Finite Group Databasebeta

← #196584

Subgroups — #196584

Order 512.

1192 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
1C2×#2970C211yesmaximal, characteristic
2#35334C211yesmaximal, characteristic
3#21380C211yesmaximal, characteristic
4C2×#1537C211yesmaximal, characteristic, complemented
5C2×#1526C211yesmaximal, characteristic, complemented
6#9126C214yesmaximal, complemented
7#9121C214yesmaximal, complemented
8#8033C211yesmaximal, characteristic, complemented
9#8033C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2970—24no—
13#2970—24no—
14C2×#446—21no—
15C2×#435—22no—
16C2×C8×D4—21no—
17C2×#432C2212yes—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C22×#97C2212yes—
23#2851C2214yes—
24#2628C412yes—
25#2628C412yes—
26#2581—21no—
27#2581—21no—
28#2580—21no—
29#2580—21no—
30#2564—22no—
31#2564—22no—
32#2316C412yes—
33#2311C412yes—
34C2×#382C2211yescharacteristic
35#1984—21no—
36#1954—21no—
37#1870—22no—
38#1749C2214yes—
39#1537—24no—
40#1526—24no—
41C2×#330C2212yes—
42C2×#330C2214yes—
43C2×#330C2212yes—
44#1295C2212yes—
45#1295C2214yes—
46#1295C2212yes—
47C22×#41C2×C411yescharacteristic
48C22×M16—21no—
49C22×M16—21no—
50C23×C8—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125C2312yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C4×#41—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×D4D412yes—
69C2×C4×D4D412yes—
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×#117C2×C412yes—
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C412yes—
75C22×C4⋊C4C2×C412yes—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#446—28no—
79#435—216no—
80C8×D4—28no—
81#432—22no—
82#432—22no—
83#432—22no—
84#432—24no—
85#432—24no—
86#432D414yes—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—24no—
94C4×M16—22no—
95C4×M16—22no—
96C4×M16—22no—
97C4×M16D414yes—
98C2×#97—22no—
99C2×#97—24no—
100C2×#97—24no—
101C2×#97C2312yes—
102C2×#97C2312yes—
103C2×#97C2311yescharacteristic
104C2×#97C2312yes—
105C2×#97C2311yescharacteristic
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#400—22no—
109#400—22no—
110#399—22no—
111#399—22no—
112#395—22no—
113#395—22no—
114#393—22no—
115#393—22no—
116#391—22no—
117#390—22no—
118#389—22no—
119#389—22no—
120#389C2×C414yes—
121#389C2×C414yes—
122#387—24no—
123#387—24no—
124#386—24no—
125#385—24no—
126#385—24no—
127#382—21no—
128#382—21no—
129#382C2×C412yes—
130#382C2312yes—
131#380—24no—
132#380C2×C414yes—
133#380C2×C414yes—
134C2×#95—21no—
135C2×#95—21no—
136#336—24no—
137#330—28no—
138#330—24no—
139#330—24no—
140C2×#41—24no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—42no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8C22⋊C411yescharacteristic
147C22×D4C22⋊C412yes—
148C22×D4C22⋊C411yescharacteristic
149C23×C4—22no—
150C23×C4C22×C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C2×M16—24no—
157C2×M16—44no—
158C2×M16—44no—
159C2×M16—42no—
160C22×C8—21no—
161C22×C8—28no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—48no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171#125—22no—
172#125—22no—
173#125—22no—
174#125—22no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195#117—22no—
196#117—22no—
197#117—24no—
198#117—24no—
199#117—24no—
200#117—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C414yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C412yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42C22×C412yes—
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42C2411yescharacteristic
238C4⋊C8—28no—
239C4⋊C8—24no—
240C4⋊C8—42no—
241C22⋊C8—28no—
242C22⋊C8—24no—
243C22⋊C8—48no—
244C22⋊C8—42no—
245#97—28no—
246#97—21no—
247#97—22no—
248#97—21no—
249#97—22no—
250#97—22no—
251#97—22no—
252#97—22no—
253#97C2×D412yes—
254#97C2×D414yes—
255#97C2×D412yes—
256C4×C8—24no—
257C4×C8—42no—
258#95—22no—
259#95—22no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—22no—
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C412yes—
268#41—48no—
269#41—48no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4#10114yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4#12012yes—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4#12312yes—
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318M16—48no—
319M16—48no—
320C2×C8—24no—
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—416no—
326C2×C8—44no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—44no—
331C2×C8—42no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—44no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—44no—
363C42C4×D412yes—
364C42C4×D412yes—
365C42#12012yes—
366C42#12112yes—
367C42#12112yes—
368C42#12312yes—
369C42C22×D411yescharacteristic
370C42C22×D411yescharacteristic
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C23—21no—
374C23—21no—
375C23—22no—
376C23—24no—
377C23—24no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23#38612yes—
382C23C2×#10114yes—
383C23C22×C22⋊C411yescharacteristic
384Q8—44no—
385Q8—84no—
386Q8—84no—
387D4—28no—
388D4—48no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—164no—
424C2×C4#38612yes—
425C2×C4#38612yes—
426C2×C4#38612yes—
427C2×C4C2×#10112yes—
428C2×C4#41312yes—
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×#12011yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12111yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C8—24no—
436C8—44no—
437C8—44no—
438C8—42no—
439C8—82no—
440C22—28no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—24no—
445C22—42no—
446C22—42no—
447C22—44no—
448C22—42no—
449C22—42no—
450C22C2×#38611yescharacteristic
451C22C22×#10111yescharacteristic
452C22C2×#41311yescharacteristic
453C22#290112yes—
454C22#290112yes—
455C4—21no—
456C4—21no—
457C4—22no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—84no—
467C4—82no—
468C4—82no—
469C4#190114yes—
470C2—44no—
471C2—42no—
472C2#1768912yes—
473C2#1769112yes—
474C2#2090311yescharacteristic
475C2#2098011yescharacteristic
476C2C2×#190111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1537C2semidirect
5. C2×#1526C2semidirect
6. #9126C2semidirect
7. #9121C2semidirect
8. #8033C2semidirect
9. #8033C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29702
#353342
#213802
C2×#15372
C2×#15262
#91262
#91212
#80332