Finite Group Databasebeta

← #224162

Subgroups — #224162

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36555C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic, complemented
3Q8×#97C211yesmaximal, characteristic, complemented
4#9893C212yesmaximal, complemented
5#9892C212yesmaximal, complemented
6#9891C211yesmaximal, characteristic, complemented
7#9891C211yesmaximal, characteristic, complemented
8#8583C212yesmaximal
9#8579C211yesmaximal, characteristic
10#8226C212yesmaximal
11C4×#329C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2864C2212yes—
14#2859C2211yescharacteristic
15#2587—22no—
16#2582C411yescharacteristic
17#2538C411yescharacteristic
18#2522—22no—
19#2497—21no—
20#2454C412yes—
21#2453C412yes—
22#2423—21no—
23#2360—21no—
24Q8×C4⋊C4C411yescharacteristic
25D4×C4⋊C4C411yescharacteristic
26#2345—21no—
27C4×#124C2211yescharacteristic
28C4×#124C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1770—22no—
34#1770—21no—
35#1767C2211yescharacteristic
36#1767C2212yes—
37#1767C2211yescharacteristic
38#1767C2212yes—
39#1745C2212yes—
40C4×#97C2211yescharacteristic
41C4×#97C2211yescharacteristic
42C4×#97C2212yes—
43C2×#329C2211yescharacteristic
44C2×#329C2212yes—
45C2×#329C2211yescharacteristic
46#1295C2211yescharacteristic
47#1295C2211yescharacteristic
48#1295C2212yes—
49#1295C2212yes—
50#1295C2212yes—
51#1295C2212yes—
52#1295C2212yes—
53C2×#125C2×C412yes—
54C2×#124C2311yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C412yes—
63C22×C4⋊C4—21no—
64C8×Q8—28no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×#97—21no—
70C2×#97—22no—
71C2×#97—21no—
72C2×#97C2311yescharacteristic
73C2×#97C2311yescharacteristic
74C2×#97C2312yes—
75C2×#97C2312yes—
76C2×#97C2311yescharacteristic
77C2×#97C2311yescharacteristic
78C2×C4×C8—21no—
79C2×C4×C8—22no—
80C2×C4×C8—21no—
81#400—22no—
82#399—22no—
83#398—22no—
84#398—22no—
85#396—22no—
86#396—22no—
87#391—21no—
88#391C2×C412yes—
89#391C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C412yes—
92#389—21no—
93#389—22no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—22no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#388—21no—
104#388—22no—
105#388—22no—
106#388—21no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#385—22no—
113#385—22no—
114#384—21no—
115#384C2×C411yescharacteristic
116#384C2×C411yescharacteristic
117#382C2312yes—
118#382C2311yescharacteristic
119#380—21no—
120#380—21no—
121#380—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C22⋊C4—22no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C43C2311yescharacteristic
134C43C2312yes—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141#329—22no—
142#329—24no—
143#329Q814yescomplemented
144C22×Q8C22⋊C411yescharacteristic
145C22×Q8C22×C411yescharacteristic
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155#125—82no—
156#124—21no—
157#124—21no—
158#124C2×Q812yes—
159#124#4112yes—
160C22⋊Q8—44no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—82no—
166C4×D4—42no—
167C4×D4—82no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C412yes—
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2411yescharacteristic
214C4⋊C8—28no—
215C4⋊C8—24no—
216C4⋊C8—42no—
217C4⋊C8—44no—
218#97—21no—
219#97—21no—
220#97—22no—
221#97—22no—
222#97—21no—
223#97—21no—
224#97—21no—
225#97—21no—
226#97—22no—
227#97—22no—
228#97—42no—
229#97—44no—
230#97—42no—
231#97C2×Q812yes—
232#97C2×Q812yes—
233C4×C8—28no—
234C4×C8—24no—
235C4×C8—42no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—21no—
252C2×Q8—21no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8#10414yes—
258C2×Q8C4×Q812yes—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4C4×Q812yes—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—82no—
319C4⋊C4—81no—
320C4⋊C4—82no—
321C4⋊C4—81no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—82no—
332C22⋊C4—81no—
333C22⋊C4—81no—
334C22⋊C4—82no—
335C42—22no—
336C42—24no—
337C42—22no—
338C42—22no—
339C42—41no—
340C42—41no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—41no—
346C42—42no—
347C42—42no—
348C42—82no—
349C42C22×Q811yescharacteristic
350C42C2×#4111yescharacteristic
351C422−1+411yescharacteristic
352C422−1+411yescharacteristic
353C23—22no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23C22×C22⋊C411yescharacteristic
361Q8—28no—
362Q8—42no—
363Q8—42no—
364D4—42no—
365D4—42no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—84no—
408C2×C4—164no—
409C2×C4#41312yes—
410C2×C4C2×#10414yes—
411C2×C4C2×#10412yes—
412C2×C4#42112yes—
413C2×C4C2×C4×Q811yescharacteristic
414C2×C4#51911yescharacteristic
415C2×C4#54811yescharacteristic
416C2×C4#54811yescharacteristic
417C2×C4D4×Q811yescharacteristic
418C2×C4D4×Q811yescharacteristic
419C8—24no—
420C8—44no—
421C8—42no—
422C8—42no—
423C8—42no—
424C8—82no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22Q8×C22⋊C411yescharacteristic
433C22C2×#41311yescharacteristic
434C22#288811yescharacteristic
435C22C22×#10411yescharacteristic
436C22C2×#42111yescharacteristic
437C22#290111yescharacteristic
438C22#290111yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—24no—
444C4—21no—
445C4—21no—
446C4—41no—
447C4—42no—
448C4—41no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—41no—
453C4—82no—
454C4—81no—
455C4—81no—
456C4—82no—
457C4#153214yes—
458C2—82no—
459C2C2×#153211yescharacteristic
460C2#1765511yescharacteristic
461C2#1768111yescharacteristic
462C2#1768711yescharacteristic
463C2#2251511yescharacteristic
464C2Q8×#10411yescharacteristic
465C2#2252911yescharacteristic
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. #21345C2semidirect
3. Q8×#97C2semidirect
4. #9893C2semidirect
5. #9892C2semidirect
6. #9891C2semidirect
7. #9891C2semidirect
143. #329Q8semidirect

Maximal subgroups

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

subgroupindex
#365552
#213452
Q8×#972
#98932
#98922
#98912
#85832
#85792
#82262
C4×#3292