Finite Group Databasebeta

← #161104

Subgroups — #161104

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34317C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic, complemented
3#22018C211yesmaximal, characteristic, complemented
4#7650C212yesmaximal, complemented
5#7642C212yesmaximal
6#7619C212yesmaximal, complemented
7#7608C212yesmaximal
8#7591C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic
12C2×#554C411yescharacteristic
13#2370C412yes—
14#2369C411yescharacteristic
15#2362—22no—
16#2362—21no—
17#2357C412yes—
18#2355C411yescharacteristic
19#2354C411yescharacteristic
20#2344—21no—
21#2332—22no—
22#2323—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#125C2212yes—
25#2293C2211yescharacteristic
26#2293C2212yes—
27#2274—21no—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1916—22no—
34#1839C2212yes—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2211yescharacteristic
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53#554—24no—
54#554—24no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—22no—
68C22×C4⋊C4—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×#97—22no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#391—21no—
78#391—21no—
79#391—22no—
80#390C2×C412yes—
81#390C2×C411yescharacteristic
82#390C2311yescharacteristic
83#389—21no—
84#389—22no—
85#389—21no—
86#389C2×C412yes—
87#389C2×C411yescharacteristic
88#389C2311yescharacteristic
89#389C2312yes—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—21no—
94#388—22no—
95#388—21no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#384C2×C411yescharacteristic
103#382C2×C412yes—
104#381—22no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—22no—
126C43C2312yes—
127C43C2311yescharacteristic
128#336—22no—
129#336—21no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8C22⋊C411yescharacteristic
144C22×D4C22×C411yescharacteristic
145C23×C4—21no—
146C23×C4—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8M1614yes—
155#125—24no—
156#125—24no—
157C22⋊Q8—28no—
158C22⋊Q8—216no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—42no—
164C4×Q8—81no—
165C4×Q8—81no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—22no—
169C4×D4M1614yes—
170#117—44no—
171#117—42no—
172#117—42no—
173#117—82no—
174#117—82no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42C22×C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2411yescharacteristic
222C4⋊C8—42no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225C4⋊C8—82no—
226#97—44no—
227C4×C8—42no—
228C4×C8—42no—
229C4×C8—81no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244C24—21no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—81no—
250C2×Q8—81no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4C2×M1612yes—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C2×C8—22no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—81no—
317C4⋊C4C2×M1614yes—
318C4⋊C4C2×M1612yes—
319C22⋊C4—28no—
320C22⋊C4—24no—
321C22⋊C4—44no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—81no—
328C22⋊C4—82no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—21no—
334C42—22no—
335C42—22no—
336C42—21no—
337C42—22no—
338C42—42no—
339C42—41no—
340C42—41no—
341C42—42no—
342C42—42no—
343C42—41no—
344C42—44no—
345C42—42no—
346C42—82no—
347C42—82no—
348C42—82no—
349C42C2×M1612yes—
350C42C2×M1612yes—
351C42#13512yes—
352C42#13512yes—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23C22×C22⋊C411yescharacteristic
361Q8—44no—
362Q8—84no—
363D4—42no—
364D4—42no—
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—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—48no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—88no—
407C2×C4#40712yes—
408C2×C4#40712yes—
409C2×C4#41612yes—
410C2×C4#41612yes—
411C2×C4#42112yes—
412C2×C4C22×M1611yescharacteristic
413C2×C4C22×M1611yescharacteristic
414C2×C4C2×#13511yescharacteristic
415C2×C4C2×#13511yescharacteristic
416C2×C4#57611yescharacteristic
417C2×C4#57711yescharacteristic
418C8—42no—
419C8—81no—
420C8—82no—
421C8—81no—
422C8—81no—
423C8—82no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22C2×#40711yescharacteristic
432C22#288411yescharacteristic
433C22#288911yescharacteristic
434C22C2×#41611yescharacteristic
435C22#289711yescharacteristic
436C22C2×#42111yescharacteristic
437C22#290111yescharacteristic
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—41no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—44no—
452C4—42no—
453C4—81no—
454C4—82no—
455C4—81no—
456C4#148814yes—
457C2—82no—
458C2#1737111yescharacteristic
459C2C2×#148811yescharacteristic
460C2#1745311yescharacteristic
461C2#1760211yescharacteristic
462C2#1761611yescharacteristic
463C2#1768211yescharacteristic
464C2#1772711yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22832C2semidirect
3. #22018C2semidirect
4. #7650C2semidirect
6. #7619C2semidirect
8. #7591C2semidirect
10. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#343172
#228322
#220182
#76502
#76422
#76192
#76082
#75912
#75742
#75702
#75602