Finite Group Databasebeta

← #112699

Subgroups — #112699

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#500C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#7897C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10C2×#500—22no—
11C2×#500—22no—
12C2×#500—22no—
13C2×#500—22no—
14C2×#500—22no—
15C2×#500C414yescomplemented
16C2×#431C2211yescharacteristic
17C22×#107—22no—
18C2×#423C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2285C2211yescharacteristic
22#1997—22no—
23#1997—22no—
24#1993—22no—
25#1990—22no—
26#1990—22no—
27#1987—22no—
28#1824—22no—
29C2×#343—22no—
30#1736—22no—
31#1548—22no—
32#1548—22no—
33C2×#333—22no—
34#1279—22no—
35#1278—22no—
36C23×Q8—21no—
37C23×Q8—21no—
38C22×M16—21no—
39C22×M16D412yes—
40C23×C8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—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⋊Q8C2×C412yes—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D412yes—
70C22×C42C2311yescharacteristic
71#500—28no—
72#500—28no—
73#500—28no—
74#500—28no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79C2×#107—28no—
80C2×#107—28no—
81C2×#107—28no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423D412yes—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8D412yes—
89C2×C22⋊C8—21no—
90C4×M16—21no—
91C4×M16—21no—
92C4×M16—21no—
93C4×M16—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2×C412yes—
101C2×C4×C8D412yes—
102C2×C4×C8D412yes—
103#384D412yes—
104#382—21no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109#343—28no—
110#340—28no—
111#340—28no—
112#333—28no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—41no—
119C22×Q8—41no—
120C22×Q8—41no—
121C22×Q8—41no—
122C22×Q8—41no—
123C22×Q8—41no—
124C22×Q8—41no—
125C22×Q8—84no—
126C22×Q8—84no—
127C23×C4—22no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4#4111yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—41no—
139C2×M16—42no—
140C2×M16—42no—
141C2×M16Q1614yes—
142C22×C8—28no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—41no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
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—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
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⋊Q8C22⋊C414yes—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
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—41no—
200C2×C4⋊C4—41no—
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⋊C4C22⋊C414yes—
210C2×C22⋊C4—41no—
211C2×C42—41no—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42#4111yescharacteristic
224#107—232no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—41no—
230C4⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233#97—41no—
234#97—41no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—41no—
239C4×C8—41no—
240C4×C8—41no—
241C4×C8—41no—
242C4×C8C22⋊C414yes—
243#95—41no—
244#95—41no—
245C24#12111yescharacteristic
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×Q8—84no—
261C2×Q8—84no—
262C2×Q8—84no—
263C2×Q8—84no—
264C2×Q8—84no—
265C2×Q8—84no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4#12311yescharacteristic
293C22×C4#12411yescharacteristic
294C22×C4#12711yescharacteristic
295C22×C4C2×Q1612yes—
296C22×C4#13512yes—
297C22×C4#13712yes—
298C22×C4#13712yes—
299M16—24no—
300M16—24no—
301M16—42no—
302M16—42no—
303M16—44no—
304M16—82no—
305C2×C8—28no—
306C2×C8—28no—
307C2×C8—28no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—44no—
315C2×C8—81no—
316C2×C8—81no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
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—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C22⋊C4—82no—
337C42—81no—
338C42—81no—
339C42C2×C22⋊C411yescharacteristic
340C42C2×C22⋊C411yescharacteristic
341C42C4×D411yescharacteristic
342C42C4×D411yescharacteristic
343C42#12011yescharacteristic
344C42#12011yescharacteristic
345C42#12011yescharacteristic
346C42#12111yescharacteristic
347C42#12111yescharacteristic
348C42#12111yescharacteristic
349C42#12111yescharacteristic
350C42#12111yescharacteristic
351C42#12111yescharacteristic
352C42#12311yescharacteristic
353C42#12411yescharacteristic
354C42#12711yescharacteristic
355C23—22no—
356C23—22no—
357C23#38611yescharacteristic
358C23#39211yescharacteristic
359C23#39411yescharacteristic
360C23#46212yes—
361C23#46412yes—
362Q8—84no—
363Q8—84no—
364Q8—84no—
365Q8—84no—
366Q8—84no—
367Q8—84no—
368Q8—84no—
369Q8—84no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4#37911yescharacteristic
392C2×C4#38611yescharacteristic
393C2×C4#38611yescharacteristic
394C2×C4#38611yescharacteristic
395C2×C4#39011yescharacteristic
396C2×C4#39011yescharacteristic
397C2×C4#39211yescharacteristic
398C2×C4#39211yescharacteristic
399C2×C4#39211yescharacteristic
400C2×C4#39411yescharacteristic
401C2×C4#39411yescharacteristic
402C2×C4#39411yescharacteristic
403C2×C4C4×Q1612yes—
404C2×C4#44112yes—
405C2×C4#45112yes—
406C2×C4#45112yes—
407C2×C4#45212yes—
408C2×C4#45212yes—
409C2×C4#45614yes—
410C2×C4#46212yes—
411C2×C4#46412yes—
412C2×C4#46712yes—
413C2×C4#47012yes—
414C2×C4#47214yes—
415C2×C4#49411yescharacteristic
416C2×C4#49511yescharacteristic
417C2×C4#49711yescharacteristic
418C2×C4#49711yescharacteristic
419C8—28no—
420C8—44no—
421C8—82no—
422C8—82no—
423C22—21no—
424C22—21no—
425C22—24no—
426C22—24no—
427C22—21no—
428C22—21no—
429C22#143811yescharacteristic
430C22#187812yes—
431C22#200512yes—
432C22#201511yescharacteristic
433C22#201611yescharacteristic
434C22#201812yes—
435C22#205612yes—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—84no—
441C4—84no—
442C4—162no—
443C4#187812yes—
444C4#188612yes—
445C4#196911yescharacteristic
446C4#197111yescharacteristic
447C4#197812yes—
448C4#200112yes—
449C4#200512yes—
450C4#201511yescharacteristic
451C4#201611yescharacteristic
452C4#204512yes—
453C2—22no—
454C2—22no—
455C2#1016412yes—
456C2#1016512yes—
457C2#1097811yescharacteristic
458C2#1102411yescharacteristic
459C2#1102711yescharacteristic
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
15. C2×#500C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5002
#218342
C2×#19972
#213242
C2×#15482
#78972
C2×#12782