Finite Group Databasebeta

← #416253

Subgroups — #416253

Order 512.

944 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1950C211yesmaximal, characteristic
2#21172C211yesmaximal, characteristic
3#20755C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5C2×#1801C211yesmaximal, characteristic
6C4×#419C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11249C212yesmaximal
9#11145C212yesmaximal
10#11128C212yesmaximal
11#10858C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×#419C2211yescharacteristic
19C2×#419C2211yescharacteristic
20C22×#103C2212yes—
21C22×#101C2212yes—
22C2×C4×M16C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1950—24no—
26#1949—22no—
27#1949—22no—
28#1832—22no—
29#1831—22no—
30#1805—22no—
31#1805—22no—
32#1801—24no—
33#1763—22no—
34C4×#103—22no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2212yes—
39#1736C2211yescharacteristic
40#1736C2212yes—
41#1736C2211yescharacteristic
42C2×#340C2212yes—
43#1729C2212yes—
44#1729C2212yes—
45C23×Q8C2311yescharacteristic
46C22×M16C2312yes—
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C23×C8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8Q814yes—
59C2×#117C2311yescharacteristic
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C2×#108—24no—
67C2×#108—24no—
68C2×#108—42no—
69#428—42no—
70#428—42no—
71C2×#107—22no—
72C2×#106—22no—
73#423—22no—
74#423—22no—
75#423—22no—
76#423D414yes—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8D412yes—
84C2×C4⋊C8D412yes—
85#419—22no—
86#419—22no—
87#419—22no—
88#419—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#101—24no—
95C2×#101—24no—
96C2×#101—42no—
97C4×M16—22no—
98C4×M16—22no—
99C2×#97—21no—
100C2×C4×C8—21no—
101#382—21no—
102#382—21no—
103#382—21no—
104#382—21no—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109#376—21no—
110C2×#95—21no—
111C2×#95—21no—
112C43—21no—
113#343—24no—
114#343—24no—
115#343—42no—
116#341—24no—
117#340—24no—
118#340—24no—
119#340—22no—
120#340—22no—
121#336—24no—
122#336—24no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8C2×Q812yes—
135C22×Q8#4112yes—
136C22×Q8#4112yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4#4111yescharacteristic
141C23×C4#4111yescharacteristic
142C23×C4C2411yescharacteristic
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—22no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—21no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16C2×D414yes—
165C2×M16#4112yes—
166C2×M16#4112yes—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C2×D414yes—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186#117—21no—
187#117—21no—
188#117—21no—
189#117—21no—
190#117—21no—
191#117—21no—
192#117—21no—
193#117—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×Q812yes—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—42no—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#108—48no—
231#107—44no—
232#106—44no—
233C4⋊C8—24no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239#103—28no—
240#103—28no—
241#103—48no—
242#101—48no—
243#97—42no—
244C4×C8—42no—
245#95—22no—
246#95—22no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24C2×#4111yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8#12514yes—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4#12612yes—
307C22×C4#12612yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×Q811yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C42−1+411yescharacteristic
321C22×C42−1+411yescharacteristic
322M16—24no—
323M16—22no—
324M16—24no—
325M16—22no—
326M16—22no—
327M16—22no—
328M16—41no—
329M16—41no—
330M16—41no—
331M16—42no—
332M16—41no—
333M16—42no—
334M16—42no—
335M16—42no—
336M16—42no—
337M16—42no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C42—44no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—44no—
381C42—44no—
382C42—82no—
383C42#12112yes—
384C42#12112yes—
385C42#12112yes—
386C42#12112yes—
387C42C22⋊Q812yes—
388C42C22⋊Q812yes—
389C42C22⋊Q812yes—
390C42C22⋊Q812yes—
391C23—22no—
392C23—22no—
393C23C2×#12111yescharacteristic
394C23C2×C22⋊Q811yescharacteristic
395C23C2×#12611yescharacteristic
396C23#55211yescharacteristic
397C23#55311yescharacteristic
398C23#55511yescharacteristic
399C23#55611yescharacteristic
400Q8—24no—
401Q8—24no—
402Q8—42no—
403Q8—42no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407Q8—42no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—162no—
437C2×C4—162no—
438C2×C4#39612yes—
439C2×C4#39612yes—
440C2×C4#39612yes—
441C2×C4#39612yes—
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12111yescharacteristic
444C2×C4C2×C22⋊Q811yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4#52511yescharacteristic
447C2×C4#52511yescharacteristic
448C2×C4C2×#12511yescharacteristic
449C2×C4#52911yescharacteristic
450C2×C4#52911yescharacteristic
451C2×C4#53311yescharacteristic
452C2×C4#53511yescharacteristic
453C2×C4#53611yescharacteristic
454C2×C4#54211yescharacteristic
455C2×C4#54211yescharacteristic
456C2×C4#54411yescharacteristic
457C2×C4#54411yescharacteristic
458C2×C4#54611yescharacteristic
459C2×C4#54611yescharacteristic
460C2×C4#54611yescharacteristic
461C2×C4#54711yescharacteristic
462C2×C4#54811yescharacteristic
463C2×C4#54811yescharacteristic
464C2×C4#54811yescharacteristic
465C2×C4#55011yescharacteristic
466C2×C4#55211yescharacteristic
467C2×C4#55211yescharacteristic
468C2×C4#55311yescharacteristic
469C2×C4#55311yescharacteristic
470C8—44no—
471C8—44no—
472C8—42no—
473C8—42no—
474C8—82no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22C2×#39611yescharacteristic
484C22#241411yescharacteristic
485C22#241511yescharacteristic
486C22#243911yescharacteristic
487C22#244211yescharacteristic
488C22#249411yescharacteristic
489C22#249511yescharacteristic
490C22#306712yes—
491C22#306912yes—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—44no—
497C4—82no—
498C4—82no—
499C4—82no—
500C4—162no—
501C4#242611yescharacteristic
502C4#242711yescharacteristic
503C4#244111yescharacteristic
504C4#244211yescharacteristic
505C4#253511yescharacteristic
506C4#253611yescharacteristic
507C4#262811yescharacteristic
508C4#285111yescharacteristic
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2#1522011yescharacteristic
514C2#2360111yescharacteristic
515C2#2360311yescharacteristic
516C2#2467512yes—
517C2#2472512yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19502
#211722
#207552
C2×#18052
C2×#18012
C4×#4192
#113032
#112492
#111452
#111282
#108582