Finite Group Databasebeta

← #224461

Subgroups — #224461

Order 512.

835 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36562C211yesmaximal, characteristic
2#21346C212yesmaximal, complemented
3#9895C212yesmaximal, complemented
4#9894C212yesmaximal, complemented
5#9891C212yesmaximal, complemented
6#8583C211yesmaximal, characteristic
7#8583C211yesmaximal, characteristic
8#8579C211yesmaximal, characteristic
9#8224C211yesmaximal, characteristic
10#8224C211yesmaximal, characteristic
11C4×#329C211yesmaximal, characteristic
12C2×#557C411yescharacteristic
13C2×#554C411yescharacteristic
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2859C2211yescharacteristic
17#2567C411yescharacteristic
18#2566C411yescharacteristic
19#2566C411yescharacteristic
20#2565C411yescharacteristic
21#2565C411yescharacteristic
22#2556—22no—
23#2556—22no—
24#2542C411yescharacteristic
25#2429—22no—
26#2360—22no—
27C4×#124C2211yescharacteristic
28C4×#124C2211yescharacteristic
29C4×#124C2211yescharacteristic
30C42×Q8C2211yescharacteristic
31#1985—22no—
32#1940—22no—
33#1916—22no—
34#1916—22no—
35#1767C2212yes—
36#1767C2212yes—
37#1767C2212yes—
38#1745C2212yes—
39#1745C2212yes—
40C4×#97C2212yes—
41C2×#329C2211yescharacteristic
42C2×#329C2211yescharacteristic
43C2×#329C2211yescharacteristic
44C2×#329C2211yescharacteristic
45#1295C2212yes—
46#1295C2212yes—
47#1295C2212yes—
48#1295C2212yes—
49#1295C2212yes—
50#1295C2212yes—
51#557—44no—
52#554—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×#125—21no—
58C2×#125C2×C411yescharacteristic
59C2×#125C2×C411yescharacteristic
60C2×#125C2×C411yescharacteristic
61C2×#125C2×C411yescharacteristic
62C2×#124C2311yescharacteristic
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D412yes—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C22×C4⋊C4—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×#97—22no—
81C2×#97C2312yes—
82C2×#97C2312yes—
83C2×#97C2312yes—
84C2×#97C2312yes—
85C2×C4×C8—22no—
86#399—22no—
87#398—22no—
88#398—22no—
89#396—22no—
90#396—22no—
91#395—22no—
92#391—22no—
93#391—22no—
94#391C2×C411yescharacteristic
95#391C2×C411yescharacteristic
96#391C2×C411yescharacteristic
97#390C2×C411yescharacteristic
98#390C2×C411yescharacteristic
99#390C2×C411yescharacteristic
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#387—22no—
109#387—22no—
110#385—22no—
111#385—22no—
112#384C2×C411yescharacteristic
113#384C2×C411yescharacteristic
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#382C2×C411yescharacteristic
117#382C2×C411yescharacteristic
118#382C2311yescharacteristic
119#382C2311yescharacteristic
120#382C2311yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C43C2311yescharacteristic
131C43C2311yescharacteristic
132C43C2311yescharacteristic
133#336—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#329—22no—
141#329—22no—
142#329—22no—
143#329Q814yescomplemented
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8C22⋊C411yescharacteristic
147C22×Q8C22×C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—44no—
159#125—42no—
160#125—42no—
161#125—42no—
162#125—42no—
163#125—44no—
164#125—44no—
165#124—22no—
166#124C2×Q812yes—
167#124#4112yes—
168C22⋊Q8—48no—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C4×Q8—24no—
173C4×Q8—44no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—44no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—42no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
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—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—44no—
235C4⋊C8—44no—
236C4⋊C8—44no—
237#97—22no—
238#97—22no—
239#97—22no—
240#97—22no—
241#97—22no—
242#97—22no—
243#97—22no—
244#97—44no—
245#97C2×Q814yes—
246C4×C8—44no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258C24—21no—
259C2×Q8—21no—
260C2×Q8—22no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—82no—
268C2×Q8C4×Q812yes—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4C4×Q812yes—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D412yes—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4#10014yes—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—82no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—24no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42C22×Q811yescharacteristic
362C42C2×#4111yescharacteristic
363C422−1+412yes—
364C23—22no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23C22×C22⋊C411yescharacteristic
370Q8—42no—
371Q8—42no—
372Q8—44no—
373Q8—84no—
374Q8—162no—
375D4—42no—
376D4—42no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—84no—
414C2×C4—162no—
415C2×C4—162no—
416C2×C4C2×#10012yes—
417C2×C4C2×#10012yes—
418C2×C4C2×#10012yes—
419C2×C4#42114yes—
420C2×C4C2×C4×Q811yescharacteristic
421C2×C4#51911yescharacteristic
422C2×C4#54812yes—
423C2×C4D4×Q812yes—
424C8—44no—
425C8—44no—
426C8—82no—
427C8—82no—
428C22—44no—
429C22—42no—
430C22—82no—
431C22—81no—
432C22—81no—
433C22Q8×C22⋊C411yescharacteristic
434C22C22×#10011yescharacteristic
435C22#288911yescharacteristic
436C22C2×#42112yes—
437C22#290112yes—
438C4—22no—
439C4—22no—
440C4—24no—
441C4—22no—
442C4—22no—
443C4—21no—
444C4—21no—
445C4—22no—
446C4—42no—
447C4—42no—
448C4—41no—
449C4—42no—
450C4—41no—
451C4—41no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C2—82no—
459C2#1766311yescharacteristic
460C2#1768012yes—
461C2#1768511yescharacteristic
462C2Q8×#10011yescharacteristic
463C2#2252912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21346C2semidirect
3. #9895C2semidirect
4. #9894C2semidirect
5. #9891C2semidirect
143. #329Q8semidirect

Maximal subgroups

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

subgroupindex
#365622
#213462
#98952
#98942
#98912
#85832
#85792
#82242
C4×#3292