Finite Group Databasebeta

← #149269

Subgroups — #149269

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2159C211yesmaximal, characteristic
2#21154C211yesmaximal, characteristic
3C2×#1944C211yesmaximal, characteristic
4C2×#1386C211yesmaximal, characteristic
5C2×#1386C211yesmaximal, characteristic
6#12557C211yesmaximal, characteristic
7C2×#1377C211yesmaximal, characteristic
8C22×M32C2211yescharacteristic
9C22×M32C2211yescharacteristic
10C2×#427C2211yescharacteristic
11C2×#427C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#97C2211yescharacteristic
15#2296—21no—
16C2×#389—21no—
17#2159—22no—
18#2159—22no—
19C2×#367—21no—
20C2×#366—21no—
21#1944—22no—
22#1944—21no—
23#1944—21no—
24#1944—22no—
25#1943—22no—
26#1735—21no—
27C2×#340—21no—
28#1386—22no—
29#1386C414yescomplemented
30#1380—22no—
31#1377—22no—
32C23×C8D411yescharacteristic
33C23×C8C2311yescharacteristic
34C2×#117—21no—
35C2×#117—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—41no—
39C22×C42D411yescharacteristic
40C2×M32—21no—
41C2×M32—21no—
42C2×M32—21no—
43C2×M32—21no—
44C2×M32—21no—
45C2×M32—21no—
46C2×M32—21no—
47C2×M32—21no—
48C2×M32C2×C412yes—
49C2×M32D412yes—
50C2×M32Q812yes—
51C2×M32Q812yes—
52C22×C16—21no—
53C22×C16—21no—
54#427—21no—
55#427—21no—
56#427—21no—
57#427—21no—
58#427—21no—
59#427—21no—
60#427C2×C412yes—
61#427Q812yes—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107D412yes—
68C2×#107D411yescharacteristic
69C2×#107D411yescharacteristic
70C2×#107D412yes—
71C2×#106—21no—
72C2×#106—21no—
73C2×#106—21no—
74C2×#106—21no—
75C2×#106—21no—
76C2×#106C2×C412yes—
77C2×#106D412yes—
78C2×#106Q812yes—
79C2×#97—21no—
80C2×#97—21no—
81C2×#97—21no—
82C2×#97—21no—
83C2×#97—21no—
84C2×#97—21no—
85#389—22no—
86#389—22no—
87#389—24no—
88#389—42no—
89#385—41no—
90#385—41no—
91C4×C4⋊C4—42no—
92C2×#95—41no—
93#367—24no—
94#366—24no—
95#340—22no—
96#340—22no—
97#340—24no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4D811yescharacteristic
101C23×C4D811yescharacteristic
102C23×C4C2×D411yescharacteristic
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8Q1612yes—
110C22×C8C22×C411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×Q811yescharacteristic
117C22×C8C2×Q811yescharacteristic
118C22×C8C2×Q811yescharacteristic
119C22×C8C2×Q811yescharacteristic
120C22×C8#4111yescharacteristic
121C22×C8#4111yescharacteristic
122C22×C8#4111yescharacteristic
123C22×C8#4111yescharacteristic
124#117—22no—
125#117—22no—
126#117—41no—
127#117—41no—
128#117—41no—
129#117—41no—
130#117—41no—
131#117—41no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—84no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—41no—
160C2×C42—41no—
161M32—22no—
162M32—22no—
163M32—22no—
164M32—22no—
165M32—22no—
166M32—41no—
167M32—41no—
168M32C4⋊C414yescomplemented
169C2×C16—22no—
170C2×C16—22no—
171C2×C16—22no—
172C2×C16—22no—
173C2×C16—41no—
174C2×C16—41no—
175#107—22no—
176#107—21no—
177#107—22no—
178#107—21no—
179#107—22no—
180#107—21no—
181#107—21no—
182#107—41no—
183#107—41no—
184#107C4⋊C414yescomplemented
185#106—22no—
186#106—22no—
187#106—22no—
188#106—22no—
189#106—22no—
190#106—41no—
191#106—41no—
192#106C4⋊C414yescomplemented
193#97—41no—
194#97—41no—
195#97—41no—
196#97—41no—
197#95—44no—
198#95—42no—
199#95—42no—
200C24C2×D811yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4C4×D411yescharacteristic
223C22×C4#12011yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C4⋊Q811yescharacteristic
229C22×C4C2×Q1611yescharacteristic
230C22×C4#13511yescharacteristic
231C22×C4#13511yescharacteristic
232C22×C4AGL(1,ℤ/8)11yescharacteristic
233C22×C4AGL(1,ℤ/8)11yescharacteristic
234C22×C4#13711yescharacteristic
235C22×C4#13711yescharacteristic
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8C2×C4⋊C411yescharacteristic
244C2×C8C2×C4⋊C411yescharacteristic
245C2×C8C2×C4⋊C411yescharacteristic
246C2×C8C4×D411yescharacteristic
247C2×C8C4×D411yescharacteristic
248C2×C8C4×Q811yescharacteristic
249C2×C8#12011yescharacteristic
250C2×C8#12111yescharacteristic
251C2×C8#12111yescharacteristic
252C2×C8C22⋊Q811yescharacteristic
253C2×C8C22⋊Q811yescharacteristic
254C2×C8C22⋊Q811yescharacteristic
255C2×C8C22⋊Q811yescharacteristic
256C2×C8C22⋊Q811yescharacteristic
257C2×C8C22⋊Q811yescharacteristic
258C2×C8C22⋊Q811yescharacteristic
259C2×C8#12311yescharacteristic
260C2×C8#12311yescharacteristic
261C2×C8#12311yescharacteristic
262C2×C8#12511yescharacteristic
263C2×C8#12511yescharacteristic
264C2×C8#12511yescharacteristic
265C2×C8#12711yescharacteristic
266C2×C8C4⋊Q811yescharacteristic
267C2×C8C4⋊Q811yescharacteristic
268C2×C8C4⋊Q811yescharacteristic
269C2×C8C4⋊Q811yescharacteristic
270C2×C8C4⋊Q811yescharacteristic
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—84no—
284C4⋊C4—84no—
285C4⋊C4—88no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C42—41no—
293C42—42no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42—42no—
298C42—81no—
299C42—81no—
300C16—24no—
301C16—42no—
302C23—22no—
303C23—22no—
304C23C4×D811yescharacteristic
305C23#44711yescharacteristic
306C23#44711yescharacteristic
307C23#46611yescharacteristic
308C23#47411yescharacteristic
309C23#47411yescharacteristic
310C23#50011yescharacteristic
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—162no—
335C2×C4—162no—
336C2×C4#38511yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4#38911yescharacteristic
339C2×C4#39211yescharacteristic
340C2×C4#39511yescharacteristic
341C2×C4#39511yescharacteristic
342C2×C4#39811yescharacteristic
343C2×C4C4×Q1611yescharacteristic
344C2×C4#44011yescharacteristic
345C2×C4#44011yescharacteristic
346C2×C4#44911yescharacteristic
347C2×C4#44911yescharacteristic
348C2×C4#45111yescharacteristic
349C2×C4#45111yescharacteristic
350C2×C4#45211yescharacteristic
351C2×C4#45211yescharacteristic
352C2×C4#46711yescharacteristic
353C2×C4#46911yescharacteristic
354C2×C4#47011yescharacteristic
355C2×C4#47711yescharacteristic
356C2×C4#47711yescharacteristic
357C2×C4#47811yescharacteristic
358C2×C4#47811yescharacteristic
359C2×C4#47911yescharacteristic
360C2×C4#47911yescharacteristic
361C2×C4#49911yescharacteristic
362C2×C4#50111yescharacteristic
363C2×C4#50111yescharacteristic
364C8—42no—
365C8#38411yescharacteristic
366C8#38511yescharacteristic
367C8#38911yescharacteristic
368C8#38911yescharacteristic
369C8#39511yescharacteristic
370C8#39811yescharacteristic
371C8#39811yescharacteristic
372C8#39911yescharacteristic
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22#186211yescharacteristic
382C22#186211yescharacteristic
383C22#194211yescharacteristic
384C22#199911yescharacteristic
385C22#205711yescharacteristic
386C22#205711yescharacteristic
387C22#208611yescharacteristic
388C22#219012yes—
389C22#219112yes—
390C4—84no—
391C4—81no—
392C4—81no—
393C4—84no—
394C4—161no—
395C4—162no—
396C4—161no—
397C4#144411yescharacteristic
398C4#186311yescharacteristic
399C4#186311yescharacteristic
400C4#194411yescharacteristic
401C4#200111yescharacteristic
402C4#205911yescharacteristic
403C4#205911yescharacteristic
404C4#208711yescharacteristic
405C2—21no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2#1132811yescharacteristic
410C2#1294312yes—
411C2#1331611yescharacteristic
412C2#1331711yescharacteristic
413C2#1345412yes—
subgroupG/H
center, socleC23#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #1386C4semidirect
168. M32C4⋊C4semidirect
184. #107C4⋊C4semidirect
192. #106C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21592
#211542
C2×#19442
C2×#13862
#125572
C2×#13772