Finite Group Databasebeta

← #117460

Subgroups — #117460

Order 512.

854 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
1#21982C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3C2×#1546C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic
5#10839C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7C2×#1276C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#391—22no—
13#2285C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×C43C2211yescharacteristic
16#1990—21no—
17#1990—21no—
18#1988—22no—
19#1947—22no—
20#1852—22no—
21#1845—21no—
22#1834—21no—
23#1823—22no—
24C2×#339—22no—
25#1546—22no—
26#1546C414yescomplemented
27C2×#332—22no—
28C4⋊#95—22no—
29C4⋊#95—22no—
30#1296—22no—
31#1276—22no—
32C23×Q8—21no—
33C22×M16—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—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⋊Q8C2×C412yes—
45C2×C4⋊Q8Q812yes—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4D412yes—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64C2×#107—24no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8D412yes—
72C2×C4⋊C8D412yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#391—44no—
78#391—44no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384D411yescharacteristic
84#384D411yescharacteristic
85#382—21no—
86#380—21no—
87#380—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C2×#95—28no—
91C43—21no—
92C43—21no—
93C43—21no—
94C43—21no—
95#340—24no—
96#340—24no—
97#340—24no—
98#339—44no—
99#332—28no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—41no—
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—22no—
112C23×C4—22no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—22no—
117C2×M16—41no—
118C2×M16—42no—
119C2×M16—42no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—41no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139#117—42no—
140#117—42no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4C4⋊C414yes—
169C2×C22⋊C4—41no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—22no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42D814yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q811yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197#107—84no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8C22⋊C414yes—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—41no—
213C4×C8—41no—
214#95—416no—
215#95—41no—
216#95—41no—
217C24#12011yescharacteristic
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—81no—
233C2×Q8—81no—
234C2×Q8—81no—
235C2×Q8—81no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4C22⋊Q811yescharacteristic
270C22×C4C22⋊Q811yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4#12311yescharacteristic
274C22×C4#12411yescharacteristic
275C22×C4#12711yescharacteristic
276C22×C4C2×D812yes—
277C22×C4#13512yes—
278C22×C4AGL(1,ℤ/8)12yes—
279C22×C4#13712yes—
280M16—44no—
281M16—82no—
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—81no—
290C2×C8—81no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C22⋊C4—82no—
313C42—24no—
314C42—24no—
315C42—44no—
316C42—44no—
317C42—44no—
318C42—44no—
319C42—42no—
320C42—42no—
321C42—81no—
322C42—81no—
323C42C2×C22⋊C411yescharacteristic
324C42#11711yescharacteristic
325C42C4×D411yescharacteristic
326C42C4×D411yescharacteristic
327C42#12111yescharacteristic
328C42#12111yescharacteristic
329C42C22⋊Q811yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42#12311yescharacteristic
334C42#12311yescharacteristic
335C42#12411yescharacteristic
336C42#12411yescharacteristic
337C42#12611yescharacteristic
338C42#12611yescharacteristic
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23#38511yescharacteristic
344C23#39211yescharacteristic
345C23#39411yescharacteristic
346C23#44712yes—
347C23#45212yes—
348Q8—84no—
349Q8—82no—
350Q8—82no—
351Q8—84no—
352Q8—82no—
353Q8—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—28no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—88no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4#38011yescharacteristic
389C2×C4#38511yescharacteristic
390C2×C4#38611yescharacteristic
391C2×C4#38811yescharacteristic
392C2×C4#38811yescharacteristic
393C2×C4#39011yescharacteristic
394C2×C4#39311yescharacteristic
395C2×C4#39411yescharacteristic
396C2×C4#39611yescharacteristic
397C2×C4#39611yescharacteristic
398C2×C4#39711yescharacteristic
399C2×C4#39911yescharacteristic
400C2×C4C4×D812yes—
401C2×C4#44012yes—
402C2×C4#45412yes—
403C2×C4#45512yes—
404C2×C4#45612yes—
405C2×C4#45612yes—
406C2×C4#45912yes—
407C2×C4#46412yes—
408C2×C4#46612yes—
409C2×C4#46812yes—
410C2×C4#47412yes—
411C2×C4#47812yes—
412C2×C4#48012yes—
413C2×C4#48212yes—
414C2×C4#49311yescharacteristic
415C2×C4#49511yescharacteristic
416C2×C4#49611yescharacteristic
417C2×C4#49711yescharacteristic
418C8—44no—
419C8—82no—
420C8—82no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—21no—
428C22—21no—
429C22#143911yescharacteristic
430C22#186212yes—
431C22#200311yescharacteristic
432C22#200411yescharacteristic
433C22#200511yescharacteristic
434C22#200511yescharacteristic
435C22#201111yescharacteristic
436C22#201511yescharacteristic
437C22#203312yes—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—48no—
443C4—84no—
444C4—161no—
445C4—162no—
446C4—161no—
447C4#185712yes—
448C4#188212yes—
449C4#192512yes—
450C4#197011yescharacteristic
451C4#197311yescharacteristic
452C4#202812yes—
453C4#203812yes—
454C4#206712yes—
455C4#209211yescharacteristic
456C4#209311yescharacteristic
457C2—22no—
458C2—22no—
459C2#1028411yescharacteristic
460C2#1028911yescharacteristic
461C2#1029611yescharacteristic
462C2#1029811yescharacteristic
463C2#1106411yescharacteristic
464C2#1112311yescharacteristic
465C2#1113611yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
26. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#219822
#218332
C2×#15462
#147142
#108392
#80432
C2×#12762