Finite Group Databasebeta

← #224101

Subgroups — #224101

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34941C211yesmaximal, characteristic
2#22821C211yesmaximal, characteristic, complemented
3#21339C211yesmaximal, characteristic, complemented
4#9875C212yesmaximal, complemented
5#9870C211yesmaximal, characteristic, complemented
6#9866C212yesmaximal, complemented
7#9864C211yesmaximal, characteristic, complemented
8#8569C212yesmaximal
9#8563C211yesmaximal, characteristic
10#8183C212yesmaximal
11C4×D4⋊C8C211yesmaximal, characteristic
12#2574—22no—
13#2567C411yescharacteristic
14#2563C411yescharacteristic
15#2552C411yescharacteristic
16#2538C411yescharacteristic
17#2508C412yes—
18#2497—21no—
19#2454C412yes—
20#2423—21no—
21#2396—22no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×#125C2212yes—
24C4×C22⋊Q8—21no—
25#2293C2212yes—
26#2285—21no—
27C42×Q8C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1916—22no—
33#1916—22no—
34#1842C2211yescharacteristic
35#1831C2212yes—
36C8×C4⋊C4—21no—
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2211yescharacteristic
45C2×D4⋊C8C2211yescharacteristic
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117—22no—
64C22×C42—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—22no—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79#400—22no—
80#399—22no—
81#398—22no—
82#398—22no—
83#396—22no—
84#396—22no—
85#395—22no—
86#393—22no—
87#391—21no—
88#391—21no—
89#390C2×C412yes—
90#390C2×C412yes—
91#389—21no—
92#389—21no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#384—21no—
105#384—21no—
106#384C2×C411yescharacteristic
107#382—22no—
108#382C2×C412yes—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2312yes—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—21no—
130C43C2312yes—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140D4⋊C8—24no—
141D4⋊C8—22no—
142D4⋊C8Q814yescomplemented
143C22×Q8—21no—
144C22×Q8C22⋊C411yescharacteristic
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157#125—24no—
158#125—42no—
159C22⋊Q8—44no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—42no—
165C4×Q8—81no—
166C4×Q8—81no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—41no—
173C4×D4C2×Q812yes—
174C4×D4#4112yes—
175#117—82no—
176#117—82no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—22no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—22no—
229C4⋊C8—42no—
230C4⋊C8—44no—
231C4⋊C8—42no—
232C4⋊C8—81no—
233C4⋊C8C2×Q812yes—
234C4×C8—21no—
235C4×C8—22no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—22no—
239C4×C8—42no—
240C4×C8—42no—
241C4×C8—42no—
242C4×C8—82no—
243C4×C8C2×Q812yes—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258C24—21no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—42no—
263C2×Q8—81no—
264C2×Q8—81no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4C4×Q812yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—81no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—41no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—81no—
340C4⋊C4—82no—
341C4⋊C4C4×Q812yes—
342C4⋊C4#13112yes—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—42no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—21no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—21no—
361C42—22no—
362C42—21no—
363C42—22no—
364C42—41no—
365C42—42no—
366C42—42no—
367C42—41no—
368C42—41no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42#13112yes—
374C42#13112yes—
375C42#13512yes—
376C42#13512yes—
377C42C22×Q811yescharacteristic
378C42C2×#4111yescharacteristic
379C422−1+411yescharacteristic
380C422−1+411yescharacteristic
381C23—22no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23C22×C22⋊C411yescharacteristic
389Q8—44no—
390Q8—84no—
391D4—22no—
392D4—22no—
393D4—41no—
394D4—41no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4#40812yes—
438C2×C4#40812yes—
439C2×C4#41612yes—
440C2×C4#41612yes—
441C2×C4#42112yes—
442C2×C4C2×C4×Q811yescharacteristic
443C2×C4#51911yescharacteristic
444C2×C4#54811yescharacteristic
445C2×C4#54811yescharacteristic
446C2×C4D4×Q811yescharacteristic
447C2×C4D4×Q811yescharacteristic
448C2×C4C2×#13111yescharacteristic
449C2×C4C2×#13111yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4#56811yescharacteristic
452C2×C4C2×#13511yescharacteristic
453C2×C4C2×#13511yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57711yescharacteristic
456C8—42no—
457C8—42no—
458C8—42no—
459C8—82no—
460C8—81no—
461C8—82no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22Q8×C22⋊C411yescharacteristic
470C22C2×#40811yescharacteristic
471C22#288011yescharacteristic
472C22C2×#41611yescharacteristic
473C22#289711yescharacteristic
474C22C2×#42111yescharacteristic
475C22#290111yescharacteristic
476C4—21no—
477C4—21no—
478C4—22no—
479C4—22no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—41no—
485C4—42no—
486C4—42no—
487C4—81no—
488C4—82no—
489C4—82no—
490C4#148214yes—
491C4#296011yescharacteristic
492C4#296211yescharacteristic
493C4#298511yescharacteristic
494C4#298611yescharacteristic
495C4#336011yescharacteristic
496C4#336111yescharacteristic
497C4#338011yescharacteristic
498C4#338111yescharacteristic
499C2—82no—
500C2C2×#148211yescharacteristic
501C2#1746811yescharacteristic
502C2#1750811yescharacteristic
503C2#1752611yescharacteristic
504C2#2250911yescharacteristic
505C2#2251911yescharacteristic
506C2#2252911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22×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
2. #22821C2semidirect
3. #21339C2semidirect
4. #9875C2semidirect
5. #9870C2semidirect
6. #9866C2semidirect
7. #9864C2semidirect
142. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#349412
#228212
#213392
#98752
#98702
#98662
#98642
#85692
#85632
#81832
C4×D4⋊C82