Finite Group Databasebeta

← #148694

Subgroups — #148694

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1801C211yesmaximal, characteristic
2#13976C211yesmaximal, characteristic
3C2×#1394C212yesmaximal
4C2×#1394C212yesmaximal
5#10763C211yesmaximal, characteristic
6C22×C4×Q8C2211yescharacteristic
7C2×#423—21no—
8C22×#101C2212yes—
9C22×#101C2212yes—
10C2×#391C2211yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#389C411yescharacteristic
13C2×#384C411yescharacteristic
14#1801—24no—
15C2×#343—21no—
16C2×#343—21no—
17#1728—21no—
18#1444—21no—
19#1444—21no—
20#1436—21no—
21#1436—21no—
22C4⋊#95C411yescharacteristic
23C4×#95C411yescharacteristic
24#1394—24no—
25#1394—24no—
26C23×Q8C2311yescharacteristic
27C22×M16—22no—
28C22×M16—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—41no—
32C2×C4×Q8—41no—
33C2×C4×Q8—42no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47#423—24no—
48#423—42no—
49C2×C4⋊C8—41no—
50C2×C4⋊C8—41no—
51C2×#101—22no—
52C2×#101—22no—
53C2×#101—22no—
54C2×#101—22no—
55C2×#101—22no—
56C2×#101—22no—
57C2×#101—22no—
58C2×#101—22no—
59C2×#101—22no—
60C2×#101—22no—
61C2×#101D414yes—
62C2×#101Q814yes—
63#391—21no—
64#391—21no—
65#391—22no—
66#391—22no—
67#391D412yes—
68#391Q812yes—
69#389—41no—
70#389—41no—
71#384—42no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81#343—24no—
82#343—42no—
83#336—41no—
84#336—41no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8—41no—
89C22×Q8C2×D411yescharacteristic
90C22×Q8C2×D412yes—
91C22×Q8C2×Q811yescharacteristic
92C22×Q8C2×Q812yes—
93C22×Q8#4111yescharacteristic
94C22×Q8#4111yescharacteristic
95C22×Q8#4112yes—
96C22×Q8#4112yes—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C2×M16—24no—
109C2×M16—24no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C2×M16—42no—
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—42no—
119C2×M16—42no—
120C22×C8—42no—
121C22×C8—42no—
122C4×Q8—44no—
123C4×Q8—44no—
124C4×Q8—44no—
125C4×Q8—44no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—81no—
148C2×C4⋊C4—81no—
149C2×C4⋊C4—81no—
150C2×C4⋊C4—81no—
151C2×C4⋊C4C4⋊C412yes—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42C4⋊C412yes—
172C4⋊C8—44no—
173C4⋊C8—44no—
174#101—24no—
175#101—24no—
176#101—24no—
177#101—24no—
178#101—42no—
179#101—42no—
180#101—42no—
181#101—42no—
182#95—24no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—42no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—42no—
192#95—41no—
193#95—81no—
194#95—81no—
195C24C2×C22⋊C411yescharacteristic
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—42no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8#12112yes—
213C2×Q8C22⋊Q812yes—
214C2×Q8C22⋊Q812yes—
215C2×Q8#12312yes—
216C2×Q8#12512yes—
217C2×Q8#12512yes—
218C2×Q8#12612yes—
219C2×Q8C4⋊Q812yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4#9911yescharacteristic
279C22×C4#9911yescharacteristic
280C22×C4#9912yes—
281C22×C4C2×C4⋊C411yescharacteristic
282C22×C4#11711yescharacteristic
283C22×C4C4×D412yes—
284C22×C4C4×Q812yes—
285C22×C4#12011yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12112yes—
289C22×C4C22⋊Q811yescharacteristic
290C22×C4C22⋊Q812yes—
291C22×C4C22⋊Q811yescharacteristic
292C22×C4#12311yescharacteristic
293C22×C4#12411yescharacteristic
294C22×C4C4⋊Q811yescharacteristic
295M16—44no—
296M16—44no—
297M16—44no—
298M16—44no—
299M16—82no—
300M16—82no—
301M16—82no—
302M16—82no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—82no—
306C2×C8—82no—
307C2×C8—82no—
308C2×C8—82no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—82no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#38011yescharacteristic
335C23#38611yescharacteristic
336C23#39111yescharacteristic
337C23C2×#9911yescharacteristic
338C23C2×#9911yescharacteristic
339C23#41011yescharacteristic
340C23#41011yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—42no—
344Q8—42no—
345Q8—42no—
346Q8—42no—
347Q8—42no—
348Q8—42no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—24no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4#35612yes—
428C2×C4#38712yes—
429C2×C4#38912yes—
430C2×C4#39411yescharacteristic
431C2×C4#39511yescharacteristic
432C2×C4#39612yes—
433C2×C4#39812yes—
434C2×C4#39811yescharacteristic
435C2×C4#40011yescharacteristic
436C8—84no—
437C8—84no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22#144111yescharacteristic
451C22#179911yescharacteristic
452C22#179911yescharacteristic
453C22#189811yescharacteristic
454C22#189911yescharacteristic
455C22#199611yescharacteristic
456C22#199611yescharacteristic
457C22C2×#35611yescharacteristic
458C22#213211yescharacteristic
459C22#213311yescharacteristic
460C22#213411yescharacteristic
461C4—22no—
462C4—21no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—21no—
467C4—21no—
468C4—22no—
469C4—21no—
470C4—44no—
471C4—42no—
472C4—41no—
473C4—82no—
474C4—82no—
475C4—81no—
476C4—81no—
477C4—161no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2#1120311yescharacteristic
483C2#1280211yescharacteristic
484C2#1280311yescharacteristic
485C2#1309311yescharacteristic
486C2#1309411yescharacteristic
487C2#1330211yescharacteristic
488C2#1330311yescharacteristic
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

Maximal subgroups

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

subgroupindex
C2×#18012
#139762
C2×#13942
#107632