Finite Group Databasebeta

← #216116

Subgroups — #216116

Order 512.

902 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C2×#1831C211yesmaximal, characteristic
3C4×#432C211yesmaximal, characteristic
4#9877C212yesmaximal, complemented
5#9647C212yesmaximal, complemented
6#8937C214yesmaximal
7#8570C214yesmaximal, complemented
8C2×#432—21no—
9C22×C4⋊C8C2212yes—
10#2864—21no—
11#2863—21no—
12#2863C412yes—
13#2862C411yescharacteristic
14#2859C411yescharacteristic
15#2857C411yescharacteristic
16#2857C411yescharacteristic
17#2855—21no—
18#2855C412yes—
19#2851—21no—
20C4×C4⋊Q8C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×#125—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8C2212yes—
25#2295—21no—
26#2295C2212yes—
27#2283—21no—
28C2×C43C2211yescharacteristic
29#1831—21no—
30#1831—21no—
31#1831C2212yes—
32#1831C2212yes—
33#1767—21no—
34C4×C4⋊C8C2212yes—
35#1749—21no—
36C4×C22⋊C8C2212yes—
37#1745—21no—
38C2×#336—21no—
39C2×#336—21no—
40C4×#97C2212yes—
41C2×Q8⋊C8C2218yes—
42#1279C2218yes—
43C23×C8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8C2312yes—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58#432—28no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—44no—
62C2×C4⋊C8C2314yes—
63C2×C22⋊C8—22no—
64C2×#97—22no—
65C2×#97—22no—
66C2×C4×C8C2314yes—
67#391—21no—
68#391—21no—
69#391—21no—
70#391—21no—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2×C412yes—
74#391C2×C412yes—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#385—22no—
96#385—22no—
97#384C2×C411yescharacteristic
98#384C2×C411yescharacteristic
99#383—22no—
100#382—21no—
101#382—21no—
102#382—21no—
103#381—22no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380C2×C412yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2312yes—
114C4×C22⋊C4—22no—
115#376—22no—
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130Q8⋊C8—48no—
131C22×Q8—21no—
132C22×Q8C22×C412yes—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—44no—
143C22×C8—44no—
144C22×C8—44no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147#125—44no—
148C22⋊Q8—28no—
149C22⋊Q8—44no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—44no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—42no—
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×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209C4⋊C8—28no—
210C4⋊C8—28no—
211C4⋊C8—44no—
212C22⋊C8—28no—
213#97—28no—
214#97—44no—
215C4×C8—24no—
216C4×C8—24no—
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—
229#95—22no—
230#95—22no—
231C24C2×C22⋊C411yescharacteristic
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4#10414yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4#13114yes—
274C22×C4#13714yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C2×C8—28no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—88no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C42—22no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42H(5,𝔽2)12yes—
335C422−1+412yes—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23#40812yes—
341C23#41912yes—
342C23C2×#10412yes—
343C23C22×C22⋊C411yescharacteristic
344Q8—48no—
345Q8—84no—
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—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4#40812yes—
397C2×C4#41912yes—
398C2×C4C2×#10412yes—
399C2×C4C2×#10412yes—
400C2×C4#42112yes—
401C2×C4#52012yes—
402C2×C4#53511yescharacteristic
403C2×C4#53611yescharacteristic
404C2×C4#53712yes—
405C2×C4C2×#13112yes—
406C2×C4#56812yes—
407C2×C4C2×#13712yes—
408C2×C4#57512yes—
409C8—48no—
410C8—84no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22#148514yes—
418C22#231111yescharacteristic
419C22C2×#40811yescharacteristic
420C22#287911yescharacteristic
421C22C2×#41911yescharacteristic
422C22#289611yescharacteristic
423C22C22×#10411yescharacteristic
424C22C2×#42111yescharacteristic
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4#299214yes—
442C4#321512yes—
443C4#323612yes—
444C2—24no—
445C2C2×#148511yescharacteristic
446C2#1746711yescharacteristic
447C2#1749611yescharacteristic
448C2#1750211yescharacteristic
449C2#2185211yescharacteristic
450C2#2185611yescharacteristic
451C2#2186611yescharacteristic
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. #9877C2semidirect
5. #9647C2semidirect
7. #8570C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×#18312
C4×#4322
#98772
#96472
#89372
#85702