Finite Group Databasebeta

← #193813

Subgroups — #193813

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#34979C211yesmaximal, characteristic
3#17464C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic
5#8962C214yesmaximal, complemented
6#8961C214yesmaximal, complemented
7#7870C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2989—24no—
11#2983C2212yes—
12#2975C2212yes—
13#2973—22no—
14#2965—24no—
15#2920—22no—
16#2920C2212yes—
17C2×#431C2212yes—
18#2611—22no—
19#2568C411yescharacteristic
20#2547C411yescharacteristic
21#2546C411yescharacteristic
22#2542C411yescharacteristic
23#2499C412yes—
24#2432C412yes—
25#2415—22no—
26#2395—22no—
27C4×C4⋊Q8C2212yes—
28C4×#124C2212yes—
29C4×C22⋊Q8—22no—
30C4×#117C2211yescharacteristic
31#1485—24no—
32#1481—24no—
33#1479C2218yes—
34#1465—22no—
35#1464—22no—
36#1461C2212yes—
37#1460C2212yes—
38#1276C2218yes—
39C22×#41C2×C411yescharacteristic
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×#124C2×C412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2×C412yes—
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2312yes—
59C2×C4×Q8C2311yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D412yes—
63C2×C4×D4C2311yescharacteristic
64C2×#117—21no—
65C2×#117—21no—
66C2×#117C2×C411yescharacteristic
67C2×#117D412yes—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70#446—28no—
71#436—28no—
72#435—216no—
73#432—24no—
74#432—24no—
75#432—22no—
76#432—22no—
77#431—24no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431D414yes—
82#431C2314yes—
83#431C2314yes—
84#423—24no—
85#423—22no—
86C2×C4⋊C8D414yes—
87#407—24no—
88#407—22no—
89C2×C22⋊C8—22no—
90C4×M16—24no—
91C4×M16—22no—
92C2×C4×C8—22no—
93#400—22no—
94#399—22no—
95#398—22no—
96#397—22no—
97#397—24no—
98#394—24no—
99#391—22no—
100#391C2×C412yes—
101#391C2×C412yes—
102#391C2×C412yes—
103#390C2×C412yes—
104#389—24no—
105#389—22no—
106#388—24no—
107#387—24no—
108#385—24no—
109#384C2×C412yes—
110#382C2×C412yes—
111#380—22no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C22⋊C4—21no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4C2×C412yes—
119#376—21no—
120C43C2312yes—
121Q8⋊C8—216no—
122#324—24no—
123#323—24no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—42no—
127C2×#41—42no—
128C22×Q8—21no—
129C22×Q8C22×C412yes—
130C22×Q8C22×C411yescharacteristic
131C22×D4C22⋊C412yes—
132C22×D4C22×C411yescharacteristic
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—28no—
140C2×M16—42no—
141C22×C8—24no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—44no—
147#125—42no—
148#125—42no—
149#124—44no—
150C22⋊Q8—24no—
151C22⋊Q8—24no—
152C22⋊Q8—48no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173#117—22no—
174#117—22no—
175#117—24no—
176#117—24no—
177#117—44no—
178#117—42no—
179#117—42no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C2×D412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C2×D412yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42C22⋊C412yes—
209C2×C42C22⋊C412yes—
210C2×C42C22×C412yes—
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42C2411yescharacteristic
221C4⋊C8—28no—
222C4⋊C8—24no—
223C4⋊C8—24no—
224C4⋊C8—42no—
225C4⋊C8C2×D418yes—
226C22⋊C8—28no—
227C22⋊C8—24no—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C22⋊C8—42no—
231#97—28no—
232#97—42no—
233C4×C8—24no—
234C4×C8—24no—
235C4×C8—24no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243C24C2×C22⋊C412yes—
244C24C2×C22⋊C411yescharacteristic
245#41—48no—
246#41—48no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—44no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—42no—
261C2×D4—42no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C4×D412yes—
287C22×C4C4×D412yes—
288C22×C4#12012yes—
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4#12312yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295M16—88no—
296M16—84no—
297C2×C8—28no—
298C2×C8—44no—
299C2×C8—84no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—44no—
320C22⋊C4—48no—
321C22⋊C4—48no—
322C22⋊C4—44no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—24no—
331C42—44no—
332C42—44no—
333C42—42no—
334C42—42no—
335C42C4×D412yes—
336C42C4×D412yes—
337C42#12011yescharacteristic
338C42#12011yescharacteristic
339C42#12112yes—
340C42#12112yes—
341C42#12311yescharacteristic
342C42#12311yescharacteristic
343C42C22×D412yes—
344C42C2×#4112yes—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23#38612yes—
355C23C22×C22⋊C411yescharacteristic
356Q8—44no—
357Q8—48no—
358Q8—82no—
359Q8—82no—
360D4—48no—
361D4—44no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4#38611yescharacteristic
399C2×C4#38611yescharacteristic
400C2×C4#38611yescharacteristic
401C2×C4#38612yes—
402C2×C4#38611yescharacteristic
403C2×C4C2×C4×D412yes—
404C2×C4C2×#12011yescharacteristic
405C2×C4C2×#12112yes—
406C2×C4C2×#12311yescharacteristic
407C2×C4#56812yes—
408C2×C4#56812yes—
409C8—48no—
410C8—84no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—44no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—82no—
423C22C2×#38611yescharacteristic
424C22#287911yescharacteristic
425C22#287911yescharacteristic
426C4—21no—
427C4—21no—
428C4—22no—
429C4—21no—
430C4—21no—
431C4—22no—
432C4—44no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4#188412yes—
440C4#293014yes—
441C4#302212yes—
442C2—21no—
443C2—21no—
444C2—42no—
445C2—82no—
446C2#2089611yescharacteristic
447C2C2×#188411yescharacteristic
448C2#2095511yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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. #17464C2semidirect
5. #8962C2semidirect
6. #8961C2semidirect
7. #7870C2semidirect

Maximal subgroups

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

subgroupindex
#602772
#349792
#174642
#173662
#89622
#89612
#78702