Finite Group Databasebeta

← #103389

Subgroups — #103389

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2093C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3#17069C211yesmaximal, characteristic
4#11357C211yesmaximal, characteristic, complemented
5C2×#1271C211yesmaximal, characteristic, complemented
6C2×#1270C211yesmaximal, characteristic
7#7025C211yesmaximal, characteristic, complemented
8C22×#107—21no—
9C22×#106—21no—
10C22×#102—21no—
11C2×#397—21no—
12C2×#394—21no—
13C2×#390C2211yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#384C2211yescharacteristic
16#2093—22no—
17#2093—22no—
18#2084—22no—
19#2084—22no—
20C2×#340—21no—
21C2×#340—22no—
22C2×#336C2211yescharacteristic
23C2×#336C2211yescharacteristic
24C2×#336C2211yescharacteristic
25#1441—21no—
26#1439—21no—
27#1439—22no—
28C4⋊#95C2211yescharacteristic
29#1271—22no—
30#1270—22no—
31C23×D4D411yescharacteristic
32C23×C8—21no—
33C23×C8—21no—
34C23×C8—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D412yes—
42C22×C4⋊C4D411yescharacteristic
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—22no—
46C22×C22⋊C4—22no—
47C22×C42C2311yescharacteristic
48C2×#107—24no—
49C2×#107—44no—
50C2×#106—24no—
51C2×#106—42no—
52C2×#106—42no—
53C2×#102—24no—
54C2×#102—44no—
55#397—42no—
56#397—42no—
57#394—42no—
58#394—42no—
59#390—21no—
60#390—21no—
61#390—21no—
62#390—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#389—21no—
67#384—21no—
68#384—21no—
69#384—21no—
70#384D412yes—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—22no—
76#340—24no—
77#340—44no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336D412yes—
83#336D412yes—
84C25—21no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—21no—
89C22×D4—21no—
90C22×D4—42no—
91C22×D4—42no—
92C22×D4SD1612yes—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C22×C8—24no—
102C22×C8—24no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—42no—
107C22×C8—42no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—42no—
114C22×C8—41no—
115C22×C8—42no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4D812yes—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—82no—
166C2×C22⋊C4—82no—
167C2×C22⋊C4—82no—
168C2×C42—42no—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173#107—48no—
174#106—48no—
175#102—48no—
176#95—44no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183#95—82no—
184C24—22no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—42no—
189C24—41no—
190C24—42no—
191C24—41no—
192C24—41no—
193C24—42no—
194C24—42no—
195C24#12011yescharacteristic
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—44no—
199C2×D4—42no—
200C2×D4—44no—
201C2×D4—42no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—82no—
205C2×D4—82no—
206C2×D4—81no—
207C2×D4—81no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12711yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×SD1611yescharacteristic
252C22×C4#13511yescharacteristic
253C22×C4#13511yescharacteristic
254C22×C4#13512yes—
255C22×C4AGL(1,ℤ/8)11yescharacteristic
256C22×C4AGL(1,ℤ/8)11yescharacteristic
257C22×C4AGL(1,ℤ/8)11yescharacteristic
258C22×C4AGL(1,ℤ/8)12yes—
259C22×C4#13711yescharacteristic
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—44no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—44no—
271C2×C8—42no—
272C2×C8—82no—
273C2×C8—81no—
274C2×C8—81no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—82no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—82no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C22⋊C4—88no—
300C22⋊C4—88no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—82no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C23—44no—
313C23—42no—
314C23—44no—
315C23—42no—
316C23—42no—
317C23—82no—
318C23—82no—
319C23—82no—
320C23—82no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23—81no—
325C23—81no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23#39211yescharacteristic
336C23#44711yescharacteristic
337C23#44911yescharacteristic
338C23#44912yes—
339C23#45011yescharacteristic
340C23#45211yescharacteristic
341C23#45312yes—
342C23#45511yescharacteristic
343C23#45611yescharacteristic
344C23#45612yes—
345C23#45812yes—
346D4—44no—
347D4—82no—
348D4—82no—
349D4—82no—
350C2×C4—22no—
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—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—162no—
402C2×C4—162no—
403C2×C4#44711yescharacteristic
404C2×C4#44811yescharacteristic
405C2×C4#44911yescharacteristic
406C2×C4#44911yescharacteristic
407C2×C4#46312yes—
408C2×C4#46511yescharacteristic
409C2×C4#46611yescharacteristic
410C2×C4#46811yescharacteristic
411C2×C4#46911yescharacteristic
412C2×C4#49511yescharacteristic
413C2×C4#49611yescharacteristic
414C8—44no—
415C8—44no—
416C8—82no—
417C22—82no—
418C22—84no—
419C22—84no—
420C22—82no—
421C22—82no—
422C22—161no—
423C22—161no—
424C22—162no—
425C22—162no—
426C22—161no—
427C22—161no—
428C22#160912yes—
429C22#160912yes—
430C22#161012yes—
431C22#161212yes—
432C22#161312yes—
433C22#161812yes—
434C22#162012yes—
435C22#162112yes—
436C22#163911yescharacteristic
437C22#164611yescharacteristic
438C22#164711yescharacteristic
439C22#165212yes—
440C22#165412yes—
441C22#165411yescharacteristic
442C22#166012yes—
443C22#166112yes—
444C22#199911yescharacteristic
445C22#200011yescharacteristic
446C22#200211yescharacteristic
447C22#200511yescharacteristic
448C22#200711yescharacteristic
449C22#201211yescharacteristic
450C22#202211yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—21no—
462C4—82no—
463C4—161no—
464C4—162no—
465C4—162no—
466C4—161no—
467C2—162no—
468C2#779312yes—
469C2#780012yes—
470C2#781912yes—
471C2#782012yes—
472C2#991911yescharacteristic
473C2#992011yescharacteristic
474C2#997111yescharacteristic
475C2#1004711yescharacteristic
476C2#1007411yescharacteristic
477C2#1008211yescharacteristic
478C2#1023111yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
4. #11357C2semidirect
5. C2×#1271C2semidirect
7. #7025C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20932
C2×#20842
#170692
#113572
C2×#12712
C2×#12702
#70252