Finite Group Databasebeta

← #224526

Subgroups — #224526

Order 512.

868 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2#22627C212yesmaximal, complemented
3#9654C212yesmaximal, complemented
4#9652C212yesmaximal
5#9651C212yesmaximal
6#8562C212yesmaximal, complemented
7#8559C211yesmaximal, characteristic
8#8154C212yesmaximal, complemented
9#8152C211yesmaximal, characteristic
10#2494—22no—
11#2494—22no—
12#2494—22no—
13#2494—22no—
14C2×#401C412yes—
15C2×#397C412yes—
16C2×#389C411yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#388C2212yes—
19C2×#387C2212yes—
20C2×#382C411yescharacteristic
21C2×#380C411yescharacteristic
22C2×#380C411yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1940—22no—
26#1917—22no—
27#1916—22no—
28#1797C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1771—22no—
32#1749C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2212yes—
36C2×#324C2212yes—
37C2×#324C2212yes—
38C2×#324C2212yes—
39C2×#323C2212yes—
40C2×#323C2211yescharacteristic
41C2×#323C2211yescharacteristic
42#1280C2212yes—
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45C24×C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2311yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2312yes—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8C2312yes—
63C2×C22⋊C8C2312yes—
64C2×C22⋊C8C2312yes—
65C2×C22⋊C8C2312yes—
66C2×#97—22no—
67C2×#97—22no—
68C2×C4×C8—22no—
69#401—22no—
70#401—22no—
71#401—22no—
72#401—22no—
73#397—22no—
74#397—22no—
75#397—22no—
76#397—22no—
77#389—22no—
78#389—21no—
79#389—21no—
80#389—22no—
81#389—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#382—22no—
91#382—21no—
92#382—21no—
93#380—22no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—22no—
99C4×C4⋊C4D412yes—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#324—24no—
121#324—24no—
122#324—42no—
123#324—42no—
124#323—24no—
125#323—41no—
126#323—41no—
127C25C22×C411yescharacteristic
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C412yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—82no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—82no—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C4⋊C8—44no—
199C4⋊C8—82no—
200C4⋊C8—82no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8#4114yes—
209#97—44no—
210#97—82no—
211C4×C8—82no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24#13112yes—
241C24C23×C411yescharacteristic
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C2×#4112yes—
304C22×C4H(5,𝔽2)11yescharacteristic
305C22×C42−1+411yescharacteristic
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—24no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23C22×C22⋊C411yescharacteristic
377C23C4×#4111yescharacteristic
378C23#52011yescharacteristic
379C23C2×#13111yescharacteristic
380C23#56811yescharacteristic
381C23#56811yescharacteristic
382C23#56811yescharacteristic
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4#41012yes—
445C2×C4#41312yes—
446C2×C4#54812yes—
447C2×C4#55012yes—
448C8—44no—
449C8—82no—
450C8—82no—
451C8—82no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22#234111yescharacteristic
476C22#287911yescharacteristic
477C22#288011yescharacteristic
478C22C2×#41011yescharacteristic
479C22#288311yescharacteristic
480C22C2×#41311yescharacteristic
481C22#288911yescharacteristic
482C22#296711yescharacteristic
483C22#296811yescharacteristic
484C22#299211yescharacteristic
485C22#299511yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—82no—
496C4—82no—
497C4—81no—
498C4—81no—
499C4—81no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—82no—
505C2#1739811yescharacteristic
506C2#1740711yescharacteristic
507C2#1741311yescharacteristic
508C2#1742411yescharacteristic
509C2#2253311yescharacteristic
510C2#2253511yescharacteristic
511C2#2253711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22627C2semidirect
3. #9654C2semidirect
6. #8562C2semidirect
8. #8154C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
#226272
#96542
#96522
#96512
#85622
#85592
#81542
#81522