⋊C4C2×Q811yescharacteristic
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×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4112yes—
186C2×C42#4112yes—
187#108—416no—
188C4⋊C8—24no—
189C4⋊C8—24no—
190C4⋊C8—24no—
191C4⋊C8—24no—
192C4⋊C8—24no—
193C4⋊C8—24no—
194C4⋊C8—42no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—24no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—44no—
202C22⋊C8—44no—
203C22⋊C8C22×C418yes—
204#97—24no—
205#97—24no—
206#97—42no—
207C4×C8—24no—
208C4×C8—24no—
209C4×C8—24no—
210C4×C8—24no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214C4×C8—41no—
215C24C2×C4⋊C411yescharacteristic
216C24C2×C4⋊C411yescharacteristic
217C24C22×Q811yescharacteristic
218#41—48no—
219#41—48no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4#10814yescomplemented
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×C4C2×C4⋊C411yescharacteristic
265C22×C4C2×C4⋊C411yescharacteristic
266C22×C4C22⋊Q812yes—
267C22×C4C22⋊Q812yes—
268C22×C4C22⋊Q812yes—
269C22×C4C22⋊Q812yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D412yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4112yes—
274M16—28no—
275M16—44no—
276M16—48no—
277M16—84no—
278C2×C8—24no—
279C2×C8—28no—
280C2×C8—28no—
281C2×C8—48no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—84no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42C4×D412yes—
313C42C4×D412yes—
314C42C4×D412yes—
315C42C4×D412yes—
316C42#12011yescharacteristic
317C42#12012yes—
318C42#12011yescharacteristic
319C42#12311yescharacteristic
320C42#12311yescharacteristic
321C42#12312yes—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23C2×#10812yes—
336C23C22×C4⋊C411yescharacteristic
337C23C2×C22⋊Q812yes—
338Q8—44no—
339D4—44no—
340D4—44no—
341D4—44no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
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#38512yes—
381C2×C4#38512yes—
382C2×C4#38511yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4#38511yescharacteristic
385C2×C4C2×#10812yes—
386C2×C4C2×#10812yes—
387C2×C4#43012yes—
388C2×C4C2×C4×D412yes—
389C2×C4C2×#12011yescharacteristic
390C2×C4C2×#12311yescharacteristic
391C8—48no—
392C8—44no—
393C8—48no—
394C8—82no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22C2×#38511yescharacteristic
413C22C22×#10811yescharacteristic
414C22C2×#43011yescharacteristic
415C4—22no—
416C4—22no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—21no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4#185512yes—
427C4#185512yes—
428C4#185812yes—
429C4#302512yes—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—42no—
434C2—81no—
435C2C2×#185511yescharacteristic
436C2C2×#185811yescharacteristic
437C2#2088811yescharacteristic
subgroupG/H
centerC42#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC22×#41Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C42 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21370C2semidirect
5. #8860C2semidirect
6. #8859C2semidirect
7. #8054C2semidirect
255. C22×C4#108semidirect

Maximal subgroups

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

subgroupindex
#602772
#213742
#213702
#173492
#88602
#88592
#80542