Finite Group Databasebeta

← #194288

Subgroups — #194288

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#8987C214yesmaximal, complemented
7#8986C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209C418yescomplemented
14C2×#494—21no—
15C2×#487—21no—
16C2×#462C2214yes—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2370—22no—
22#2369—21no—
23#2365—21no—
24#2355—22no—
25#2355C2212yes—
26Q8×C4⋊C4C2212yes—
27#2329—22no—
28#2317—21no—
29C4×C22⋊Q8—22no—
30#2285C2211yescharacteristic
31#2274—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990C2212yes—
34#1990C2212yes—
35#1988—22no—
36#1988C2214yes—
37#1987—22no—
38#1824—22no—
39C2×#340—21no—
40C2×#340—21no—
41#1481—24no—
42C2×Q8⋊C8C2214yes—
43#1296C2212yes—
44#1296C2212yes—
45#1279C2214yes—
46C23×Q8—21no—
47C22×Q16—24no—
48C23×C8—21no—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52#533D414yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8C2312yes—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8D412yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#494—28no—
78#487—28no—
79#462—28no—
80#462C2×C4116yescomplemented
81#451—216no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431C2×C414yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#103—24no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#391—24no—
98#391—22no—
99#389—22no—
100#389—22no—
101#389—24no—
102#389—22no—
103#387—24no—
104#387—24no—
105#387—24no—
106#385—24no—
107#384—22no—
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#382—22no—
111#382—21no—
112#380—21no—
113#380D412yes—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4C2312yes—
126C4×C22⋊C4—24no—
127C2×#95—21no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132Q8⋊C8—28no—
133C22×Q8—21no—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8—82no—
139C22×Q8—82no—
140C22×Q8C2×D412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×Q16—216no—
148C2×Q16—84no—
149C2×Q16—84no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—42no—
156C4⋊Q8—24no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8C22⋊C414yes—
168C4⋊Q8C22⋊C414yes—
169C4⋊Q8C22×C414yes—
170#125—24no—
171#125—24no—
172C22⋊Q8—24no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C22⋊Q8C22⋊C418yes—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—84no—
181C4×Q8C22×C414yes—
182C4×Q8C2×D414yes—
183#117—24no—
184#117—24no—
185#117—42no—
186#117—42no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4Q1614yescomplemented
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—82no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8C22×C414yes—
240#103—216no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243C4×C8—24no—
244C4×C8—22no—
245C4×C8—22no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254C24#12011yescharacteristic
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—81no—
265C2×Q8—81no—
266C2×Q8—88no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—88no—
270C2×Q8C2×C22⋊C414yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12312yes—
300C22×C4C2×Q1612yes—
301C22×C4#13512yes—
302C22×C4#13712yes—
303C22×C4#13712yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307Q16—816no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—84no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4C2×C22⋊C414yes—
347C4⋊C4C2×Q1614yes—
348C22⋊C4—28no—
349C22⋊C4—44no—
350C42—24no—
351C42—24no—
352C42—44no—
353C42—44no—
354C42—82no—
355C42—84no—
356C42—84no—
357C42C2×C22⋊C412yes—
358C42C4×D412yes—
359C42C4×D412yes—
360C42#12112yes—
361C42#12112yes—
362C42#12312yes—
363C42C23×C411yescharacteristic
364C42C22×D411yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23#38612yes—
372C23#45112yes—
373C23#45212yes—
374C23C2×#12011yescharacteristic
375Q8—28no—
376Q8—44no—
377Q8—48no—
378Q8—84no—
379Q8—88no—
380Q8—84no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—88no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#38612yes—
416C2×C4#38612yes—
417C2×C4#38612yes—
418C2×C4C4×Q1614yes—
419C2×C4#44114yes—
420C2×C4#45112yes—
421C2×C4#45212yes—
422C2×C4#46214yes—
423C2×C4#46414yes—
424C2×C4C22×C22⋊C411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×C4×D411yescharacteristic
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×#12111yescharacteristic
429C2×C4C2×#12311yescharacteristic
430C2×C4C22×Q1611yescharacteristic
431C2×C4C2×#13511yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C2×C4#57612yes—
435C2×C4#57712yes—
436C8—44no—
437C8—82no—
438C8—82no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22#187814yes—
446C22C2×#38611yescharacteristic
447C22C2×#45111yescharacteristic
448C22C2×#45211yescharacteristic
449C22#300811yescharacteristic
450C22#301011yescharacteristic
451C22#302211yescharacteristic
452C22#302211yescharacteristic
453C4—22no—
454C4—21no—
455C4—21no—
456C4—22no—
457C4—44no—
458C4—44no—
459C4—44no—
460C4—84no—
461C4—82no—
462C4—82no—
463C4—82no—
464C4#187814yes—
465C4#289712yes—
466C4C2×C4×Q1611yescharacteristic
467C4C2×#44111yescharacteristic
468C4C2×#46211yescharacteristic
469C4C2×#46411yescharacteristic
470C4#309612yes—
471C2—24no—
472C2#1890511yescharacteristic
473C2#1890711yescharacteristic
474C2#1892511yescharacteristic
475C2#1892711yescharacteristic
476C2#2091411yescharacteristic
477C2C2×#187811yescharacteristic
478C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ 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
6. #8987C2semidirect
7. #8986C2semidirect
13. #3209C4semidirect
80. #462C2×C4semidirect
211. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#342132
#218342
C2×#19882
C2×#14812
#89872
#89862
#80392
#78922