Finite Group Databasebeta

← #224102

Subgroups — #224102

Order 512.

810 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34950C211yesmaximal, characteristic
2Q8×C4⋊C8C211yesmaximal, characteristic
3#21339C211yesmaximal, characteristic
4#9874C212yesmaximal
5#9870C211yesmaximal, characteristic
6#9864C211yesmaximal, characteristic
7#9864C212yesmaximal
8#8568C212yesmaximal
9#8564C211yesmaximal, characteristic
10#8182C212yesmaximal
11C4×Q8⋊C8C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2586—22no—
14#2582C411yescharacteristic
15#2567C411yescharacteristic
16#2567C411yescharacteristic
17#2551C411yescharacteristic
18#2507—21no—
19#2507C412yes—
20#2451—21no—
21#2451C412yes—
22#2399—22no—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2212yes—
26#2295—21no—
27#2295C2212yes—
28C42×Q8C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1939—22no—
34#1916—22no—
35#1842C2211yescharacteristic
36#1842C2212yes—
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2211yescharacteristic
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2211yescharacteristic
50#1279C2212yes—
51#1279C2212yes—
52#1279C2211yescharacteristic
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C412yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C8×Q8—44no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—22no—
77C2×C4×C8—21no—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2312yes—
80C2×C4×C8C2311yescharacteristic
81#398—22no—
82#398—22no—
83#398—22no—
84#398—22no—
85#395—22no—
86#395—22no—
87#395—22no—
88#395—22no—
89#391—21no—
90#391—21no—
91#391—21no—
92#391—22no—
93#391—22no—
94#391C2×C412yes—
95#391C2×C412yes—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—22no—
101#389—21no—
102#389—22no—
103#389C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#384—21no—
113#384C2×C412yes—
114#384C2×C411yescharacteristic
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2312yes—
131#376—22no—
132C43—21no—
133C43C2311yescharacteristic
134C43C2312yes—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—21no—
140#336—22no—
141Q8⋊C8—24no—
142Q8⋊C8—22no—
143Q8⋊C8Q814yescomplemented
144C22×Q8—22no—
145C22×Q8—21no—
146C22×Q8C22⋊C411yescharacteristic
147C22×Q8C22×C411yescharacteristic
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—42no—
158C4⋊Q8—44no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—41no—
167C4×Q8—42no—
168C4×Q8—82no—
169C4×Q8—81no—
170C4×Q8—82no—
171C4×Q8—81no—
172C4×Q8SD1614yes—
173C4×Q8Q1614yes—
174C4×Q8C2×Q812yes—
175C4×Q8#4112yes—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C412yes—
210C2×C42C22×C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2411yescharacteristic
221C4⋊C8—22no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—22no—
225C4⋊C8—21no—
226C4⋊C8—42no—
227C4⋊C8—44no—
228C4⋊C8—44no—
229C4⋊C8—42no—
230C4⋊C8—81no—
231C4⋊C8—82no—
232C4⋊C8C2×Q812yes—
233C4×C8—21no—
234C4×C8—22no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—22no—
238C4×C8—42no—
239C4×C8—42no—
240C4×C8—44no—
241C4×C8—42no—
242C4×C8C2×Q812yes—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—21no—
255#95—22no—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259C2×Q8—21no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—42no—
269C2×Q8—81no—
270C2×Q8—82no—
271C2×Q8—81no—
272C2×Q8#10314yes—
273C2×Q8C4×Q812yes—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—81no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4#10318yes—
337C4⋊C4#10314yes—
338C4⋊C4C4×Q812yes—
339C4⋊C4#13112yes—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—21no—
345C42—22no—
346C42—21no—
347C42—22no—
348C42—21no—
349C42—22no—
350C42—21no—
351C42—22no—
352C42—22no—
353C42—41no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—41no—
358C42—42no—
359C42—41no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—81no—
364C42#13112yes—
365C42#13112yes—
366C42C2×SD1614yes—
367C42C2×SD1612yes—
368C42C2×Q1614yes—
369C42C2×Q1612yes—
370C42C22×Q811yescharacteristic
371C42C2×#4111yescharacteristic
372C422−1+411yescharacteristic
373C422−1+411yescharacteristic
374C23C22×C22⋊C411yescharacteristic
375Q8—28no—
376Q8—22no—
377Q8—22no—
378Q8—41no—
379Q8—41no—
380Q8—84no—
381Q8—88no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4#40812yes—
420C2×C4#40812yes—
421C2×C4C2×#10312yes—
422C2×C4C2×#10314yes—
423C2×C4C2×#10314yes—
424C2×C4C2×#10312yes—
425C2×C4#42112yes—
426C2×C4C2×C4×Q811yescharacteristic
427C2×C4#51911yescharacteristic
428C2×C4#54811yescharacteristic
429C2×C4#54811yescharacteristic
430C2×C4D4×Q811yescharacteristic
431C2×C4D4×Q811yescharacteristic
432C2×C4C2×#13111yescharacteristic
433C2×C4C2×#13111yescharacteristic
434C2×C4C2×#13111yescharacteristic
435C2×C4#56811yescharacteristic
436C2×C4C22×SD1611yescharacteristic
437C2×C4C22×Q1611yescharacteristic
438C2×C4#57611yescharacteristic
439C2×C4#57711yescharacteristic
440C8—42no—
441C8—42no—
442C8—42no—
443C8—82no—
444C8—82no—
445C8—81no—
446C22Q8×C22⋊C411yescharacteristic
447C22C2×#40811yescharacteristic
448C22#288011yescharacteristic
449C22C22×#10311yescharacteristic
450C22#289711yescharacteristic
451C22C2×#42111yescharacteristic
452C22#290111yescharacteristic
453C4—21no—
454C4—21no—
455C4—24no—
456C4—22no—
457C4—22no—
458C4—24no—
459C4—42no—
460C4—42no—
461C4—41no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—81no—
466C4—82no—
467C4—82no—
468C4#148114yes—
469C4#296011yescharacteristic
470C4#296211yescharacteristic
471C4#298511yescharacteristic
472C4#298611yescharacteristic
473C4#336211yescharacteristic
474C4#336311yescharacteristic
475C4Q8×SD1611yescharacteristic
476C4Q8×Q1611yescharacteristic
477C2C2×#148111yescharacteristic
478C2#1746811yescharacteristic
479C2#1750911yescharacteristic
480C2#1752211yescharacteristic
481C2#2250911yescharacteristic
482C2Q8×#10311yescharacteristic
483C2#2252911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socle, prime radicalC23C22×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
143. Q8⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#349502
Q8×C4⋊C82
#213392
#98742
#98702
#98642
#85682
#85642
#81822
C4×Q8⋊C82