Finite Group Databasebeta

← #122151

Subgroups — #122151

Order 512.

642 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37561C211yesmaximal, characteristic
2#10858C211yesmaximal, characteristic
3#8042C211yesmaximal, characteristic
4#8035C211yesmaximal, characteristic
5#7914C211yesmaximal, characteristic
6#7899C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2845—21no—
13#2844—21no—
14#2666—22no—
15#2660—21no—
16#2659C412yes—
17#2647—21no—
18#2646C412yes—
19#2570—21no—
20#2550—21no—
21#2442—21no—
22#2440—21no—
23#2438—22no—
24#2285C2211yescharacteristic
25#2283C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#343—21no—
28C2×#343—21no—
29#1296—21no—
30#1296—21no—
31#1295—21no—
32#1295—21no—
33#1280—22no—
34#1279—22no—
35#1279—21no—
36#1279—21no—
37C22×M16—21no—
38C22×M16—21no—
39C2×C4⋊Q8—21no—
40C2×#125—21no—
41C2×C22⋊Q8—22no—
42C2×#117—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42C2311yescharacteristic
46#432—21no—
47#432—21no—
48#432—21no—
49#432Q812yescomplemented
50#431—21no—
51#431—21no—
52#431—21no—
53#431Q812yescomplemented
54#423—21no—
55#423—21no—
56#423—21no—
57#423Q812yescomplemented
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C4×M16—21no—
65C4×M16—21no—
66C4×M16—21no—
67C4×M16Q812yescomplemented
68C2×#97—21no—
69C2×#97—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72#400—22no—
73#400—22no—
74#400—22no—
75#400—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#398—22no—
82#398—22no—
83#398—22no—
84#395—22no—
85#395—22no—
86#395—22no—
87#395—22no—
88#391—22no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—22no—
94#387—22no—
95#387—22no—
96#385—22no—
97#385—22no—
98#384—21no—
99#384D411yescharacteristic
100#384D411yescharacteristic
101#383—21no—
102#382—21no—
103#382—21no—
104#382—21no—
105#382—21no—
106#382—21no—
107#382—21no—
108#382D411yescharacteristic
109#382D411yescharacteristic
110#381—21no—
111#380—22no—
112#380—21no—
113#380C2×C411yescharacteristic
114#380C2×C412yes—
115#380C2×C411yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D412yes—
119C4×C4⋊C4D411yescharacteristic
120C4×C22⋊C4—21no—
121#376—21no—
122C2×#95C2×C412yes—
123C2×#95D412yes—
124#343—42no—
125#343—42no—
126C22×Q8—22no—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—41no—
137C2×M16—41no—
138C2×M16—41no—
139C2×M16—41no—
140C2×M16—41no—
141C2×M16—41no—
142C2×M16—41no—
143C2×M16—41no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8—41no—
147C22×C8—41no—
148C4⋊Q8—81no—
149C4⋊Q8—81no—
150#125—81no—
151#125—81no—
152C22⋊Q8—84no—
153#117—82no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q811yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42C2×Q811yescharacteristic
194C2×C42C2×Q811yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211#97—22no—
212#97—22no—
213#97—41no—
214#97—41no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—41no—
218C4×C8—41no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95—21no—
238#95—21no—
239C24C2×C22⋊C411yescharacteristic
240C2×Q8—82no—
241C2×Q8—82no—
242C2×Q8—82no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C4×Q811yescharacteristic
268C22×C4C4×Q811yescharacteristic
269C22×C4C4×Q811yescharacteristic
270C22×C4C4×Q811yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4C22⋊Q811yescharacteristic
279C22×C4#12411yescharacteristic
280C22×C4#12411yescharacteristic
281C22×C4C4⋊Q811yescharacteristic
282C22×C4C4⋊Q811yescharacteristic
283M16—24no—
284M16—42no—
285M16—42no—
286M16—42no—
287M16—42no—
288M16—42no—
289M16—81no—
290M16—81no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—81no—
298C2×C8—81no—
299C2×C8—81no—
300C2×C8—81no—
301C2×C8—81no—
302C2×C8—81no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—81no—
318C4⋊C4—81no—
319C4⋊C4—81no—
320C4⋊C4—81no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—82no—
338C42C22⋊Q811yescharacteristic
339C42C22⋊Q811yescharacteristic
340C42C22⋊Q811yescharacteristic
341C42C22⋊Q811yescharacteristic
342C42C22⋊Q811yescharacteristic
343C42C22⋊Q811yescharacteristic
344C42C22⋊Q811yescharacteristic
345C42C22⋊Q811yescharacteristic
346C42#12411yescharacteristic
347C42#12411yescharacteristic
348C42#12411yescharacteristic
349C42#12411yescharacteristic
350C42C4⋊Q811yescharacteristic
351C42C4⋊Q811yescharacteristic
352C42C4⋊Q811yescharacteristic
353C42C4⋊Q811yescharacteristic
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23#37911yescharacteristic
361C23#39111yescharacteristic
362C23#39111yescharacteristic
363Q8—164no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—162no—
415C2×C4—162no—
416C2×C4#39111yescharacteristic
417C2×C4#39111yescharacteristic
418C2×C4#39111yescharacteristic
419C2×C4#39111yescharacteristic
420C2×C4#39311yescharacteristic
421C2×C4#39311yescharacteristic
422C2×C4#39311yescharacteristic
423C2×C4#39311yescharacteristic
424C2×C4#39511yescharacteristic
425C2×C4#39511yescharacteristic
426C2×C4#39511yescharacteristic
427C2×C4#39511yescharacteristic
428C2×C4#45311yescharacteristic
429C2×C4#45411yescharacteristic
430C2×C4#45511yescharacteristic
431C2×C4#45611yescharacteristic
432C8—44no—
433C8—82no—
434C8—82no—
435C8—82no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22#144611yescharacteristic
447C22#179111yescharacteristic
448C22#179111yescharacteristic
449C22#179211yescharacteristic
450C22#179311yescharacteristic
451C22#179511yescharacteristic
452C22#179611yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—82no—
464C4—82no—
465C4—81no—
466C4—81no—
467C4—82no—
468C4—81no—
469C4—81no—
470C4—82no—
471C4#202711yescharacteristic
472C4#202811yescharacteristic
473C4#206011yescharacteristic
474C4#206111yescharacteristic
475C2—41no—
476C2—41no—
477C2#848311yescharacteristic
478C2#848411yescharacteristic
479C2#848511yescharacteristic
480C2#848611yescharacteristic
481C2#1138811yescharacteristic
482C2#1139111yescharacteristic
483C2#1139211yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ 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
49. #432Q8semidirect
53. #431Q8semidirect
57. #423Q8semidirect
67. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#375612
#108582
#80422
#80352
#79142
#78992
#78962