Finite Group Databasebeta

← #482142

Subgroups — #482142

Order 512.

1016 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35702C211yesmaximal, characteristic
2#20638C211yesmaximal, characteristic
3#17350C211yesmaximal, characteristic
4#11906C214yesmaximal, complemented
5#11904C214yesmaximal, complemented
6#11724C212yesmaximal, complemented
7#11721C212yesmaximal, complemented
8C2×D4×Q8C411yescharacteristic
9C2×#518C2211yescharacteristic
10#2972—21no—
11#2905—21no—
12C22×#100—21no—
13#2628C2211yescharacteristic
14#2626C412yes—
15#2580C411yescharacteristic
16#2564C2212yes—
17#2563C411yescharacteristic
18#2452—21no—
19#2450—22no—
20#2439—21no—
21#2437—22no—
22#2409C2212yes—
23#2396C2211yescharacteristic
24C2×#393C412yes—
25#2349—21no—
26#2349C411yescharacteristic
27D4×C22⋊C4—21no—
28#1800—22no—
29#1800C2214yes—
30#1742C2212yes—
31#1460—21no—
32#1459C2214yes—
33#1458C2212yes—
34C2×#323—22no—
35#1354C2214yescomplemented
36#1354C2218yescomplemented
37#1353C2214yescomplemented
38C23×Q8C2×C412yes—
39C23×D4D411yescharacteristic
40C22×M16—22no—
41D4×Q8—44no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8C2×C412yes—
47#518—44no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×D4—21no—
50C2×C4×D4—21no—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D412yes—
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2312yes—
57C2×#117D411yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C22⋊C4—22no—
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4D412yes—
62C8×D4—44no—
63#431—22no—
64#424—44no—
65#423—24no—
66C2×#100—24no—
67C2×#100—44no—
68C2×C22⋊C8—24no—
69#400—21no—
70#400C2311yescharacteristic
71#399—21no—
72#399C2×C412yes—
73#399C2×C411yescharacteristic
74#398—21no—
75#396—22no—
76#396—22no—
77#396—21no—
78#395C2312yes—
79#395C2311yescharacteristic
80#393—22no—
81#393—22no—
82#393C2×C412yes—
83#393C2×C412yes—
84#393C2×C412yes—
85#393C2×C412yes—
86#393C2×C411yescharacteristic
87#391—22no—
88#390—21no—
89#390—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#387—22no—
94#386—22no—
95#386—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#382—21no—
102#382C2×C411yescharacteristic
103#380—22no—
104#380—21no—
105#380—22no—
106#380C2×C412yes—
107#379—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—21no—
111C2×#95—22no—
112C2×#95C2×C412yes—
113#343—22no—
114#343C2314yes—
115#324C2314yes—
116#323—22no—
117#323—24no—
118C25C22⋊C412yes—
119C22×Q8—22no—
120C22×Q8—44no—
121C22×Q8—84no—
122C22×Q8C22×C411yescharacteristic
123C22×Q8C22×C412yes—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4C22⋊C411yescharacteristic
127C22×D4C22⋊C411yescharacteristic
128C22×D4SD1614yes—
129C22×D4Q1614yes—
130C22×D4C2×D412yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C2×M16—24no—
139C2×M16—42no—
140C2×M16—44no—
141C2×M16—44no—
142C22×C8—42no—
143C22×C8—44no—
144C4⋊Q8—42no—
145C4⋊Q8—44no—
146C22⋊Q8—48no—
147C22⋊Q8—44no—
148C22⋊Q8—84no—
149C4×Q8—42no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—44no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—44no—
156C4×D4—44no—
157C4×D4—44no—
158#117—24no—
159#117—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2411yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—84no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22×C411yescharacteristic
192C2×C22⋊C4C22×C412yes—
193C2×C22⋊C4C2×D411yescharacteristic
194C2×C22⋊C4C2×D412yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C4⋊C8—48no—
201C4⋊C8—42no—
202#100—48no—
203C22⋊C8—24no—
204C22⋊C8—44no—
205C4×C8—42no—
206#95—21no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95C22×C411yescharacteristic
218#95C22×C412yes—
219#95C22×C411yescharacteristic
220C24—42no—
221C24—42no—
222C24—82no—
223C24—82no—
224C24#10318yes—
225C24C2×C22⋊C411yescharacteristic
226C24C2×C22⋊C412yes—
227C24C2×C22⋊C412yes—
228C2×Q8—42no—
229C2×Q8—44no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—44no—
233C2×Q8—82no—
234C2×Q8—82no—
235C2×Q8—88no—
236C2×Q8—88no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—48no—
242C2×D4—48no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4#10314yes—
246C2×D4#10314yes—
247C2×D4C2×SD1614yes—
248C2×D4C2×Q1614yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×SD1612yes—
277C22×C4C2×Q1612yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281M16—44no—
282M16—44no—
283M16—84no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—84no—
287C2×C8—164no—
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C4⋊C4—82no—
300C4⋊C4—81no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4#9914yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—82no—
316C22⋊C4—81no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C42—22no—
321C42—24no—
322C42—22no—
323C42—42no—
324C42—42no—
325C23—21no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—21no—
330C23—22no—
331C23—48no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—41no—
336C23—42no—
337C23—41no—
338C23—88no—
339C23—162no—
340C23—162no—
341C23—164no—
342C23C2×#10312yes—
343C23C2×#10314yes—
344C23C2×#10314yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—88no—
349Q8—84no—
350D4—28no—
351D4—28no—
352D4—44no—
353D4—44no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—88no—
390C2×C4—162no—
391C2×C4—162no—
392C2×C4—164no—
393C2×C4C2×#9914yes—
394C2×C4C2×#9912yes—
395C2×C4C2×#10312yes—
396C2×C4#42112yes—
397C2×C4C22×SD1611yescharacteristic
398C2×C4C22×Q1611yescharacteristic
399C8—84no—
400C8—84no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—24no—
405C22—24no—
406C22—42no—
407C22—44no—
408C22—44no—
409C22—44no—
410C22—42no—
411C22—42no—
412C22—41no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—82no—
417C22—81no—
418C22—168no—
419C22C22×#9911yescharacteristic
420C22C22×#10311yescharacteristic
421C22C2×#42111yescharacteristic
422C4—24no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—42no—
431C4—81no—
432C4—82no—
433C4—81no—
434C4—82no—
435C4—81no—
436C4#152014yes—
437C2—21no—
438C2—21no—
439C2—44no—
440C2—44no—
441C2—42no—
442C2C2×#152011yescharacteristic
443C2#3334811yescharacteristic
444C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11906C2semidirect
5. #11904C2semidirect
6. #11724C2semidirect
7. #11721C2semidirect
35. #1354C22semidirect
36. #1354C22semidirect
37. #1353C22semidirect

Maximal subgroups

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

subgroupindex
#357022
#206382
#173502
#119062
#119042
#117242
#117212