Finite Group Databasebeta

← #192307

Subgroups — #192307

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34070C211yesmaximal, characteristic
2#21386C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic
5#13848C211yesmaximal, characteristic, complemented
6#8873C214yesmaximal, complemented
7#8870C214yesmaximal, complemented
8#8014C211yesmaximal, characteristic, complemented
9#7906C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×C4×#41—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2328—21no—
15#2327—21no—
16#2326—22no—
17#2325—22no—
18#2325C2214yes—
19#2297—21no—
20#2290—22no—
21#2290—22no—
22#2285C2211yescharacteristic
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1988—21no—
28#1982—21no—
29#1948—22no—
30#1874C2214yes—
31#1853C2214yes—
32C4×#106—22no—
33C4×#102—22no—
34#1502C418yescomplemented
35C2×#327C2212yes—
36C2×#327C2212yes—
37#1444C2212yes—
38#1442C2212yes—
39#1441—22no—
40C4×#95—21no—
41#1294C2212yes—
42#1294C2212yes—
43#1280C2214yes—
44C22×#41—21no—
45#532—22no—
46#532—22no—
47#532C2×C414yes—
48#532Q814yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×#127D411yescharacteristic
51C2×#124—21no—
52C2×#121—21no—
53C2×#121C2312yes—
54C4×#41—44no—
55C4×#41—44no—
56C2×C4×Q8—22no—
57C2×C4×D4—22no—
58C2×C4×D4—24no—
59C2×C4×D4—21no—
60C2×C4×D4D412yes—
61C2×#117—22no—
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C42—22no—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72#431—22no—
73#431—22no—
74#431—22no—
75#431C2×C414yescomplemented
76C2×#106—24no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×#102—24no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#391—22no—
85#390—22no—
86#388—24no—
87#388—24no—
88#387—24no—
89#387—24no—
90#386—24no—
91#386—24no—
92#386—24no—
93#385—24no—
94#385—24no—
95#384D412yes—
96#382—21no—
97#380—21no—
98#380C2312yes—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—44no—
106C4×C4⋊C4C2×C414yescomplemented
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110#376—21no—
111#376—21no—
112#376—21no—
113#376—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95D412yes—
119C2×#95C2312yes—
120C2×#95C2311yescharacteristic
121C2×#95C2311yescharacteristic
122#340—24no—
123#340—24no—
124#327C2×C418yescomplemented
125#327C2×C418yescomplemented
126C2×#41—24no—
127C2×#41—24no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C22×Q8—21no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C2×D412yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—22no—
150C4⋊Q8C4⋊C414yescomplemented
151#127—22no—
152#127C4⋊C414yescomplemented
153#124—24no—
154#124—24no—
155#121—24no—
156#121—24no—
157#121—22no—
158#121—22no—
159#121C22×C414yes—
160#121C2×Q814yes—
161C4×Q8—48no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—48no—
166C4×D4—48no—
167C4×D4—48no—
168C4×D4C4⋊C418yescomplemented
169#117—48no—
170#117—42no—
171#117—42no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D414yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—48no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42SD1614yes—
210C2×C42C22×C412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×Q812yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218#106—48no—
219C4⋊C8—24no—
220C4⋊C8—42no—
221#102—48no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8C22×C414yes—
226C4×C8—24no—
227C4×C8—42no—
228#95—24no—
229#95—24no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—41no—
242#95—41no—
243#95C22×C412yes—
244#95C22×C412yes—
245C24—22no—
246C24C22×D411yescharacteristic
247#41—48no—
248#41—48no—
249#41—48no—
250#41—48no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—44no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—82no—
267C2×D4C2×C4⋊C414yes—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—48no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4C4×D412yes—
305C22×C4C4×D412yes—
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4C22⋊Q812yes—
311C22×C4C22⋊Q812yes—
312C22×C4C2×SD1614yes—
313C22×C4C2×SD1612yes—
314C22×C4#13512yes—
315C22×C4AGL(1,ℤ/8)12yes—
316C22×C4AGL(1,ℤ/8)12yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×Q811yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C2×C8—24no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—42no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4—416no—
337C4⋊C4—44no—
338C4⋊C4—82no—
339C4⋊C4C2×C4⋊C414yes—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—28no—
343C22⋊C4—48no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C42—24no—
349C42—24no—
350C42—48no—
351C42—48no—
352C42—42no—
353C42—42no—
354C42—44no—
355C42—82no—
356C42—82no—
357C42C2×C4⋊C412yes—
358C42C2×C4⋊C412yes—
359C42C4×D412yes—
360C42C4×D412yes—
361C42C22⋊Q812yes—
362C42C22⋊Q812yes—
363C42#12312yes—
364C42#12312yes—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—42no—
369C23—42no—
370C23—44no—
371C23—44no—
372C23C2×C4×D411yescharacteristic
373C23C2×#12011yescharacteristic
374C23C2×C22⋊Q811yescharacteristic
375C23C22×SD1611yescharacteristic
376C23C2×#13511yescharacteristic
377C23C2×AGL(1,ℤ/8)11yescharacteristic
378C23C2×AGL(1,ℤ/8)11yescharacteristic
379Q8—48no—
380D4—44no—
381D4—44no—
382D4—44no—
383D4—44no—
384D4—44no—
385D4—48no—
386D4—82no—
387D4—84no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—416no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—82no—
425C2×C4—816no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#38512yes—
434C2×C4#38512yes—
435C2×C4#38512yes—
436C2×C4#38512yes—
437C2×C4C4×SD1614yes—
438C2×C4#44214yes—
439C2×C4#44912yes—
440C2×C4#44912yes—
441C2×C4#45012yes—
442C2×C4#45012yes—
443C2×C4#47614yes—
444C2×C4#47814yes—
445C2×C4C22×C4⋊C411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×C22⋊Q811yescharacteristic
448C2×C4C2×#12311yescharacteristic
449C8—44no—
450C8—82no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—44no—
460C22—44no—
461C22—44no—
462C22—84no—
463C22—84no—
464C22C2×#38511yescharacteristic
465C22C2×C4×SD1611yescharacteristic
466C22C2×#44211yescharacteristic
467C22C2×#45011yescharacteristic
468C22C2×#44911yescharacteristic
469C22#301812yes—
470C22#302512yes—
471C22C2×#47611yescharacteristic
472C22C2×#47811yescharacteristic
473C4—22no—
474C4—22no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—44no—
480C4—44no—
481C4—44no—
482C4—82no—
483C4—84no—
484C4—82no—
485C4—84no—
486C4#186414yes—
487C4#186414yes—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—84no—
493C2#1812312yes—
494C2#1812612yes—
495C2#2086311yescharacteristic
496C2C2×#186411yescharacteristic
497C2#2088811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21359C2semidirect
5. #13848C2semidirect
6. #8873C2semidirect
7. #8870C2semidirect
8. #8014C2semidirect
9. #7906C2semidirect
34. #1502C4semidirect
75. #431C2×C4semidirect
106. C4×C4⋊C4C2×C4semidirect
124. #327C2×C4semidirect
125. #327C2×C4semidirect
150. C4⋊Q8C4⋊C4semidirect
152. #127C4⋊C4semidirect
168. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#340702
#213862
#213592
C2×#15022
#138482
#88732
#88702
#80142
#79062