Finite Group Databasebeta

← #148002

Subgroups — #148002

Order 512.

858 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#356C212yesmaximal
2C2×#1801C211yesmaximal, characteristic
3#14024C211yesmaximal, characteristic
4C2×#1394C212yesmaximal
5#10763C211yesmaximal, characteristic
6C22×C4⋊Q8C2211yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C2×#423—21no—
9C22×#101C2212yes—
10C22×#101C2212yes—
11C2×#391C411yescharacteristic
12C2×#391C411yescharacteristic
13C2×#391C2211yescharacteristic
14C2×#356—24no—
15C2×#356—24no—
16C2×#356—24no—
17#1801—24no—
18C2×#343—21no—
19C2×#343—21no—
20#1728—21no—
21#1441—21no—
22#1441—21no—
23#1436—21no—
24#1436—21no—
25C4⋊#95C411yescharacteristic
26C4×#95C411yescharacteristic
27#1394—24no—
28C23×Q8C2×C411yescharacteristic
29C23×Q8C2311yescharacteristic
30C22×M16—22no—
31C22×M16—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8Q812yes—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—41no—
43C2×C4×Q8—41no—
44C2×C4×Q8—42no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56#423—24no—
57#423—42no—
58C2×C4⋊C8—41no—
59C2×C4⋊C8—41no—
60C2×#101—22no—
61C2×#101—22no—
62C2×#101—22no—
63C2×#101—22no—
64C2×#101—22no—
65C2×#101—22no—
66C2×#101—22no—
67C2×#101—22no—
68C2×#101—22no—
69C2×#101—22no—
70C2×#101D414yes—
71C2×#101D414yes—
72#391—22no—
73#391—22no—
74#391—41no—
75#391—41no—
76#391—42no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87#356—216no—
88#343—24no—
89#343—42no—
90#336—41no—
91#336—41no—
92C22×Q8—21no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8C4⋊C412yes—
104C22×Q8C2×D411yescharacteristic
105C22×Q8C2×D412yes—
106C22×Q8C2×D412yes—
107C22×Q8C2×Q811yescharacteristic
108C22×Q8#4111yescharacteristic
109C22×Q8#4111yescharacteristic
110C22×Q8#4112yes—
111C22×Q8#4112yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—42no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—42no—
133C2×M16—42no—
134C2×M16—42no—
135C22×C8—42no—
136C22×C8—42no—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4×Q8—44no—
146C4×Q8—44no—
147C4×Q8—44no—
148C4×Q8—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42C4⋊C412yes—
192C4⋊C8—44no—
193C4⋊C8—44no—
194#101—24no—
195#101—24no—
196#101—24no—
197#101—24no—
198#101—42no—
199#101—42no—
200#101—42no—
201#101—42no—
202#95—24no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—42no—
208#95—42no—
209#95—41no—
210#95—41no—
211#95—42no—
212#95—41no—
213#95—41no—
214C24C2×C22⋊C411yescharacteristic
215C2×Q8—24no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8#12012yes—
242C2×Q8#12112yes—
243C2×Q8C22⋊Q812yes—
244C2×Q8C22⋊Q812yes—
245C2×Q8#12312yes—
246C2×Q8#12312yes—
247C2×Q8#12412yes—
248C2×Q8#12612yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4#9911yescharacteristic
308C22×C4#9911yescharacteristic
309C22×C4#9912yes—
310C22×C4C2×C4⋊C411yescharacteristic
311C22×C4#11711yescharacteristic
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4#12011yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4C22⋊Q811yescharacteristic
320C22×C4C22⋊Q811yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4#12411yescharacteristic
323C22×C4#12711yescharacteristic
324M16—44no—
325M16—44no—
326M16—44no—
327M16—44no—
328M16—82no—
329M16—82no—
330M16—82no—
331M16—82no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—82no—
335C2×C8—82no—
336C2×C8—82no—
337C2×C8—82no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—82no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—24no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—82no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23#38011yescharacteristic
365C23#38611yescharacteristic
366C23#39011yescharacteristic
367C23C2×#9911yescharacteristic
368C23C2×#9911yescharacteristic
369C23#41011yescharacteristic
370C23#41011yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—42no—
374Q8—42no—
375Q8—42no—
376Q8—42no—
377Q8—42no—
378Q8—42no—
379Q8—84no—
380Q8—84no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4#35612yescomplemented
460C2×C4#38512yes—
461C2×C4#38812yes—
462C2×C4#39211yescharacteristic
463C2×C4#39311yescharacteristic
464C2×C4#39412yes—
465C2×C4#39612yes—
466C2×C4#39711yescharacteristic
467C2×C4#39911yescharacteristic
468C8—84no—
469C8—84no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22#143911yescharacteristic
483C22#179911yescharacteristic
484C22#179911yescharacteristic
485C22#189811yescharacteristic
486C22#189911yescharacteristic
487C22#196411yescharacteristic
488C22#196411yescharacteristic
489C22C2×#35611yescharacteristic
490C22#213211yescharacteristic
491C22#213311yescharacteristic
492C22#213511yescharacteristic
493C4—21no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—21no—
498C4—22no—
499C4—21no—
500C4—21no—
501C4—22no—
502C4—44no—
503C4—41no—
504C4—42no—
505C4—82no—
506C4—82no—
507C4—81no—
508C4—81no—
509C4—161no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—21no—
514C2#1104311yescharacteristic
515C2#1280211yescharacteristic
516C2#1280411yescharacteristic
517C2#1309411yescharacteristic
518C2#1309511yescharacteristic
519C2#1322711yescharacteristic
520C2#1322811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
459. C2×C4#356semidirect

Maximal subgroups

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

subgroupindex
C22×#3562
C2×#18012
#140242
C2×#13942
#107632