Finite Group Databasebeta

← #112694

Subgroups — #112694

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#494C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#7897C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C23×Q16—22no—
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#494—22no—
12C2×#494—22no—
13C2×#494—22no—
14C2×#494—22no—
15C2×#494—22no—
16C2×#494C414yescomplemented
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2285C2211yescharacteristic
22#1997—22no—
23#1997—22no—
24#1993—22no—
25#1990—22no—
26#1981—22no—
27#1956—22no—
28#1787—22no—
29C2×#343—22no—
30#1736—22no—
31#1549—22no—
32#1549—22no—
33C2×#332—22no—
34#1279—22no—
35#1278—22no—
36C23×Q8—21no—
37C23×Q8D412yes—
38C22×Q16—28no—
39C22×Q16—28no—
40C22×Q16—28no—
41C22×Q16—416no—
42C22×M16—21no—
43C22×M16D412yes—
44C23×C8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8C2×C412yes—
66C2×C4⋊Q8D412yes—
67C2×C4⋊Q8D412yes—
68C2×C4⋊Q8D412yes—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#494—28no—
76#494—28no—
77#494—28no—
78#494—28no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86#423D412yes—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D412yes—
90C2×#103—28no—
91C2×C22⋊C8—21no—
92C4×M16—21no—
93C4×M16—21no—
94C4×M16—21no—
95C4×M16—21no—
96C2×#97—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2×C412yes—
103C2×C4×C8D412yes—
104C2×C4×C8D412yes—
105#384D412yes—
106#382—21no—
107#380—21no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111#343—28no—
112#340—28no—
113#332—28no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—48no—
123C22×Q8—48no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8C22⋊C414yes—
128C23×C4—22no—
129C23×C4—41no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C2×Q16—232no—
134C2×Q16—432no—
135C2×Q16—432no—
136C2×Q16—432no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—41no—
144C2×M16—42no—
145C2×M16—42no—
146C2×M16Q1614yescomplemented
147C22×C8—28no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—41no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C4⋊Q8C22⋊C414yes—
186#117—42no—
187#117—42no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—41no—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—41no—
234C4⋊C8—41no—
235#103—88no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238#97—41no—
239#97—41no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8—41no—
244C4×C8—41no—
245C4×C8—41no—
246C4×C8—41no—
247C4×C8C22⋊C414yes—
248#95—41no—
249#95—41no—
250C24#12111yescharacteristic
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—48no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—48no—
266C2×Q8—84no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×Q8—84no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4#12411yescharacteristic
299C22×C4#12711yescharacteristic
300C22×C4C2×Q1612yes—
301C22×C4#13512yes—
302C22×C4#13712yes—
303C22×C4#13712yes—
304Q16—464no—
305M16—24no—
306M16—24no—
307M16—42no—
308M16—42no—
309M16—44no—
310M16—82no—
311C2×C8—28no—
312C2×C8—28no—
313C2×C8—28no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—44no—
321C2×C8—81no—
322C2×C8—81no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C22⋊C4—82no—
343C42—81no—
344C42—81no—
345C42C2×C22⋊C411yescharacteristic
346C42C2×C22⋊C411yescharacteristic
347C42C4×D411yescharacteristic
348C42C4×D411yescharacteristic
349C42#12011yescharacteristic
350C42#12011yescharacteristic
351C42#12011yescharacteristic
352C42#12111yescharacteristic
353C42#12111yescharacteristic
354C42#12111yescharacteristic
355C42#12111yescharacteristic
356C42#12111yescharacteristic
357C42#12111yescharacteristic
358C42#12311yescharacteristic
359C42#12411yescharacteristic
360C42#12711yescharacteristic
361C23—22no—
362C23—22no—
363C23#38611yescharacteristic
364C23#39211yescharacteristic
365C23#39411yescharacteristic
366C23#46212yes—
367C23#46412yes—
368Q8—48no—
369Q8—48no—
370Q8—48no—
371Q8—84no—
372Q8—84no—
373Q8—84no—
374Q8—84no—
375Q8—84no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#37911yescharacteristic
398C2×C4#38611yescharacteristic
399C2×C4#38611yescharacteristic
400C2×C4#38611yescharacteristic
401C2×C4#39011yescharacteristic
402C2×C4#39011yescharacteristic
403C2×C4#39211yescharacteristic
404C2×C4#39211yescharacteristic
405C2×C4#39211yescharacteristic
406C2×C4#39411yescharacteristic
407C2×C4#39411yescharacteristic
408C2×C4#39411yescharacteristic
409C2×C4C4×Q1612yes—
410C2×C4#44112yes—
411C2×C4#45112yes—
412C2×C4#45112yes—
413C2×C4#45212yes—
414C2×C4#45212yes—
415C2×C4#45314yes—
416C2×C4#46212yes—
417C2×C4#46412yes—
418C2×C4#46712yes—
419C2×C4#47012yes—
420C2×C4#47314yes—
421C2×C4#49411yescharacteristic
422C2×C4#49511yescharacteristic
423C2×C4#49711yescharacteristic
424C2×C4#49711yescharacteristic
425C8—28no—
426C8—44no—
427C8—82no—
428C8—82no—
429C22—21no—
430C22—21no—
431C22—24no—
432C22—24no—
433C22—21no—
434C22—21no—
435C22#143811yescharacteristic
436C22#187812yes—
437C22#200212yes—
438C22#201511yescharacteristic
439C22#201611yescharacteristic
440C22#202012yes—
441C22#205612yes—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—84no—
447C4—84no—
448C4—162no—
449C4#187812yes—
450C4#188712yes—
451C4#196911yescharacteristic
452C4#197111yescharacteristic
453C4#197912yes—
454C4#200112yes—
455C4#200212yes—
456C4#201511yescharacteristic
457C4#201611yescharacteristic
458C4#204712yes—
459C2—22no—
460C2—22no—
461C2#1015412yes—
462C2#1015512yes—
463C2#1097511yescharacteristic
464C2#1102411yescharacteristic
465C2#1102911yescharacteristic
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
16. C2×#494C4semidirect
146. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#4942
#218042
C2×#19972
#213242
C2×#15492
#78972
C2×#12782