Finite Group Databasebeta

← #206743

Subgroups — #206743

Order 512.

868 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic
3C4×#432C211yesmaximal, characteristic
4#9867C214yesmaximal, complemented
5#9866C214yesmaximal, complemented
6#7900C212yesmaximal
7#7900C212yesmaximal
8C2×#432C2212yes—
9C2×#431C2212yes—
10#2864—21no—
11#2864—21no—
12#2859C411yescharacteristic
13#2859C411yescharacteristic
14#2859C411yescharacteristic
15#2859C411yescharacteristic
16#2851—21no—
17#2851—21no—
18#2851C412yes—
19#2851C412yes—
20C4×#125—21no—
21C4×#125—21no—
22C4×#125C2212yes—
23C4×#125C2212yes—
24#2296—21no—
25#2296—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1831—21no—
31#1797—21no—
32#1797—21no—
33#1767C2212yes—
34C4×C4⋊C8C2212yes—
35C4×C4⋊C8C2212yes—
36C4×C4⋊C8C2212yes—
37#1749—21no—
38C4×C22⋊C8—21no—
39#1745—21no—
40C4×#97—21no—
41#1279C2218yes—
42#1279C2218yes—
43C2×#125—21no—
44C2×#125—21no—
45C2×#125C2×C412yes—
46C2×#125C2×C412yes—
47C2×#117—21no—
48C2×#117—21no—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42C2311yescharacteristic
56#432—24no—
57#432Q818yes—
58#431—28no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2314yes—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×#97—22no—
66C2×#97—22no—
67C2×C4×C8—22no—
68C2×C4×C8C2314yes—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C412yes—
84#389C2×C412yes—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97#384C2×C411yescharacteristic
98#384C2×C411yescharacteristic
99#384C2×C411yescharacteristic
100#384C2×C411yescharacteristic
101#382—21no—
102#382—21no—
103#382—21no—
104#382—21no—
105#382C2×C412yes—
106#382C2×C412yes—
107#380—21no—
108#380—21no—
109#380C2×C412yes—
110#380C2×C412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2312yes—
118C4×C4⋊C4C2312yes—
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43C2311yescharacteristic
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—24no—
139C22×C8—24no—
140#125—28no—
141#125—44no—
142#125—44no—
143#125—44no—
144#117—44no—
145#117—44no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q814yes—
204C2×C42#4114yes—
205C2×C42C2411yescharacteristic
206C4⋊C8—24no—
207C4⋊C8—44no—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C4⋊C8C2×Q818yes—
211C22⋊C8—28no—
212C22⋊C8—44no—
213#97—28no—
214#97—44no—
215C4×C8—28no—
216C4×C8—44no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229C24C2×C22⋊C411yescharacteristic
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C4×Q814yes—
264C22×C4C22⋊Q814yes—
265C22×C4#12412yes—
266C22×C4C4⋊Q812yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C2×C8—28no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—44no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—41no—
321C42—44no—
322C42—44no—
323C42C4×Q814yes—
324C42C22⋊Q814yes—
325C42#12412yes—
326C42C4⋊Q812yes—
327C42#13114yes—
328C42C22×Q812yes—
329C42C2×#4112yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23#39112yes—
337C23C22×C22⋊C411yescharacteristic
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4#39112yes—
387C2×C4#39112yes—
388C2×C4#39112yes—
389C2×C4#40812yes—
390C2×C4#40812yes—
391C2×C4#42112yes—
392C2×C4#42112yes—
393C2×C4C2×C4×Q812yes—
394C2×C4C2×C22⋊Q812yes—
395C2×C4C2×#12411yescharacteristic
396C2×C4C2×C4⋊Q811yescharacteristic
397C2×C4C2×#13112yes—
398C2×C4#56812yes—
399C2×C4#57512yes—
400C2×C4#57512yes—
401C8—48no—
402C8—84no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22C2×#39111yescharacteristic
411C22C2×#40811yescharacteristic
412C22#287911yescharacteristic
413C22#289611yescharacteristic
414C22#289611yescharacteristic
415C22C2×#42111yescharacteristic
416C22C2×#42111yescharacteristic
417C4—22no—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—82no—
434C4#296214yes—
435C4#308014yes—
436C2—42no—
437C2#1747711yescharacteristic
438C2#1748211yescharacteristic
439C2#1749411yescharacteristic
440C2#1749911yescharacteristic
441C2#2130211yescharacteristic
442C2#2130611yescharacteristic
443C2#2131411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9867C2semidirect
5. #9866C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#208082
C4×#4322
#98672
#98662
#79002