Finite Group Databasebeta

← #224121

Subgroups — #224121

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34950C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic
3#21339C211yesmaximal, characteristic
4#9873C212yesmaximal
5#9870C211yesmaximal, characteristic
6#9869C212yesmaximal
7#9864C211yesmaximal, characteristic
8#8570C212yesmaximal
9#8564C211yesmaximal, characteristic
10#8178C212yesmaximal
11C4×Q8⋊C8C211yesmaximal, characteristic
12#2586—22no—
13#2582C411yescharacteristic
14#2567C411yescharacteristic
15#2567C411yescharacteristic
16#2551C411yescharacteristic
17#2507—21no—
18#2507C412yes—
19#2451—21no—
20#2451C412yes—
21#2399—22no—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2212yes—
25#2295—21no—
26#2295C2212yes—
27C42×Q8C2211yescharacteristic
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1916—22no—
34#1842C2211yescharacteristic
35#1831C2212yes—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
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#1279C2212yes—
50#1279C2211yescharacteristic
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
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—21no—
74C2×C4×C8—21no—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78#398—22no—
79#398—22no—
80#398—22no—
81#398—22no—
82#395—22no—
83#395—22no—
84#395—22no—
85#395—22no—
86#391—21no—
87#391—21no—
88#391—21no—
89#391—22no—
90#391—22no—
91#391C2×C412yes—
92#391C2×C412yes—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—22no—
98#389—21no—
99#389—22no—
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#384—21no—
110#384C2×C412yes—
111#384C2×C411yescharacteristic
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2312yes—
128#376—22no—
129C43—21no—
130C43C2312yes—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—21no—
139#336—22no—
140Q8⋊C8—24no—
141Q8⋊C8—22no—
142Q8⋊C8Q814yescomplemented
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8C22⋊C411yescharacteristic
146C22×Q8C22×C411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C4⋊Q8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—42no—
157C4⋊Q8—44no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—41no—
168C4×Q8—42no—
169C4×Q8—82no—
170C4×Q8—81no—
171C4×Q8—82no—
172C4×Q8—81no—
173C4×Q8C2×Q812yes—
174C4×Q8#4112yes—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊C411yescharacteristic
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×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220C4⋊C8—21no—
221C4⋊C8—22no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—22no—
225C4⋊C8—42no—
226C4⋊C8—44no—
227C4⋊C8—42no—
228C4⋊C8—81no—
229C4⋊C8C2×Q812yes—
230#97—82no—
231C4×C8—21no—
232C4×C8—22no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—22no—
236C4×C8—42no—
237C4×C8—42no—
238C4×C8—42no—
239C4×C8C2×Q812yes—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—22no—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—42no—
267C2×Q8—81no—
268C2×Q8—82no—
269C2×Q8—81no—
270C2×Q8C4×Q812yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—81no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4C4×Q812yes—
336C4⋊C4#13112yes—
337C42—22no—
338C42—22no—
339C42—21no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—21no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—21no—
349C42—21no—
350C42—22no—
351C42—22no—
352C42—41no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—41no—
357C42—42no—
358C42—41no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—81no—
363C42#13112yes—
364C42#13112yes—
365C42#13712yes—
366C42#13712yes—
367C42C22×Q811yescharacteristic
368C42C2×#4111yescharacteristic
369C422−1+411yescharacteristic
370C422−1+411yescharacteristic
371C23C22×C22⋊C411yescharacteristic
372Q8—22no—
373Q8—22no—
374Q8—41no—
375Q8—41no—
376Q8—44no—
377Q8—84no—
378Q8—88no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4#40812yes—
419C2×C4#40812yes—
420C2×C4#41912yes—
421C2×C4#41912yes—
422C2×C4#42112yes—
423C2×C4C2×C4×Q811yescharacteristic
424C2×C4#51911yescharacteristic
425C2×C4#54811yescharacteristic
426C2×C4#54811yescharacteristic
427C2×C4D4×Q811yescharacteristic
428C2×C4D4×Q811yescharacteristic
429C2×C4C2×#13111yescharacteristic
430C2×C4C2×#13111yescharacteristic
431C2×C4C2×#13111yescharacteristic
432C2×C4#56811yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57611yescharacteristic
436C2×C4#57711yescharacteristic
437C8—42no—
438C8—42no—
439C8—42no—
440C8—82no—
441C8—81no—
442C8—82no—
443C22Q8×C22⋊C411yescharacteristic
444C22C2×#40811yescharacteristic
445C22#288011yescharacteristic
446C22C2×#41911yescharacteristic
447C22#289711yescharacteristic
448C22C2×#42111yescharacteristic
449C22#290111yescharacteristic
450C4—21no—
451C4—21no—
452C4—22no—
453C4—22no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—41no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—81no—
463C4—82no—
464C4—82no—
465C4#149714yes—
466C4#296011yescharacteristic
467C4#296211yescharacteristic
468C4#298511yescharacteristic
469C4#298611yescharacteristic
470C4#336911yescharacteristic
471C4#337511yescharacteristic
472C4#338511yescharacteristic
473C4#338711yescharacteristic
474C2C2×#149711yescharacteristic
475C2#1748311yescharacteristic
476C2#1751411yescharacteristic
477C2#1752211yescharacteristic
478C2#2250911yescharacteristic
479C2#2252511yescharacteristic
480C2#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
142. Q8⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#349502
#228252
#213392
#98732
#98702
#98692
#98642
#85702
#85642
#81782
C4×Q8⋊C82