Finite Group Databasebeta

← #495190

Subgroups — #495190

Order 512.

1203 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2C2×#2133C212yesmaximal
3C22×#356C212yesmaximal, complemented
4#20639C211yesmaximal, characteristic, complemented
5#20399C211yesmaximal, characteristic, complemented
6#13228C214yesmaximal, complemented
7#13227C214yesmaximal, complemented
8C2×#536C411yescharacteristic
9C2×#536C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2905—21no—
13C22×#101C2212yes—
14C22×#101C2212yes—
15#2873—21no—
16#2546C412yes—
17#2546C2212yes—
18#2545C412yes—
19#2545C2212yes—
20#2429—22no—
21#2429—22no—
22#2413—21no—
23#2413—21no—
24#2413—21no—
25#2413—21no—
26#2310C411yescharacteristic
27#2297C411yescharacteristic
28#2296C411yescharacteristic
29#2133—28no—
30C2×#356—28no—
31C2×#356C2218yes—
32C2×#356C2214yes—
33C2×#355C2214yes—
34#1801—22no—
35#1801C2214yes—
36C4×#101—22no—
37C4×#101C2214yes—
38#1742—21no—
39#1742—21no—
40C22×#41C2×C411yescharacteristic
41C23×Q8C2×C411yescharacteristic
42C23×Q8C2311yescharacteristic
43C22×M16—22no—
44C22×M16—22no—
45#536—24no—
46#536—24no—
47#536—44no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8C2×C412yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C412yes—
55C2×C4⋊Q8C2312yes—
56C2×C4⋊Q8C2311yescharacteristic
57C2×#124C2×C411yescharacteristic
58C2×#124C2×C412yes—
59C2×#124C2311yescharacteristic
60C2×#123—21no—
61C2×#123—21no—
62C2×#123—21no—
63C2×#123—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—22no—
67#519—44no—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8D412yes—
71C2×C4×Q8C2312yes—
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C42C2×C411yescharacteristic
82#424—44no—
83#423—24no—
84C2×#101—22no—
85C2×#101—24no—
86C2×#101—24no—
87C2×#101—24no—
88C2×#101—22no—
89C2×#101D418yes—
90C2×#101C2314yes—
91C2×#101C2314yes—
92#405—44no—
93C4×M16—24no—
94#398—22no—
95#398—22no—
96#398—22no—
97#398—22no—
98#396—22no—
99#396—22no—
100#396—22no—
101#396—22no—
102#391—22no—
103#391C2×C412yes—
104#391C2×C412yes—
105#391C2×C412yes—
106#390C2×C412yes—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#386—21no—
117#386—21no—
118#385—21no—
119#385—21no—
120#381—21no—
121#381—21no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C22⋊C4—22no—
125#356—216no—
126#356—28no—
127#355—28no—
128#343—24no—
129#343—24no—
130C2×#41—24no—
131C2×#41—42no—
132C2×#41—42no—
133C22×Q8—21no—
134C22×Q8—24no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8C22×C412yes—
140C22×Q8C22×C411yescharacteristic
141C22×Q8C22×C411yescharacteristic
142C22×Q8C2×D411yescharacteristic
143C22×Q8C2×D411yescharacteristic
144C22×Q8C2×D414yes—
145C22×Q8#4114yes—
146C22×Q8C2411yescharacteristic
147C22×D4—21no—
148C22×D4C22×C411yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×M16—28no—
156C2×M16—24no—
157C2×M16—24no—
158C2×M16—44no—
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—44no—
162C2×M16—44no—
163C22×C8—42no—
164C22×C8—42no—
165C4⋊Q8—28no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8—24no—
170C4⋊Q8—22no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175#124—24no—
176#124—22no—
177#124—22no—
178#124—44no—
179#123—44no—
180#123—44no—
181#123—42no—
182#123—42no—
183#123—44no—
184C22⋊Q8—48no—
185C22⋊Q8—44no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8—44no—
191C4×Q8—44no—
192#117—24no—
193#117—24no—
194#117—24no—
195#117—24no—
196#117—42no—
197#117—42no—
198#117—42no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4C22⋊C411yescharacteristic
231C2×C22⋊C4C22⋊C411yescharacteristic
232C2×C22⋊C4C22⋊C411yescharacteristic
233C2×C22⋊C4C22⋊C411yescharacteristic
234C2×C42—24no—
235C2×C42—21no—
236C2×C42C22⋊C412yes—
237C2×C42C22⋊C412yes—
238C2×C42C22⋊C412yes—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C412yes—
242C2×C42C22×C412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C4⋊C8—48no—
246#101—28no—
247#101—24no—
248#101—24no—
249#101—24no—
250#101—44no—
251#101C2×D418yes—
252#97—44no—
253C4×C8—44no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262C24—21no—
263C24C2×C22⋊C411yescharacteristic
264#41—84no—
265#41—84no—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—41no—
284C2×Q8—42no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×Q8C4×D414yes—
290C2×Q8#12114yes—
291C2×Q8#12412yes—
292C2×Q8#12712yes—
293C2×Q8C22×D412yes—
294C2×Q8C2×#4112yes—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C4×D414yes—
328C22×C4#12114yes—
329C22×C4#12412yes—
330C22×C4#12712yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334M16—28no—
335M16—44no—
336M16—48no—
337M16—44no—
338M16—44no—
339M16—84no—
340C2×C8—44no—
341C2×C8—48no—
342C2×C8—44no—
343C2×C8—84no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4#9914yes—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—22no—
384C42—22no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23#39012yes—
396C23C2×#9912yes—
397C23#41012yes—
398C23C22×C22⋊C411yescharacteristic
399Q8—44no—
400Q8—44no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404Q8—42no—
405Q8—84no—
406Q8—82no—
407Q8—84no—
408D4—82no—
409D4—84no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—48no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4#39012yes—
452C2×C4#39012yes—
453C2×C4#39012yes—
454C2×C4C2×#9914yes—
455C2×C4C2×C4×D412yes—
456C2×C4C2×#12112yes—
457C2×C4C2×#12411yescharacteristic
458C2×C4C2×#12711yescharacteristic
459C8—48no—
460C8—84no—
461C22—22no—
462C22—24no—
463C22—42no—
464C22—42no—
465C22—81no—
466C22—81no—
467C22—82no—
468C22—82no—
469C22—81no—
470C22#196414yes—
471C22C2×#39011yescharacteristic
472C22C22×#9911yescharacteristic
473C22C2×#41011yescharacteristic
474C4—24no—
475C4—22no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—21no—
480C4—22no—
481C4—22no—
482C4—44no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—81no—
489C4—82no—
490C2—21no—
491C2—21no—
492C2—24no—
493C2—161no—
494C2C2×#196411yescharacteristic
495C2#3337212yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#356C2semidirect
4. #20639C2semidirect
5. #20399C2semidirect
6. #13228C2semidirect
7. #13227C2semidirect

Maximal subgroups

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

subgroupindex
#356672
C2×#21332
C22×#3562
#206392
#203992
#132282
#132272