Finite Group Databasebeta

← #122167

Subgroups — #122167

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2606C211yesmaximal, characteristic
2#10851C211yesmaximal, characteristic
3C2×#1300C211yesmaximal, characteristic
4#8022C211yesmaximal, characteristic
5#8018C211yesmaximal, characteristic
6C2×#1288C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×C4×M16C2211yescharacteristic
11#2606—22no—
12#2606—22no—
13#2606—22no—
14#2606C414yescomplemented
15C2×#398—21no—
16C2×#398—21no—
17C2×#389—21no—
18C2×#389—21no—
19C2×#387—21no—
20C2×#387—21no—
21#2285C2211yescharacteristic
22C2×#383C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×#376C2211yescharacteristic
25C2×#339—21no—
26C2×#339—21no—
27#1300—22no—
28#1294—21no—
29#1294—21no—
30#1294—21no—
31#1294—21no—
32#1288—22no—
33#1285—22no—
34C22×M16—21no—
35C2×#117—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C42—21no—
44C22×C42D411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42C2311yescharacteristic
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432Q812yes—
54#432Q812yes—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C4×M16—21no—
60C4×M16—21no—
61C4×M16—21no—
62C4×M16Q812yes—
63C2×#97—21no—
64C2×#97—21no—
65C2×#97—21no—
66C2×C4×C8—21no—
67#398—24no—
68#398—24no—
69#398—24no—
70#398—24no—
71#389—24no—
72#389—24no—
73#389—24no—
74#389—24no—
75#387—24no—
76#387—24no—
77#387—24no—
78#387—24no—
79#384Q812yes—
80#383—21no—
81#383—21no—
82#383C2×C412yes—
83#383D412yes—
84#382—21no—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4D412yes—
93#376—21no—
94#376—21no—
95#376C2×C412yes—
96#376D412yes—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#339—42no—
108#339—42no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127#117—41no—
128#117—41no—
129#117—41no—
130#117—41no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—81no—
148C2×C4⋊C4—81no—
149C2×C4⋊C4—81no—
150C2×C4⋊C4—81no—
151C2×C22⋊C4—41no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—44no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42C22⋊C412yes—
163C2×C42C22⋊C412yes—
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×Q811yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42C2×Q811yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—41no—
188#97—41no—
189#97—41no—
190C4×C8—22no—
191C4×C8—41no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—41no—
207#95—41no—
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95C22⋊C412yes—
212C24#12011yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4#12411yescharacteristic
263C22×C4#12411yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4C4⋊Q811yescharacteristic
266C22×C4C4⋊Q811yescharacteristic
267C22×C4C4⋊Q811yescharacteristic
268M16—24no—
269M16—42no—
270M16—42no—
271M16—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—82no—
278C2×C8—82no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—48no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C42—42no—
294C42—42no—
295C42—44no—
296C42—44no—
297C42—81no—
298C42—81no—
299C42—81no—
300C42—81no—
301C42C4×Q811yescharacteristic
302C42C4×Q811yescharacteristic
303C42C4×Q811yescharacteristic
304C42C4×Q811yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42C22⋊Q811yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42#12411yescharacteristic
312C42#12411yescharacteristic
313C42#12411yescharacteristic
314C42C4⋊Q811yescharacteristic
315C42C4⋊Q811yescharacteristic
316C42C4⋊Q811yescharacteristic
317C23—22no—
318C23—22no—
319C23#37911yescharacteristic
320C23#39311yescharacteristic
321C23#39511yescharacteristic
322C23#45311yescharacteristic
323C23#45411yescharacteristic
324C23#45511yescharacteristic
325C23#45611yescharacteristic
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4#39111yescharacteristic
384C2×C4#39111yescharacteristic
385C2×C4#39111yescharacteristic
386C2×C4#39111yescharacteristic
387C2×C4#39111yescharacteristic
388C2×C4#39111yescharacteristic
389C2×C4#39311yescharacteristic
390C2×C4#39311yescharacteristic
391C2×C4#39311yescharacteristic
392C2×C4#39511yescharacteristic
393C2×C4#39511yescharacteristic
394C2×C4#39511yescharacteristic
395C8—44no—
396C8—82no—
397C8—82no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22#144611yescharacteristic
407C22#179211yescharacteristic
408C22#179311yescharacteristic
409C22#202711yescharacteristic
410C22#202811yescharacteristic
411C22#203112yes—
412C22#203112yes—
413C22#206011yescharacteristic
414C22#206111yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—44no—
428C4—44no—
429C4—82no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—162no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—21no—
438C2#1038012yes—
439C2#1038112yes—
440C2#1138711yescharacteristic
441C2#1138711yescharacteristic
442C2#1139211yescharacteristic
subgroupG/H
center, socleC23#393
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2606C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26062
#108512
C2×#13002
#80222
#80182
C2×#12882
C2×#12852