Finite Group Databasebeta

← #207290

Subgroups — #207290

Order 512.

890 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35304C211yesmaximal, characteristic
2#20814C211yesmaximal, characteristic
3#13805C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9911C212yesmaximal
6#9910C212yesmaximal
7#9909C212yesmaximal
8#8026C211yesmaximal, characteristic
9#8022C211yesmaximal, characteristic
10#8017C211yesmaximal, characteristic
11#8013C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2851C2212yes—
16#2660C412yescomplemented
17#2647C412yescomplemented
18#2612—22no—
19#2606—21no—
20#2605—21no—
21#2569—21no—
22#2549—21no—
23#2438—22no—
24#2331C412yescomplemented
25#2330C412yescomplemented
26#2285C2211yescharacteristic
27#2283C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#376C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1843—21no—
32#1832—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1768—22no—
36#1436—22no—
37#1434—21no—
38#1434C2212yes—
39C4×#95—21no—
40C4×#95C2212yes—
41#1294C2211yescharacteristic
42#1294C2211yescharacteristic
43#1294C2211yescharacteristic
44#1294C2211yescharacteristic
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C2×C4⋊Q8C2×C412yes—
52C2×#125C2×C412yes—
53C2×C22⋊Q8—22no—
54C2×#117—21no—
55C22×C4⋊C4—21no—
56C22×C42—21no—
57C22×C42—22no—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60#432—22no—
61#432—22no—
62#431—22no—
63#431—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C22⋊C8C2311yescharacteristic
69C2×C22⋊C8C2311yescharacteristic
70C2×C22⋊C8C2312yes—
71C2×#97—21no—
72C2×C4×C8—21no—
73#400—22no—
74#400—22no—
75#398—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#395—22no—
80#395—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#385—22no—
98#385—22no—
99#384C2×C412yes—
100#384C2×C412yes—
101#383—21no—
102#382—21no—
103#382—22no—
104#382C2×C412yes—
105#382C2×C412yes—
106#382Q814yescomplemented
107#381—21no—
108#380—22no—
109#380—21no—
110#380C2311yescharacteristic
111#380C2312yes—
112#380C2311yescharacteristic
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C22⋊C4—21no—
118#376—21no—
119#376—22no—
120#376C2×C412yes—
121#376C2×C411yescharacteristic
122#376C2×C411yescharacteristic
123#376D411yescharacteristic
124#376D411yescharacteristic
125#376D412yes—
126#376Q814yescomplemented
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D411yescharacteristic
133C2×#95D412yes—
134C2×#95D411yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2312yes—
137C2×#95C2311yescharacteristic
138#336—22no—
139#336—22no—
140C22×Q8—22no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C4⋊Q8—44no—
153#125—44no—
154C22⋊Q8—82no—
155C22⋊Q8—82no—
156#117—81no—
157#117—81no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—81no—
171C2×C4⋊C4—81no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C42—24no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—44no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42—44no—
187C2×C42—82no—
188C2×C42C22×C412yes—
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×Q812yes—
197C2×C42C2×Q812yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C4⋊C8—44no—
201C4⋊C8—44no—
202C4⋊C8—42no—
203C4⋊C8—42no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210#97—42no—
211C4×C8—42no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—24no—
224#95—24no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—44no—
232#95—44no—
233#95—82no—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C411yescharacteristic
239#95C22⋊C412yes—
240#95C22⋊C411yescharacteristic
241#95C22⋊C411yescharacteristic
242#95C22⋊C412yes—
243#95C22⋊C411yescharacteristic
244#95C2×Q812yes—
245#95C2×Q812yes—
246#95C2×Q814yes—
247C24C22×D411yescharacteristic
248C2×Q8—44no—
249C2×Q8—82no—
250C2×Q8—82no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C22⋊Q812yes—
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q812yes—
297C22×C4C22⋊Q812yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C22×Q811yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—84no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—42no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C42—24no—
334C42—24no—
335C42—42no—
336C42—42no—
337C42—48no—
338C42—82no—
339C42—81no—
340C42—81no—
341C42—88no—
342C42C4×Q812yes—
343C42C4×Q812yes—
344C42C4×Q812yes—
345C42C4×Q812yes—
346C42#12412yes—
347C42#12412yes—
348C42C4⋊Q812yes—
349C42C4⋊Q812yes—
350C23—22no—
351C23—22no—
352C23C22×C22⋊C411yescharacteristic
353C23C2×C22⋊Q811yescharacteristic
354C23C2×C22⋊Q811yescharacteristic
355C23#57611yescharacteristic
356C23#57711yescharacteristic
357C23#57711yescharacteristic
358C23#57811yescharacteristic
359Q8—164no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4#39112yes—
423C2×C4#39112yes—
424C2×C4#39112yes—
425C2×C4#39112yes—
426C2×C4#41014yes—
427C2×C4C2×C4×Q811yescharacteristic
428C2×C4C2×C4×Q811yescharacteristic
429C2×C4C2×#12411yescharacteristic
430C2×C4C2×C4⋊Q811yescharacteristic
431C8—81no—
432C8—81no—
433C8—82no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22C2×#39111yescharacteristic
443C22C2×#41012yes—
444C22#289711yescharacteristic
445C22#289811yescharacteristic
446C22#290112yes—
447C22#308111yescharacteristic
448C22#308211yescharacteristic
449C22#308311yescharacteristic
450C22#308411yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—24no—
460C4—44no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—84no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4—82no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—21no—
474C2#1760412yes—
475C2#1760512yes—
476C2#2131511yescharacteristic
477C2#2133011yescharacteristic
478C2#2133211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2660C4semidirect
17. #2647C4semidirect
24. #2331C4semidirect
25. #2330C4semidirect
106. #382Q8semidirect
126. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#353042
#208142
#138052
#99132
#99112
#99102
#99092
#80262
#80222
#80172
#80132