Finite Group Databasebeta

← #117565

Subgroups — #117565

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21787C211yesmaximal, characteristic
2C2×#1962C211yesmaximal, characteristic
3C2×#1574C211yesmaximal, characteristic, complemented
4C2×#1543C211yesmaximal, characteristic
5#7880C211yesmaximal, characteristic, complemented
6C2×#1276C211yesmaximal, characteristic, complemented
7C2×#1276C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×#104—22no—
15C2×C43C2211yescharacteristic
16#1962—22no—
17#1962—22no—
18#1954—21no—
19#1954—21no—
20#1954—22no—
21#1780—22no—
22#1574—22no—
23#1574—22no—
24C2×#335—21no—
25C2×#334—21no—
26#1543—22no—
27#1543—22no—
28C2×#333C411yescharacteristic, complemented
29C2×#333C411yescharacteristic, complemented
30C2×#331C411yescharacteristic, complemented
31C2×#331C411yescharacteristic, complemented
32#1277—22no—
33#1276—22no—
34#1276—22no—
35C22×#41D412yes—
36C22×M16—21no—
37#532—21no—
38#532—21no—
39#532—21no—
40#532—21no—
41#532—21no—
42#532—21no—
43#532—21no—
44#532—21no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#124Q812yes—
48C2×#121—21no—
49C2×#121—21no—
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431D412yes—
71#431D412yes—
72#423—21no—
73#423—21no—
74#423—21no—
75#423D412yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2×C411yescharacteristic
79C2×C4⋊C8C2×C411yescharacteristic
80C2×C4⋊C8C2×C411yescharacteristic
81C2×C4⋊C8C2×C411yescharacteristic
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8D412yes—
85C2×#104—28no—
86C2×#104—48no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C43—21no—
98C43—21no—
99C43—21no—
100C43D412yes—
101#335—24no—
102#334—24no—
103#333—22no—
104#333—22no—
105#331—22no—
106#331—22no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—22no—
111C2×#41—44no—
112C2×#41—44no—
113C22×Q8C4⋊C412yes—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4C4⋊C412yes—
118C23×C4—22no—
119C23×C4—41no—
120C23×C4C2×D411yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C2×M16—22no—
124C2×M16—41no—
125C2×M16—44no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—41no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138#127—21no—
139#127—21no—
140#127—21no—
141#127—21no—
142#124—22no—
143#124—22no—
144#124—22no—
145#124—22no—
146#121—41no—
147#121—41no—
148#121—41no—
149#121—41no—
150#121—41no—
151#121—41no—
152#121—41no—
153#121—41no—
154C4×D4—41no—
155C4×D4—41no—
156C4×D4—41no—
157C4×D4—41no—
158C4×D4—41no—
159C4×D4—41no—
160C4×D4—41no—
161C4×D4—41no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—24no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×Q811yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—22no—
200C4⋊C8—21no—
201C4⋊C8—22no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8C22⋊C412yes—
209C4⋊C8C22⋊C412yes—
210#104—416no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219C22⋊C8—84no—
220C22⋊C8—84no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—41no—
224C4×C8—41no—
225C4×C8—41no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8—41no—
229C24—22no—
230C24#12311yescharacteristic
231#41—44no—
232#41—44no—
233#41—48no—
234#41—82no—
235#41—82no—
236#41—84no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—44no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—44no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—84no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4C2×C4⋊C411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4C4×D411yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4C22⋊Q811yescharacteristic
284C22×C4C22⋊Q811yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4#12311yescharacteristic
287C22×C4#12311yescharacteristic
288C22×C4#12411yescharacteristic
289C22×C4#12411yescharacteristic
290C22×C4#12611yescharacteristic
291C22×C4#12611yescharacteristic
292M16—44no—
293M16—82no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—82no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C42—24no—
322C42—24no—
323C42—24no—
324C42—42no—
325C42—42no—
326C42—44no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—44no—
331C42C2×C22⋊C411yescharacteristic
332C42#11711yescharacteristic
333C42C4×D411yescharacteristic
334C42C4×D411yescharacteristic
335C42#12011yescharacteristic
336C42#12111yescharacteristic
337C42#12111yescharacteristic
338C42#12111yescharacteristic
339C42#12111yescharacteristic
340C42C22⋊Q811yescharacteristic
341C42C22⋊Q811yescharacteristic
342C42C22⋊Q811yescharacteristic
343C42C22⋊Q811yescharacteristic
344C42#12311yescharacteristic
345C42#12411yescharacteristic
346C42#12711yescharacteristic
347C23—22no—
348C23—21no—
349C23—21no—
350C23—22no—
351C23—44no—
352C23—42no—
353C23—42no—
354C23—44no—
355C23—42no—
356C23#38511yescharacteristic
357C23#39411yescharacteristic
358C23#39711yescharacteristic
359Q8—44no—
360Q8—82no—
361D4—44no—
362D4—82no—
363D4—84no—
364D4—84no—
365D4—81no—
366D4—81no—
367D4—82no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—81no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—84no—
406C2×C4#38011yescharacteristic
407C2×C4#38511yescharacteristic
408C2×C4#38611yescharacteristic
409C2×C4#38811yescharacteristic
410C2×C4#38811yescharacteristic
411C2×C4#39011yescharacteristic
412C2×C4#39211yescharacteristic
413C2×C4#39311yescharacteristic
414C2×C4#39411yescharacteristic
415C2×C4#39611yescharacteristic
416C2×C4#39611yescharacteristic
417C2×C4#39911yescharacteristic
418C2×C4#44311yescharacteristic
419C2×C4#44311yescharacteristic
420C2×C4#44312yes—
421C2×C4#44411yescharacteristic
422C2×C4#44411yescharacteristic
423C2×C4#44412yes—
424C2×C4#45312yes—
425C2×C4#45512yes—
426C8—42no—
427C8—42no—
428C8—81no—
429C8—81no—
430C8—81no—
431C8—82no—
432C8—81no—
433C8—82no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—42no—
441C22—42no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—84no—
446C22—84no—
447C22#143911yescharacteristic
448C22#185612yes—
449C22#186611yescharacteristic
450C22#186611yescharacteristic
451C22#186612yes—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—48no—
459C4—82no—
460C4#187611yescharacteristic
461C4#187611yescharacteristic
462C4#192912yes—
463C4#193711yescharacteristic
464C4#193811yescharacteristic
465C4#197411yescharacteristic
466C4#197511yescharacteristic
467C4#200211yescharacteristic
468C4#200411yescharacteristic
469C4#203712yes—
470C2—41no—
471C2—41no—
472C2—162no—
473C2#893412yes—
474C2#893512yes—
475C2#1106811yescharacteristic
476C2#1112511yescharacteristic
477C2#1113511yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1574C2semidirect
5. #7880C2semidirect
6. C2×#1276C2semidirect
7. C2×#1276C2semidirect
28. C2×#333C4semidirect
29. C2×#333C4semidirect
30. C2×#331C4semidirect
31. C2×#331C4semidirect

Maximal subgroups

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

subgroupindex
#217872
C2×#19622
C2×#15742
C2×#15432
#78802
C2×#12762