Finite Group Databasebeta

← #190876

Subgroups — #190876

Order 512.

964 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
1Q8×#117C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#8710C212yesmaximal
5#8568C212yesmaximal, complemented
6#8564C212yesmaximal, complemented
7#8311C212yesmaximal
8#8182C211yesmaximal, characteristic, complemented
9#8176C211yesmaximal, characteristic, complemented
10#8175C211yesmaximal, characteristic, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C2×#433—21no—
14#2355—22no—
15#2355C412yes—
16Q8×C4⋊C4C412yes—
17#2345—22no—
18#2345C412yes—
19Q8×C22⋊C4C412yes—
20#2295—21no—
21#2295—21no—
22#2295C2211yescharacteristic
23#2295C2211yescharacteristic
24#2285—21no—
25#2285C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27C42×Q8C2211yescharacteristic
28C4×#117—21no—
29C4×#117C2211yescharacteristic
30#1923—22no—
31#1916—22no—
32#1842C2212yes—
33#1832—21no—
34#1798—21no—
35#1797C2212yes—
36#1768—21no—
37#1767C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1744C2211yescharacteristic
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C23×Q8C2×C411yescharacteristic
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2311yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×#117—21no—
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#433—44no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8—22no—
80C2×C4×C8C2312yes—
81C2×C4×C8C2312yes—
82#391—22no—
83#391—22no—
84#391C2×C412yes—
85#391C2×C412yes—
86#389—22no—
87#389—22no—
88#389C2×C412yes—
89#389C2×C412yes—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#384—22no—
97#384C2×C412yes—
98#382—22no—
99#382—21no—
100#380—21no—
101#380—22no—
102#380—22no—
103#380C2×C412yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4D412yes—
123#376—21no—
124#376—21no—
125#376D411yescharacteristic
126#376D411yescharacteristic
127C43—21no—
128C43—21no—
129C43C2311yescharacteristic
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137Q8⋊C8—24no—
138Q8⋊C8—24no—
139Q8⋊C8—24no—
140Q8⋊C8—24no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—24no—
145C22×Q8—42no—
146C22×Q8C22×C411yescharacteristic
147C22×Q8C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C4×Q8—28no—
158C4×Q8—28no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8#4114yes—
170C4×Q8#4114yes—
171#117—42no—
172#117—44no—
173#117SD1614yes—
174#117Q1614yes—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C42—28no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2411yescharacteristic
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—44no—
228C4⋊C8—44no—
229C4⋊C8—44no—
230C22⋊C8—44no—
231C22⋊C8—44no—
232#97—42no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—44no—
238C4×C8—42no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C24C2×C22⋊C411yescharacteristic
255C2×Q8—24no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—44no—
262C2×Q8—48no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8#11712yes—
266C2×Q8#11712yes—
267C2×Q8#13114yes—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—82no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×SD1612yes—
294C22×C4C2×Q1612yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
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—
309C2×C8—42no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4#10318yes—
329C4⋊C4#11712yes—
330C4⋊C4#11712yes—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4#10318yes—
334C42—24no—
335C42—24no—
336C42—21no—
337C42—21no—
338C42—24no—
339C42—24no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—42no—
349C42—42no—
350C42—44no—
351C42—44no—
352C42—42no—
353C42—42no—
354C42—82no—
355C42—82no—
356C42C2×SD1612yes—
357C42C2×SD1612yes—
358C42C2×Q1612yes—
359C42C2×Q1612yes—
360C42C2×#4112yes—
361C42C2×#4112yes—
362C23—22no—
363C23—21no—
364C23—22no—
365C23—21no—
366C23C2×#10312yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—22no—
369Q8—24no—
370Q8—24no—
371Q8—48no—
372Q8—41no—
373Q8—41no—
374Q8#43314yescomplemented
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—84no—
414C2×C4C2×#10314yes—
415C2×C4C2×#10314yes—
416C2×C4C2×#10312yes—
417C2×C4C2×#11711yescharacteristic
418C2×C4C2×#11711yescharacteristic
419C2×C4#52512yes—
420C2×C4#52512yes—
421C2×C4C2×#13112yes—
422C2×C4#56812yes—
423C2×C4C22×SD1611yescharacteristic
424C2×C4C22×Q1611yescharacteristic
425C2×C4#57512yes—
426C8—44no—
427C8—82no—
428C8—82no—
429C8—82no—
430C22—24no—
431C22—22no—
432C22—41no—
433C22—42no—
434C22—41no—
435C22#228011yescharacteristic
436C22#288012yes—
437C22C22×#10311yescharacteristic
438C22#289611yescharacteristic
439C22#290112yes—
440C4—22no—
441C4—24no—
442C4—24no—
443C4—24no—
444C4—22no—
445C4—41no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—41no—
451C4—42no—
452C4—82no—
453C4—82no—
454C4C2×#43311yescharacteristic
455C4#292111yescharacteristic
456C4#292412yes—
457C4#310312yes—
458C4#310412yes—
459C2—42no—
460C2#1750912yes—
461C2#1751612yes—
462C2#2071611yescharacteristic
463C2#2073411yescharacteristic
464C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8568C2semidirect
6. #8564C2semidirect
8. #8182C2semidirect
9. #8176C2semidirect
10. #8175C2semidirect
11. C4×Q8⋊C8C2semidirect
374. Q8#433semidirect

Maximal subgroups

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

subgroupindex
Q8×#1172
#214302
#208122
#87102
#85682
#85642
#83112
#81822
#81762
#81752
C4×Q8⋊C82