Finite Group Databasebeta

← #161097

Subgroups — #161097

Order 512.

918 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34332C211yesmaximal, characteristic
2#22700C211yesmaximal, characteristic
3#22008C211yesmaximal, characteristic
4#7648C212yesmaximal, complemented
5#7641C212yesmaximal, complemented
6#7617C212yesmaximal, complemented
7#7607C212yesmaximal, complemented
8#7591C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×#554C411yescharacteristic
13C2×#436—21no—
14#2370—21no—
15#2369C411yescharacteristic
16#2360—22no—
17#2360C412yes—
18Q8×C4⋊C4C411yescharacteristic
19D4×C4⋊C4—21no—
20D4×C4⋊C4C411yescharacteristic
21#2345C412yes—
22#2330—21no—
23#2329—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C22⋊Q8C2212yes—
26#2295C2211yescharacteristic
27C2×#389C2212yes—
28#2275—21no—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1923—22no—
34#1851—22no—
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1744C2211yescharacteristic
42#1728C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53C22×M16—22no—
54#554—24no—
55#554—24no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C412yes—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C412yes—
71C22×C42C2312yes—
72#436—44no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#391—21no—
85#391—22no—
86#391C2×C411yescharacteristic
87#391C2311yescharacteristic
88#389—21no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389C2×C411yescharacteristic
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2312yes—
99#389C2311yescharacteristic
100#387—21no—
101#387—22no—
102#387—21no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384—21no—
110#384C2×C411yescharacteristic
111#380—22no—
112#380C2×C412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130#376—21no—
131C2×#95—22no—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8C22×C411yescharacteristic
145C22×D4C22⋊C411yescharacteristic
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C22×C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C2×M16—44no—
152C2×M16—44no—
153C2×M16—82no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C4⋊Q8—22no—
161C4⋊Q8M1614yescomplemented
162#125—28no—
163C22⋊Q8—216no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C4×Q8—22no—
167C4×Q8—81no—
168C4×Q8—81no—
169C4×Q8—82no—
170C4×Q8M1614yescomplemented
171C4×D4—28no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—42no—
175C4×D4—44no—
176C4×D4SD1614yes—
177C4×D4Q1614yes—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—82no—
215C2×C42C22×C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—82no—
229C4⋊C8—82no—
230C22⋊C8—44no—
231C22⋊C8—82no—
232C22⋊C8—82no—
233C4×C8—42no—
234C4×C8—42no—
235C4×C8—81no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252C24C2×C22⋊C412yes—
253C2×Q8—21no—
254C2×Q8—24no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—81no—
258C2×Q8—81no—
259C2×Q8C2×M1612yes—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4#10314yes—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—28no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×SD1614yes—
293C22×C4C2×Q1614yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297M16—88no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C4⋊C4—28no—
317C4⋊C4—28no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—81no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—81no—
338C4⋊C4#10314yes—
339C4⋊C4C2×M1612yes—
340C4⋊C4C2×M1614yes—
341C22⋊C4—28no—
342C22⋊C4—24no—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4#10318yes—
346C42—21no—
347C42—21no—
348C42—24no—
349C42—22no—
350C42—41no—
351C42—42no—
352C42—42no—
353C42—41no—
354C42—42no—
355C42—41no—
356C42—42no—
357C42—82no—
358C42—82no—
359C42—82no—
360C42C2×M1612yes—
361C42C2×M1612yes—
362C42C2×SD1612yes—
363C42C2×Q1612yes—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23C2×#10314yes—
371C23C22×C22⋊C411yescharacteristic
372Q8—42no—
373Q8—42no—
374Q8—84no—
375D4—28no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—88no—
422C2×C4—164no—
423C2×C4#40712yes—
424C2×C4#40712yes—
425C2×C4C2×#10312yes—
426C2×C4C2×#10314yes—
427C2×C4C2×#10312yes—
428C2×C4#42112yes—
429C2×C4C22×M1611yescharacteristic
430C2×C4C22×M1611yescharacteristic
431C2×C4C22×SD1611yescharacteristic
432C2×C4C22×Q1611yescharacteristic
433C2×C4#57611yescharacteristic
434C2×C4#57711yescharacteristic
435C8—42no—
436C8—81no—
437C8—82no—
438C8—81no—
439C8—81no—
440C8—82no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22C2×#40711yescharacteristic
449C22#288311yescharacteristic
450C22#288911yescharacteristic
451C22C22×#10311yescharacteristic
452C22#289711yescharacteristic
453C22C2×#42111yescharacteristic
454C22#290111yescharacteristic
455C4—22no—
456C4—22no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—24no—
462C4—21no—
463C4—21no—
464C4—41no—
465C4—42no—
466C4—44no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—81no—
471C4—82no—
472C4—81no—
473C4#148714yes—
474C2—44no—
475C2#1737011yescharacteristic
476C2C2×#148711yescharacteristic
477C2#1745311yescharacteristic
478C2#1759911yescharacteristic
479C2#1761411yescharacteristic
480C2#1768211yescharacteristic
481C2#1772311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7648C2semidirect
5. #7641C2semidirect
6. #7617C2semidirect
7. #7607C2semidirect
8. #7591C2semidirect
9. #7577C2semidirect
10. #7570C2semidirect
11. #7562C2semidirect
161. C4⋊Q8M16semidirect
170. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
#343322
#227002
#220082
#76482
#76412
#76172
#76072
#75912
#75772
#75702
#75622