Finite Group Databasebeta

← #224115

Subgroups — #224115

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34951C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic
3C4×C8×Q8C211yesmaximal, characteristic
4#9871C212yesmaximal
5#9870C211yesmaximal, characteristic
6#9867C212yesmaximal
7#9865C211yesmaximal, characteristic
8#8570C212yesmaximal
9#8564C211yesmaximal, characteristic
10#8176C212yesmaximal
11C4×Q8⋊C8C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2585—22no—
14#2582C411yescharacteristic
15#2568C411yescharacteristic
16#2552C411yescharacteristic
17#2546C411yescharacteristic
18#2511—21no—
19#2509C412yes—
20#2453C412yes—
21#2446—21no—
22#2402—22no—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#125C2212yes—
26#2295—21no—
27#2295C2212yes—
28C42×Q8C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1917—22no—
32#1842C2211yescharacteristic
33#1831C2212yes—
34#1770—21no—
35C8×C4⋊C4—22no—
36C8×C4⋊C4—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C42×C8C2211yescharacteristic
42C42×C8C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2211yescharacteristic
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C8×Q8—28no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2311yescharacteristic
71C2×#97—22no—
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75C2×C4×C8—22no—
76C2×C4×C8—21no—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2312yes—
79C2×C4×C8C2311yescharacteristic
80#400—22no—
81#400—22no—
82#400—22no—
83#400—22no—
84#398—22no—
85#398—22no—
86#398—22no—
87#398—22no—
88#391—21no—
89#391—21no—
90#391—21no—
91#391—21no—
92#391—21no—
93#391C2×C412yes—
94#391C2×C412yes—
95#389—22no—
96#389—22no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2×C412yes—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#384—21no—
112#384C2×C411yescharacteristic
113#382C2×C412yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
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⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2312yes—
131#376—22no—
132C43—21no—
133C43C2312yes—
134C43C2311yescharacteristic
135#336—22no—
136#336—22no—
137#336—21no—
138#336—22no—
139#336—21no—
140#336—22no—
141Q8⋊C8—24no—
142Q8⋊C8—22no—
143Q8⋊C8Q814yescomplemented
144C22×Q8—21no—
145C22×Q8C22⋊C411yescharacteristic
146C22×Q8C22×C411yescharacteristic
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—44no—
156#125—24no—
157#125—42no—
158C4×Q8—24no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—41no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8C2×Q812yes—
172C4×Q8#4112yes—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C412yes—
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×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2411yescharacteristic
217C4⋊C8—28no—
218C4⋊C8—24no—
219C4⋊C8—21no—
220C4⋊C8—22no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—22no—
224C4⋊C8—42no—
225C4⋊C8—81no—
226C4⋊C8C2×Q812yes—
227#97—82no—
228C4×C8—28no—
229C4×C8—24no—
230C4×C8—22no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—22no—
235C4×C8—42no—
236C4×C8—44no—
237C4×C8—42no—
238C4×C8C2×Q812yes—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—21no—
253#95—22no—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8#10414yes—
268C2×Q8C4×Q812yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—41no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—81no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4C4×Q812yes—
333C4⋊C4#13112yes—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—21no—
339C42—22no—
340C42—21no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—21no—
346C42—21no—
347C42—22no—
348C42—24no—
349C42—22no—
350C42—41no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—42no—
358C42—42no—
359C42—81no—
360C42#13112yes—
361C42#13112yes—
362C42#13512yes—
363C42#13512yes—
364C42C22×Q811yescharacteristic
365C42C2×#4111yescharacteristic
366C422−1+411yescharacteristic
367C422−1+411yescharacteristic
368C23C22×C22⋊C411yescharacteristic
369Q8—28no—
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374Q8—84no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4#40812yes—
414C2×C4#40812yes—
415C2×C4#41612yes—
416C2×C4#41612yes—
417C2×C4C2×#10414yes—
418C2×C4C2×#10412yes—
419C2×C4C2×C4×Q811yescharacteristic
420C2×C4#51911yescharacteristic
421C2×C4#54811yescharacteristic
422C2×C4#54811yescharacteristic
423C2×C4D4×Q811yescharacteristic
424C2×C4D4×Q811yescharacteristic
425C2×C4C2×#13111yescharacteristic
426C2×C4C2×#13111yescharacteristic
427C2×C4C2×#13111yescharacteristic
428C2×C4#56811yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57811yescharacteristic
433C8—24no—
434C8—44no—
435C8—42no—
436C8—42no—
437C8—81no—
438C8—82no—
439C22Q8×C22⋊C411yescharacteristic
440C22C2×#40811yescharacteristic
441C22#288011yescharacteristic
442C22C2×#41611yescharacteristic
443C22#289811yescharacteristic
444C22C22×#10411yescharacteristic
445C22#290111yescharacteristic
446C4—21no—
447C4—21no—
448C4—24no—
449C4—22no—
450C4—22no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—41no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—82no—
459C4—81no—
460C4—82no—
461C4#149414yes—
462C4#296011yescharacteristic
463C4#296211yescharacteristic
464C4#298511yescharacteristic
465C4#298611yescharacteristic
466C4#336411yescharacteristic
467C4#336511yescharacteristic
468C4#338111yescharacteristic
469C4#338311yescharacteristic
470C2C2×#149411yescharacteristic
471C2#1748411yescharacteristic
472C2#1751011yescharacteristic
473C2#1751811yescharacteristic
474C2#2250911yescharacteristic
475C2#2252111yescharacteristic
476C2Q8×#10411yescharacteristic
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
#349512
#228252
C4×C8×Q82
#98712
#98702
#98672
#98652
#85702
#85642
#81762
C4×Q8⋊C82