Finite Group Databasebeta

← #117410

Subgroups — #117410

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5#10857C211yesmaximal, characteristic
6#7897C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#384C2211yescharacteristic
15#1993—22no—
16#1990—22no—
17#1990—21no—
18#1990—21no—
19#1990—22no—
20#1948—22no—
21#1853—22no—
22#1844—21no—
23#1834—21no—
24C2×#343—22no—
25#1736—22no—
26C2×#340—22no—
27#1548—22no—
28#1548C414yescomplemented
29C2×#333—22no—
30#1279—22no—
31#1278—22no—
32C23×Q8—21no—
33C22×M16—21no—
34C22×M16D412yes—
35C23×C8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8D412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D412yes—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431C2×C412yes—
58#431D412yes—
59C2×#106—24no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423C2×C412yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D412yes—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8D412yes—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384D411yescharacteristic
86#384D411yescharacteristic
87#382—21no—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92#343—44no—
93#340—24no—
94#340—28no—
95#340—24no—
96#340—24no—
97#333—28no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—82no—
105C22×Q8—82no—
106C23×C4—22no—
107C23×C4—41no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—41no—
118C2×M16—42no—
119C2×M16—42no—
120C2×M16SD1614yes—
121C22×C8—24no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—42no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8C4⋊C414yes—
145#117—42no—
146#117—42no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—82no—
175C2×C22⋊C4—41no—
176C2×C42—41no—
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×Q811yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189#106—84no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8C22⋊C414yes—
196C22⋊C8—22no—
197C22⋊C8—22no—
198#97—41no—
199#97—41no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—41no—
205C4×C8—41no—
206C4×C8—41no—
207C4×C8—41no—
208#95—41no—
209#95—41no—
210C24#12111yescharacteristic
211C2×Q8—24no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—84no—
216C2×Q8—84no—
217C2×Q8—84no—
218C2×Q8—84no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4#11711yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4#12711yescharacteristic
256C22×C4C2×SD1612yes—
257C22×C4#13512yes—
258C22×C4#13712yes—
259C22×C4#13712yes—
260M16—24no—
261M16—24no—
262M16—42no—
263M16—42no—
264M16—44no—
265M16—82no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—81no—
279C2×C8—81no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C22⋊C4—82no—
302C42—81no—
303C42—81no—
304C42C2×C22⋊C411yescharacteristic
305C42C2×C4⋊C411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42#12111yescharacteristic
311C42C22⋊Q811yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42#12311yescharacteristic
315C42#12311yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12411yescharacteristic
319C42#12411yescharacteristic
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23#38811yescharacteristic
325C23#39211yescharacteristic
326C23#39611yescharacteristic
327C23#46112yes—
328C23#46412yes—
329Q8—84no—
330Q8—84no—
331Q8—84no—
332Q8—84no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—44no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—164no—
360C2×C4#38011yescharacteristic
361C2×C4#38511yescharacteristic
362C2×C4#38511yescharacteristic
363C2×C4#38611yescharacteristic
364C2×C4#38811yescharacteristic
365C2×C4#39011yescharacteristic
366C2×C4#39311yescharacteristic
367C2×C4#39411yescharacteristic
368C2×C4#39411yescharacteristic
369C2×C4#39611yescharacteristic
370C2×C4#39711yescharacteristic
371C2×C4#39911yescharacteristic
372C2×C4C4×SD1612yes—
373C2×C4#44112yes—
374C2×C4#44812yes—
375C2×C4#45212yes—
376C2×C4#45512yes—
377C2×C4#45612yes—
378C2×C4#46512yes—
379C2×C4#47012yes—
380C2×C4#47112yes—
381C2×C4#47212yes—
382C2×C4#47512yes—
383C2×C4#47912yes—
384C2×C4#48312yes—
385C2×C4#48512yes—
386C2×C4#49211yescharacteristic
387C2×C4#49511yescharacteristic
388C2×C4#49711yescharacteristic
389C2×C4#49711yescharacteristic
390C8—44no—
391C8—44no—
392C8—44no—
393C8—82no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22#143911yescharacteristic
403C22#192712yes—
404C22#200411yescharacteristic
405C22#200511yescharacteristic
406C22#201311yescharacteristic
407C22#201511yescharacteristic
408C22#201811yescharacteristic
409C22#201911yescharacteristic
410C22#206612yes—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—84no—
416C4—162no—
417C4—161no—
418C4—161no—
419C4#186512yes—
420C4#188212yes—
421C4#193112yes—
422C4#196811yescharacteristic
423C4#197111yescharacteristic
424C4#202812yes—
425C4#203512yes—
426C4#206312yes—
427C4#209611yescharacteristic
428C4#209711yescharacteristic
429C2—22no—
430C2—22no—
431C2#1016111yescharacteristic
432C2#1016211yescharacteristic
433C2#1016411yescharacteristic
434C2#1016711yescharacteristic
435C2#1106411yescharacteristic
436C2#1112211yescharacteristic
437C2#1114411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#219812
#213242
C2×#19902
C2×#15482
#108572
#78972
C2×#12782