Finite Group Databasebeta

← #115753

Subgroups — #115753

Order 512.

816 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21933C211yesmaximal, characteristic
2#20700C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1821C211yesmaximal, characteristic
5C2×#1305C211yesmaximal, characteristic
6#8016C211yesmaximal, characteristic
7C2×#1293C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1851—22no—
16#1850—22no—
17#1824—22no—
18#1824—22no—
19#1824—22no—
20#1823C414yescomplemented
21#1821—22no—
22#1821—22no—
23#1798—22no—
24C2×#343—22no—
25#1736—22no—
26C2×#340—22no—
27#1305—22no—
28#1294—22no—
29#1293—22no—
30C24×C4—21no—
31C24×C4C2311yescharacteristic
32C22×M16—21no—
33C22×M16—21no—
34C22×M16D412yes—
35C23×C8D412yes—
36C2×#117D412yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4Q812yes—
52C22×C22⋊C4—21no—
53C2×C4⋊C8—24no—
54#407—21no—
55#407—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407D412yes—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8C2×C412yes—
78#381—21no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380C2×C412yes—
83#380D412yes—
84#380D412yes—
85#379—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C2×#95—21no—
89#343—28no—
90#340—28no—
91#340—28no—
92#336—24no—
93C25#4111yescharacteristic
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—44no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4—84no—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×Q811yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—44no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—44no—
126C2×M16—44no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—44no—
133C22×C8—41no—
134#117—44no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
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—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C22⋊C414yes—
166C2×C4⋊C4C4⋊C414yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—82no—
173C2×C42—22no—
174C4⋊C8—84no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—41no—
195C24—22no—
196C24—22no—
197C24—42no—
198C24—42no—
199C24C4×D411yescharacteristic
200C24#12111yescharacteristic
201C24C22⋊Q811yescharacteristic
202C24#12311yescharacteristic
203C24#12311yescharacteristic
204C24#12311yescharacteristic
205C24#12611yescharacteristic
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4—84no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C4⋊C411yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12411yescharacteristic
263C22×C4#12411yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12611yescharacteristic
266C22×C4#12711yescharacteristic
267M16—44no—
268M16—44no—
269M16—82no—
270M16—82no—
271M16—82no—
272M16—82no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—82no—
282C2×C8—82no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—82no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C42—44no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—48no—
324C23—48no—
325C23#38511yescharacteristic
326C23#38611yescharacteristic
327C23#38811yescharacteristic
328C23#39411yescharacteristic
329C23#39611yescharacteristic
330C23#39711yescharacteristic
331C23#39911yescharacteristic
332C23#47112yes—
333C23#47312yes—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
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—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—164no—
376C2×C4#38011yescharacteristic
377C2×C4#38511yescharacteristic
378C2×C4#38811yescharacteristic
379C2×C4#39011yescharacteristic
380C2×C4#39211yescharacteristic
381C2×C4#39311yescharacteristic
382C2×C4#39411yescharacteristic
383C2×C4#39611yescharacteristic
384C2×C4#45814yes—
385C8—82no—
386C8—82no—
387C8—82no—
388C8—82no—
389C22—24no—
390C22—24no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—42no—
409C22—44no—
410C22—44no—
411C22—42no—
412C22—42no—
413C22—44no—
414C22—44no—
415C22#143911yescharacteristic
416C22#189012yes—
417C22#190312yes—
418C22#190812yes—
419C22#204611yescharacteristic
420C22#204711yescharacteristic
421C22#206412yes—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—84no—
429C4—84no—
430C4—162no—
431C4#202212yes—
432C4#203012yes—
433C2—21no—
434C2—22no—
435C2—21no—
436C2—41no—
437C2—41no—
438C2—41no—
439C2—41no—
440C2#952114yes—
441C2#1105811yescharacteristic
442C2#1106211yescharacteristic
443C2#1107011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1823C4semidirect

Maximal subgroups

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

subgroupindex
#219332
#207002
C2×#18242
C2×#18212
C2×#13052
#80162
C2×#12932