Finite Group Databasebeta

← #198178

Subgroups — #198178

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#60251C211yesmaximal, characteristic, complemented
3#21845C211yesmaximal, characteristic, complemented
4#17583C211yesmaximal, characteristic
5#17403C211yesmaximal, characteristic
6#9242C218yesmaximal, complemented
7#8006C211yesmaximal, characteristic, complemented
8#7928C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3228—22no—
12#3228—22no—
13#3228—22no—
14#3228—22no—
15C2×#500C411yescharacteristic, complemented
16C2×#498C411yescharacteristic, complemented
17C2×#486C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#482C2214yes—
20C2×#478C414yescomplemented
21#2987—22no—
22#2985—22no—
23#2984—22no—
24#2983—22no—
25#2962—22no—
26#2961—22no—
27#2924C2214yes—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#432C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2283C2211yescharacteristic
33#1995—22no—
34#1986—22no—
35#1826C2214yes—
36#1517—22no—
37#1514—22no—
38#1507C2214yes—
39#1471C2214yes—
40#1469—22no—
41#1466—22no—
42#1293C2214yes—
43#1282C2214yes—
44C22×#41—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533D412yes—
50#533D412yes—
51#533D412yes—
52#533D412yes—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#127C2×C411yescharacteristic
65C2×#125C2×C412yes—
66C2×#124—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×#121C2312yes—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2×C412yes—
75C2×#117C2312yes—
76C22×C42C2311yescharacteristic
77#500—22no—
78#500—22no—
79#498—22no—
80#498—22no—
81#486—22no—
82#486—22no—
83#486—22no—
84#486—22no—
85#482—28no—
86#482—28no—
87#478—28no—
88#478—28no—
89#446—28no—
90#446—28no—
91#446—28no—
92#446—24no—
93C8×Q8—24no—
94#433—28no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432—24no—
99#432—21no—
100#432—21no—
101#432D412yes—
102#432D412yes—
103#432D412yes—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—24no—
110#431—21no—
111#431—21no—
112#431D412yes—
113C2×#107C2×C414yes—
114C2×#106C2×C414yes—
115#424—24no—
116#423—21no—
117#423—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2×C412yes—
120C2×#102C2×C414yes—
121C2×#102C2×C414yes—
122#407C2312yes—
123#407C2312yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126#405—24no—
127C4×M16—21no—
128C4×M16—21no—
129C2×#97—21no—
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C411yescharacteristic
132#383—21no—
133#382—21no—
134#381—21no—
135#380—21no—
136C4×C4⋊C4—21no—
137C4×C22⋊C4C2312yes—
138#376—21no—
139#339—28no—
140#327—28no—
141#323—28no—
142C2×#41—41no—
143C2×#41—41no—
144C2×#41—41no—
145C2×#41—41no—
146C2×#41—84no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22×C412yes—
154C23×C4—22no—
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4C2411yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160C22×C8—22no—
161C22×C8C22×C412yes—
162C22×C8C22×C412yes—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C22⋊C412yes—
172C4⋊Q8C22⋊C412yes—
173C4⋊Q8C22⋊C412yes—
174C4⋊Q8C22⋊C412yes—
175#127—21no—
176#127—21no—
177#127—21no—
178#127—21no—
179#125—22no—
180#125—22no—
181#125C22⋊C414yes—
182#125C22⋊C414yes—
183#124—41no—
184#124—41no—
185#124—41no—
186#124—41no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195#121—22no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—41no—
200#121—41no—
201#121—41no—
202#121—41no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—41no—
219#117—22no—
220#117—22no—
221#117C2×D414yes—
222#117C2×D414yes—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250#107—28no—
251#106—28no—
252C4⋊C8—28no—
253C4⋊C8—28no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262#102—28no—
263#102—28no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8C2×D414yes—
271C22⋊C8C2×D414yes—
272#97—28no—
273#97—22no—
274#97—22no—
275#97—22no—
276#97—41no—
277C4×C8—28no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—22no—
283C4×C8—41no—
284#95—22no—
285#95—41no—
286#95—41no—
287#95—41no—
288C24—22no—
289C24C2×#4111yescharacteristic
290#41—84no—
291#41—84no—
292#41—84no—
293#41—84no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—82no—
301C2×Q8—82no—
302C2×Q8—82no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—44no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4—84no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—84no—
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4#12311yescharacteristic
334C22×C4#12311yescharacteristic
335C22×C4#12311yescharacteristic
336C22×C4#12311yescharacteristic
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343M16—44no—
344M16—44no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—44no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—82no—
368C4⋊C4C2×C22⋊C414yes—
369C4⋊C4C2×C22⋊C414yes—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C42—24no—
378C42—24no—
379C42—44no—
380C42C2×C22⋊C411yescharacteristic
381C42C2×C22⋊C411yescharacteristic
382C42C2×C22⋊C411yescharacteristic
383C42C2×C22⋊C411yescharacteristic
384C42C4×D411yescharacteristic
385C42C4×D411yescharacteristic
386C42C4×D411yescharacteristic
387C42C4×D411yescharacteristic
388C42#12011yescharacteristic
389C42#12011yescharacteristic
390C42#12011yescharacteristic
391C42#12011yescharacteristic
392C42#12111yescharacteristic
393C42#12111yescharacteristic
394C42#12111yescharacteristic
395C42#12111yescharacteristic
396C23—22no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—44no—
401C23—82no—
402C23—82no—
403C23C2×C4×D411yescharacteristic
404C23C2×#12111yescharacteristic
405C23C2×#12311yescharacteristic
406Q8—44no—
407Q8—82no—
408Q8—82no—
409Q8—82no—
410Q8—82no—
411Q8—82no—
412D4—44no—
413D4—82no—
414D4—82no—
415D4—82no—
416D4—82no—
417D4—84no—
418D4—84no—
419D4—82no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4#38611yescharacteristic
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#38611yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4#38611yescharacteristic
455C2×C4#38611yescharacteristic
456C2×C4#38611yescharacteristic
457C2×C4C22×C22⋊C411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×#12011yescharacteristic
460C2×C4C2×#12111yescharacteristic
461C8—44no—
462C8—44no—
463C8—42no—
464C8—42no—
465C8—82no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—164no—
479C22C2×#38611yescharacteristic
480C22#295712yes—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—44no—
492C4—44no—
493C4—82no—
494C4—162no—
495C4#190312yes—
496C4#288412yes—
497C2—22no—
498C2—41no—
499C2—41no—
500C2—162no—
501C2#2091911yescharacteristic
502C2#2098611yescharacteristic
503C2C2×#190311yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60251C2semidirect
3. #21845C2semidirect
6. #9242C2semidirect
7. #8006C2semidirect
8. #7928C2semidirect
15. C2×#500C4semidirect
16. C2×#498C4semidirect
17. C2×#486C4semidirect
18. C2×#486C4semidirect
20. C2×#478C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#602512
#218452
#175832
#174032
#92422
#80062
#79282