Finite Group Databasebeta

← #148415

Subgroups — #148415

Order 512.

702 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2159C211yesmaximal, characteristic
2C22×#362C211yesmaximal, characteristic
3C2×#2141C211yesmaximal, characteristic
4C2×#1969C211yesmaximal, characteristic
5C2×#1387C211yesmaximal, characteristic
6C2×#1376C211yesmaximal, characteristic
7#10859C211yesmaximal, characteristic
8C23×Q16C2211yescharacteristic
9C22×M32C2211yescharacteristic
10C22×M32C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#103—21no—
15C22×C4×C8C2211yescharacteristic
16C2×#391—21no—
17#2159—22no—
18#2159—22no—
19C2×#367—21no—
20C2×#366—21no—
21C2×#362—22no—
22C2×#362—22no—
23C2×#362C414yescomplemented
24#2141—22no—
25#2141—22no—
26C2×#358—21no—
27C2×#358—21no—
28#1969—22no—
29#1969—22no—
30C2×#339—21no—
31#1729—21no—
32#1387—22no—
33#1376—22no—
34C23×Q8—21no—
35C22×Q16—21no—
36C22×Q16—21no—
37C22×Q16—21no—
38C22×Q16—21no—
39C22×Q16—21no—
40C22×Q16—21no—
41C22×Q16—21no—
42C22×Q16—21no—
43C22×Q16—21no—
44C22×Q16—21no—
45C22×Q16—21no—
46C22×Q16C2×C412yes—
47C22×Q16D412yes—
48C22×Q16Q812yes—
49C22×M16—21no—
50C23×C8D411yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C42D411yescharacteristic
55C2×M32—21no—
56C2×M32—21no—
57C2×M32—21no—
58C2×M32—21no—
59C2×M32—21no—
60C2×M32—21no—
61C2×M32—21no—
62C2×M32—21no—
63C2×M32C2×C412yes—
64C2×M32D412yes—
65C2×M32D412yes—
66C2×M32D412yes—
67C22×C16—21no—
68C22×C16—21no—
69C2×#108—21no—
70C2×#108—21no—
71C2×#108—21no—
72C2×#108—21no—
73C2×#108—21no—
74C2×#108C2×C412yes—
75#427—21no—
76#427—21no—
77#427—21no—
78#427—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107D411yescharacteristic
85C2×#107D411yescharacteristic
86C2×#107D412yes—
87C2×#106—21no—
88C2×#103—24no—
89C2×#103—42no—
90C2×#103—42no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97#391—24no—
98#391—42no—
99C2×#95—42no—
100#367—24no—
101#366—24no—
102#362—28no—
103#358—24no—
104#358—24no—
105#339—42no—
106#336—42no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8—41no—
115C22×Q8—41no—
116C22×Q8—41no—
117C22×Q8—41no—
118C22×Q8—41no—
119C22×Q8—41no—
120C22×Q8—41no—
121C23×C4—21no—
122C23×C4—41no—
123C23×C4—41no—
124C23×C4C2×D411yescharacteristic
125C2×Q16—22no—
126C2×Q16—22no—
127C2×Q16—22no—
128C2×Q16—22no—
129C2×Q16—22no—
130C2×Q16—22no—
131C2×Q16—22no—
132C2×Q16—22no—
133C2×Q16—22no—
134C2×Q16—22no—
135C2×Q16—22no—
136C2×Q16—22no—
137C2×Q16—22no—
138C2×Q16—22no—
139C2×Q16—22no—
140C2×Q16—41no—
141C2×Q16—41no—
142C2×Q16—41no—
143C2×Q16—41no—
144C2×Q16—41no—
145C2×Q16—41no—
146C2×Q16—41no—
147C2×Q16—41no—
148C2×Q16—41no—
149C2×Q16—41no—
150C2×Q16—41no—
151C2×Q16—41no—
152C2×Q16C4⋊C414yes—
153C2×M16—22no—
154C2×M16—41no—
155C2×M16—41no—
156C2×M16—41no—
157C2×M16—41no—
158C2×M16—41no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—41no—
166C22×C8D812yes—
167C22×C8C22×C411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×Q811yescharacteristic
175C22×C8#4111yescharacteristic
176C22×C8#4111yescharacteristic
177C22×C8#4111yescharacteristic
178C22×C8#4111yescharacteristic
179C22×C8#4111yescharacteristic
180C22×C8#4111yescharacteristic
181#117—41no—
182#117—41no—
183#117—41no—
184#117—41no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—41no—
200C2×C42D812yes—
201M32—22no—
202M32—22no—
203M32—22no—
204M32—22no—
205M32—22no—
206M32—41no—
207M32—41no—
208M32C22⋊C414yes—
209C2×C16—22no—
210C2×C16—22no—
211C2×C16—22no—
212C2×C16—22no—
213C2×C16—41no—
214C2×C16—41no—
215#108—22no—
216#108—22no—
217#108—41no—
218#108—41no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—22no—
223#107—21no—
224#107—22no—
225#107—41no—
226#107—41no—
227#106—41no—
228#106—41no—
229#103—48no—
230C4×C8—41no—
231C4×C8—41no—
232C4×C8—41no—
233C4×C8—41no—
234#95—48no—
235C24AGL(1,ℤ/8)11yescharacteristic
236C2×Q8—24no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—81no—
253C2×Q8—81no—
254C2×Q8—81no—
255C2×Q8—81no—
256C2×Q8—81no—
257C2×Q8—81no—
258C2×Q8—81no—
259C2×Q8—81no—
260C2×Q8—81no—
261C2×Q8—81no—
262C2×Q8—81no—
263C2×Q8—81no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—84no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4C4×D411yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4C22⋊Q811yescharacteristic
287C22×C4#12311yescharacteristic
288C22×C4#12711yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4#13511yescharacteristic
292C22×C4#13511yescharacteristic
293C22×C4AGL(1,ℤ/8)11yescharacteristic
294C22×C4#13711yescharacteristic
295C22×C4#13711yescharacteristic
296Q16—24no—
297Q16—24no—
298Q16—42no—
299Q16—42no—
300Q16—42no—
301Q16—42no—
302Q16—42no—
303Q16—42no—
304M16—42no—
305M16—42no—
306M16—81no—
307M16—81no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—81no—
317C2×C8—81no—
318C2×C8C2×C22⋊C411yescharacteristic
319C2×C8C2×C4⋊C411yescharacteristic
320C2×C8#11711yescharacteristic
321C2×C8C4×D411yescharacteristic
322C2×C8C4×D411yescharacteristic
323C2×C8C4×D411yescharacteristic
324C2×C8#12011yescharacteristic
325C2×C8#12011yescharacteristic
326C2×C8#12111yescharacteristic
327C2×C8#12111yescharacteristic
328C2×C8#12111yescharacteristic
329C2×C8#12111yescharacteristic
330C2×C8#12111yescharacteristic
331C2×C8#12111yescharacteristic
332C2×C8C22⋊Q811yescharacteristic
333C2×C8C22⋊Q811yescharacteristic
334C2×C8C22⋊Q811yescharacteristic
335C2×C8C22⋊Q811yescharacteristic
336C2×C8C22⋊Q811yescharacteristic
337C2×C8#12311yescharacteristic
338C2×C8#12311yescharacteristic
339C2×C8#12311yescharacteristic
340C2×C8#12311yescharacteristic
341C2×C8#12411yescharacteristic
342C2×C8#12411yescharacteristic
343C2×C8#12411yescharacteristic
344C2×C8#12611yescharacteristic
345C2×C8#12611yescharacteristic
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C22⋊C4—81no—
355C22⋊C4—81no—
356C42—22no—
357C42—22no—
358C42—41no—
359C42—41no—
360C42—81no—
361C42—81no—
362C16—24no—
363C16—42no—
364C23—22no—
365C23—22no—
366C23#44011yescharacteristic
367C23#44711yescharacteristic
368C23#45911yescharacteristic
369C23#46811yescharacteristic
370C23#47811yescharacteristic
371C23#48211yescharacteristic
372C23#49611yescharacteristic
373Q8—44no—
374Q8—44no—
375Q8—82no—
376Q8—82no—
377Q8—82no—
378Q8—82no—
379Q8—82no—
380Q8—82no—
381C2×C4—24no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—162no—
400C2×C4—162no—
401C2×C4#38511yescharacteristic
402C2×C4#38811yescharacteristic
403C2×C4#39011yescharacteristic
404C2×C4#39211yescharacteristic
405C2×C4#39411yescharacteristic
406C2×C4#39611yescharacteristic
407C2×C4#39911yescharacteristic
408C2×C4C4×D811yescharacteristic
409C2×C4C4×D811yescharacteristic
410C2×C4#44011yescharacteristic
411C2×C4#44711yescharacteristic
412C2×C4#45211yescharacteristic
413C2×C4#45211yescharacteristic
414C2×C4#45911yescharacteristic
415C2×C4#46411yescharacteristic
416C2×C4#46411yescharacteristic
417C2×C4#46611yescharacteristic
418C2×C4#46611yescharacteristic
419C2×C4#46811yescharacteristic
420C2×C4#47411yescharacteristic
421C2×C4#47411yescharacteristic
422C2×C4#47811yescharacteristic
423C2×C4#48011yescharacteristic
424C2×C4#48011yescharacteristic
425C2×C4#48211yescharacteristic
426C2×C4#49311yescharacteristic
427C2×C4#49511yescharacteristic
428C2×C4#49711yescharacteristic
429C8—42no—
430C8—82no—
431C8#38011yescharacteristic
432C8#38511yescharacteristic
433C8#38611yescharacteristic
434C8#38811yescharacteristic
435C8#39311yescharacteristic
436C8#39411yescharacteristic
437C8#39611yescharacteristic
438C8#39711yescharacteristic
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22#186211yescharacteristic
448C22#192511yescharacteristic
449C22#197311yescharacteristic
450C22#201111yescharacteristic
451C22#203311yescharacteristic
452C22#206711yescharacteristic
453C22#209211yescharacteristic
454C22#221212yes—
455C22#221412yes—
456C4—44no—
457C4—84no—
458C4—161no—
459C4—162no—
460C4—161no—
461C4#143911yescharacteristic
462C4#186211yescharacteristic
463C4#192511yescharacteristic
464C4#197011yescharacteristic
465C4#201511yescharacteristic
466C4#203311yescharacteristic
467C4#206711yescharacteristic
468C4#209311yescharacteristic
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2#1112311yescharacteristic
474C2#1313512yes—
475C2#1333211yescharacteristic
476C2#1333311yescharacteristic
477C2#1347912yes—
subgroupG/H
center, socleC23#459
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. C2×#362C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21592
C22×#3622
C2×#21412
C2×#19692
C2×#13872
C2×#13762
#108592