Finite Group Databasebeta

← #117413

Subgroups — #117413

Order 512.

788 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3#21202C211yesmaximal, characteristic
4C2×#1546C211yesmaximal, characteristic
5#10857C211yesmaximal, characteristic
6#7897C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#419—22no—
12C2×C4×M16C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×#384C2211yescharacteristic
16#1990—22no—
17#1990—22no—
18#1960—22no—
19#1956—21no—
20#1956—21no—
21#1947—22no—
22#1852—22no—
23#1845—21no—
24#1834—21no—
25C2×#343—22no—
26C2×#340—22no—
27#1546—22no—
28#1546C414yescomplemented
29C2×#332—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×#107—24no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423C2×C412yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D412yes—
67#419—44no—
68#419—44no—
69C2×#103—24no—
70C2×#103—24no—
71C2×C22⋊C8—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C2×#97—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8D412yes—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384D411yescharacteristic
90#384D411yescharacteristic
91#382—21no—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96#343—44no—
97#340—24no—
98#340—28no—
99#332—28no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—41no—
104C22×Q8—41no—
105C22×Q8—41no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C23×C4—22no—
111C23×C4—41no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—42no—
123C2×M16—42no—
124C2×M16D814yes—
125C22×C8—24no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—42no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—41no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8C4⋊C414yes—
149#117—42no—
150#117—42no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—82no—
179C2×C22⋊C4—41no—
180C2×C42—41no—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×Q811yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193#107—84no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8C22⋊C414yes—
200#103—48no—
201#103—48no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#97—41no—
205#97—41no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—41no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214#95—41no—
215#95—41no—
216C24#12111yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—81no—
232C2×Q8—81no—
233C2×Q8—81no—
234C2×Q8—81no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4#11711yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12611yescharacteristic
266C22×C4#12611yescharacteristic
267C22×C4#12711yescharacteristic
268C22×C4C2×D812yes—
269C22×C4#13512yes—
270C22×C4AGL(1,ℤ/8)12yes—
271C22×C4#13712yes—
272M16—24no—
273M16—24no—
274M16—42no—
275M16—42no—
276M16—44no—
277M16—82no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—81no—
291C2×C8—81no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—81no—
312C4⋊C4—81no—
313C22⋊C4—82no—
314C42—81no—
315C42—81no—
316C42C2×C22⋊C411yescharacteristic
317C42C2×C4⋊C411yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42#12111yescharacteristic
321C42#12111yescharacteristic
322C42#12111yescharacteristic
323C42C22⋊Q811yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C42#12311yescharacteristic
330C42#12411yescharacteristic
331C42#12411yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23#38811yescharacteristic
337C23#39211yescharacteristic
338C23#39611yescharacteristic
339C23#45912yes—
340C23#46412yes—
341Q8—48no—
342Q8—82no—
343Q8—82no—
344Q8—82no—
345Q8—82no—
346Q8—84no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—44no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—164no—
374C2×C4#38011yescharacteristic
375C2×C4#38511yescharacteristic
376C2×C4#38511yescharacteristic
377C2×C4#38611yescharacteristic
378C2×C4#38811yescharacteristic
379C2×C4#39011yescharacteristic
380C2×C4#39311yescharacteristic
381C2×C4#39411yescharacteristic
382C2×C4#39411yescharacteristic
383C2×C4#39611yescharacteristic
384C2×C4#39711yescharacteristic
385C2×C4#39911yescharacteristic
386C2×C4C4×D812yes—
387C2×C4#44012yes—
388C2×C4#44712yes—
389C2×C4#45212yes—
390C2×C4#45512yes—
391C2×C4#45612yes—
392C2×C4#46612yes—
393C2×C4#46812yes—
394C2×C4#47112yes—
395C2×C4#47312yes—
396C2×C4#47412yes—
397C2×C4#47812yes—
398C2×C4#48012yes—
399C2×C4#48212yes—
400C2×C4#49311yescharacteristic
401C2×C4#49511yescharacteristic
402C2×C4#49611yescharacteristic
403C2×C4#49711yescharacteristic
404C8—44no—
405C8—44no—
406C8—44no—
407C8—82no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22#143911yescharacteristic
417C22#192512yes—
418C22#200411yescharacteristic
419C22#200511yescharacteristic
420C22#201111yescharacteristic
421C22#201511yescharacteristic
422C22#201911yescharacteristic
423C22#202011yescharacteristic
424C22#206712yes—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—84no—
430C4—161no—
431C4—162no—
432C4—161no—
433C4#186212yes—
434C4#188212yes—
435C4#193012yes—
436C4#197011yescharacteristic
437C4#197311yescharacteristic
438C4#202812yes—
439C4#203312yes—
440C4#206412yes—
441C4#209211yescharacteristic
442C4#209311yescharacteristic
443C2—22no—
444C2—22no—
445C2#1016811yescharacteristic
446C2#1017111yescharacteristic
447C2#1017311yescharacteristic
448C2#1017411yescharacteristic
449C2#1106411yescharacteristic
450C2#1112311yescharacteristic
451C2#1114511yescharacteristic
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. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#219822
C2×#19902
#212022
C2×#15462
#108572
#78972
C2×#12782