Finite Group Databasebeta

← #239210

Subgroups — #239210

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4#21311C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6C22×#332C211yesmaximal, characteristic
7#10541C214yesmaximal, complemented
8#10538C214yesmaximal, complemented
9#7877C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#462—21no—
17C2×#462—21no—
18C2×#451—22no—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1989—22no—
25#1988C2212yes—
26#1988C2214yes—
27#1987—22no—
28#1987C2212yes—
29#1939C2214yes—
30#1850—22no—
31#1842—21no—
32#1831—21no—
33#1736—21no—
34#1736—21no—
35#1735—21no—
36#1735—21no—
37#1550—24no—
38C2×#333C2212yes—
39C2×#332—24no—
40C2×#332C2214yes—
41C2×#332C2212yes—
42#1277C2214yes—
43C23×Q8—21no—
44C22×Q16—22no—
45C22×Q16—22no—
46C22×M16—21no—
47C23×C8—21no—
48C23×C8—21no—
49#533—22no—
50#533—22no—
51#533—22no—
52#533D414yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8D414yes—
60C2×C4⋊Q8C2312yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#494—24no—
75#494—24no—
76#487—24no—
77#487—24no—
78#462—28no—
79#462—28no—
80#451—216no—
81#431—22no—
82#431—22no—
83#423—22no—
84#423—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×#103—22no—
99C2×#103—22no—
100C2×C22⋊C8D412yes—
101C2×C4×C8C2312yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4C2312yes—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106#376—21no—
107C43—21no—
108#340—24no—
109#340—24no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#336—24no—
115#333—24no—
116#332—28no—
117#332—24no—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8—82no—
125C22×Q8—82no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C2×Q16—28no—
131C2×Q16—28no—
132C2×Q16—84no—
133C2×Q16—84no—
134C2×M16—42no—
135C2×M16—42no—
136C2×M16—42no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—82no—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—21no—
151C4⋊Q8—22no—
152C4⋊Q8—21no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8C2×D414yes—
163C4⋊Q8C2×D412yes—
164C4⋊Q8C2×D412yes—
165#125—24no—
166#125—42no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—42no—
174C4×Q8—42no—
175#117—22no—
176#117—24no—
177#117—22no—
178#117—22no—
179#117—42no—
180#117Q1614yescomplemented
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—41no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—41no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224C4⋊C8—84no—
225#103—28no—
226#103—28no—
227C22⋊C8—24no—
228C4×C8—22no—
229C4×C8—22no—
230#95—22no—
231#95—22no—
232C24#12111yescharacteristic
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—44no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—44no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×Q8—88no—
248C2×Q8—82no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—84no—
259C22×C4#12012yes—
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12112yes—
263C22×C4#12112yes—
264C22×C4#12111yescharacteristic
265C22×C4#12312yes—
266C22×C4#12312yes—
267C22×C4#12412yes—
268C22×C4C2×Q1612yes—
269C22×C4#13512yes—
270C22×C4#13712yes—
271C22×C4#13712yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4111yescharacteristic
275Q16—88no—
276Q16—88no—
277M16—84no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—44no—
290C2×C8—84no—
291C2×C8—84no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4C2×Q1614yes—
312C22⋊C4—24no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C42—24no—
316C42—42no—
317C42—42no—
318C42—44no—
319C42—42no—
320C42—82no—
321C42#12012yes—
322C42#12112yes—
323C42#12112yes—
324C42#12312yes—
325C42#12312yes—
326C42#12412yes—
327C42C22×D411yescharacteristic
328C42C22×D411yescharacteristic
329C42C2×#4111yescharacteristic
330C42C2×#4111yescharacteristic
331C23—22no—
332C23—22no—
333C23#39412yes—
334C23#46712yes—
335C23#47012yes—
336C23C2×#12111yescharacteristic
337Q8—42no—
338Q8—44no—
339Q8—42no—
340Q8—44no—
341Q8—82no—
342Q8—82no—
343Q8—88no—
344Q8—82no—
345Q8—84no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—88no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4#39412yes—
368C2×C4#39412yes—
369C2×C4#39412yes—
370C2×C4#46214yes—
371C2×C4#46214yes—
372C2×C4#46414yes—
373C2×C4#46414yes—
374C2×C4#46712yes—
375C2×C4#47012yes—
376C2×C4C2×#12011yescharacteristic
377C2×C4C2×#12111yescharacteristic
378C2×C4C2×#12111yescharacteristic
379C2×C4C2×#12311yescharacteristic
380C2×C4C2×#12311yescharacteristic
381C2×C4C2×#12411yescharacteristic
382C2×C4C22×Q1611yescharacteristic
383C2×C4C2×#13511yescharacteristic
384C2×C4C2×#13711yescharacteristic
385C2×C4C2×#13711yescharacteristic
386C2×C4#57612yes—
387C2×C4#57712yes—
388C8—42no—
389C8—42no—
390C8—82no—
391C8—82no—
392C8—82no—
393C22—24no—
394C22—22no—
395C22—22no—
396C22—24no—
397C22#205614yes—
398C22C2×#39411yescharacteristic
399C22#304512yes—
400C22C2×#46711yescharacteristic
401C22C2×#47011yescharacteristic
402C22#306912yes—
403C4—22no—
404C4—21no—
405C4—21no—
406C4—22no—
407C4—44no—
408C4—84no—
409C4—82no—
410C4—82no—
411C4—82no—
412C4#205614yes—
413C4#300811yescharacteristic
414C4#300911yescharacteristic
415C4C2×#46211yescharacteristic
416C4C2×#46211yescharacteristic
417C4C2×#46411yescharacteristic
418C4C2×#46411yescharacteristic
419C4#313612yes—
420C2—24no—
421C2#1941112yes—
422C2#1944712yes—
423C2#2350911yescharacteristic
424C2#2352211yescharacteristic
425C2C2×#205611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
7. #10541C2semidirect
8. #10538C2semidirect
180. #117Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#219722
#213132
#213112
C2×#15502
C22×#3322
#105412
#105382
#78772