Finite Group Databasebeta

← #148043

Subgroups — #148043

Order 512.

748 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2145C212yesmaximal
2#21228C211yesmaximal, characteristic
3#20799C211yesmaximal, characteristic, complemented
4C2×#1384C212yesmaximal, complemented
5C2×#1377C211yesmaximal, characteristic, complemented
6C22×#135C2211yescharacteristic
7C22×M32C2212yes—
8C2×#427—21no—
9C2×#427C2211yescharacteristic
10C2×#427C2212yes—
11C2×#416—21no—
12C22×#97C2211yescharacteristic
13#2297—21no—
14#2285—21no—
15#2145—24no—
16#2145—24no—
17C2×#360—24no—
18#1973—22no—
19#1972C412yescomplemented
20#1971C412yescomplemented
21#1847—24no—
22#1384—24no—
23#1377—22no—
24C22×#41—21no—
25C2×#135—22no—
26C2×#135—22no—
27C2×#135—21no—
28C2×#135—21no—
29C2×#135—22no—
30C2×#135—22no—
31C2×#135D412yes—
32C2×#135Q812yes—
33C22×Q16C2×C411yescharacteristic
34C22×SD16—21no—
35C22×D8C2×C411yescharacteristic
36C23×C8D411yescharacteristic
37C23×C8C2311yescharacteristic
38C2×#117—21no—
39C2×#117—22no—
40C22×C42D411yescharacteristic
41C2×M32—22no—
42C2×M32—22no—
43C2×M32—22no—
44C2×M32—22no—
45C2×M32—22no—
46C2×M32D414yes—
47C22×C16—22no—
48#427—22no—
49#427—22no—
50#427—22no—
51#427—22no—
52#427—22no—
53#427—22no—
54#427—42no—
55#427—42no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—21no—
59C2×#106—22no—
60C2×#106C2×C412yes—
61C2×#106D414yes—
62#416—42no—
63#416—42no—
64C2×#103—22no—
65C2×#102—22no—
66C2×#97—22no—
67C2×#97—22no—
68C2×#97C2×C412yes—
69C2×#97D412yes—
70#391—22no—
71#390—22no—
72#386—42no—
73#384—22no—
74#382—22no—
75#380—42no—
76C4×C4⋊C4—42no—
77C4×C22⋊C4—42no—
78#360—216no—
79C2×#41—22no—
80C2×#41—22no—
81C2×#41—41no—
82C2×#41—41no—
83C2×#41—41no—
84C2×#41—41no—
85C2×#41—41no—
86C2×#41—41no—
87C2×#41—41no—
88C2×#41—41no—
89C2×#41—41no—
90C2×#41—41no—
91C22×Q8—21no—
92C22×D4—21no—
93C22×D4—41no—
94C23×C4—41no—
95C23×C4D812yes—
96C23×C4C2×D411yescharacteristic
97#135—24no—
98#135—24no—
99#135—24no—
100#135—24no—
101#135—42no—
102#135—42no—
103#135—42no—
104#135—42no—
105C2×Q16—21no—
106C2×Q16—22no—
107C2×Q16—22no—
108C2×Q16C4⋊C412yescomplemented
109C2×SD16—22no—
110C2×SD16—22no—
111C2×SD16—42no—
112C2×SD16—42no—
113C2×D8—21no—
114C2×D8—22no—
115C2×D8—22no—
116C2×D8C4⋊C412yescomplemented
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8C22⋊C412yes—
122C22×C8D812yes—
123C22×C8SD1612yes—
124C22×C8C22×C411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D412yes—
127C22×C8C2×D412yes—
128C22×C8C2×D411yescharacteristic
129C22×C8C2×Q811yescharacteristic
130C22×C8#4111yescharacteristic
131C22×C8#4111yescharacteristic
132C22×C8#4112yes—
133C22×C8#4112yes—
134#117—42no—
135#117—41no—
136#117—41no—
137#117—42no—
138#117—41no—
139#117—42no—
140#117—41no—
141#117—42no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—44no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—44no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—41no—
153C2×C42—42no—
154C2×C42C22⋊C412yes—
155C2×C42SD1612yes—
156M32—24no—
157M32—24no—
158M32—42no—
159M32—42no—
160C2×C16—24no—
161C2×C16—42no—
162C2×C16—42no—
163#107—44no—
164#107—42no—
165#107—42no—
166#107—42no—
167#106—24no—
168#106—24no—
169#106—24no—
170#106—44no—
171#103—44no—
172#102—44no—
173#97—24no—
174#97—24no—
175#95—44no—
176#95—44no—
177C24—41no—
178C24C2×D811yescharacteristic
179#41—42no—
180#41—42no—
181#41—42no—
182#41—42no—
183#41—42no—
184#41—42no—
185#41—42no—
186#41—42no—
187#41—81no—
188#41—81no—
189#41—81no—
190#41—81no—
191#41—81no—
192#41—81no—
193#41—81no—
194#41—81no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—81no—
210C2×D4—81no—
211C2×D4—81no—
212C2×D4—81no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—82no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4#10214yes—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12711yescharacteristic
233C22×C4C2×D811yescharacteristic
234C22×C4C2×SD1611yescharacteristic
235C22×C4C2×SD1611yescharacteristic
236C22×C4AGL(1,ℤ/8)11yescharacteristic
237C22×C4AGL(1,ℤ/8)11yescharacteristic
238C22×C4#13711yescharacteristic
239C22×C4#13711yescharacteristic
240Q16—24no—
241Q16—42no—
242SD16—44no—
243SD16—44no—
244D8—24no—
245D8—42no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8#10214yes—
253C2×C8C2×C4⋊C411yescharacteristic
254C2×C8#11711yescharacteristic
255C2×C8C4×D412yes—
256C2×C8C4×D412yes—
257C2×C8#12012yes—
258C2×C8#12112yes—
259C2×C8#12111yescharacteristic
260C2×C8#12112yes—
261C2×C8#12112yes—
262C2×C8C22⋊Q812yes—
263C2×C8C22⋊Q812yes—
264C2×C8C22⋊Q811yescharacteristic
265C2×C8#12312yes—
266C2×C8#12312yes—
267C2×C8#12412yes—
268C2×C8#12612yes—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—88no—
274C22⋊C4—88no—
275C22⋊C4—82no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C22⋊C4—82no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—82no—
284C42—81no—
285C42—81no—
286C42—82no—
287C16—44no—
288C23—42no—
289C23—42no—
290C23—84no—
291C23—81no—
292C23—81no—
293C23C2×#10211yescharacteristic
294C23#44711yescharacteristic
295C23#45911yescharacteristic
296C23#47411yescharacteristic
297C23#48011yescharacteristic
298C23#48611yescharacteristic
299C23#49311yescharacteristic
300Q8—42no—
301Q8—42no—
302Q8—81no—
303Q8—81no—
304D4—42no—
305D4—42no—
306D4—82no—
307D4—82no—
308D4—82no—
309D4—82no—
310D4—81no—
311D4—81no—
312C2×C4—24no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—84no—
324C2×C4—88no—
325C2×C4—82no—
326C2×C4—84no—
327C2×C4#38011yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#39011yescharacteristic
330C2×C4#39211yescharacteristic
331C2×C4#39311yescharacteristic
332C2×C4#39711yescharacteristic
333C2×C4#39911yescharacteristic
334C2×C4C2×#10211yescharacteristic
335C2×C4#41711yescharacteristic
336C2×C4#41711yescharacteristic
337C2×C4#44711yescharacteristic
338C2×C4#44811yescharacteristic
339C2×C4#44811yescharacteristic
340C2×C4#45911yescharacteristic
341C2×C4#46111yescharacteristic
342C2×C4#46111yescharacteristic
343C2×C4#47411yescharacteristic
344C2×C4#47611yescharacteristic
345C2×C4#47611yescharacteristic
346C2×C4#48011yescharacteristic
347C2×C4#48111yescharacteristic
348C2×C4#48111yescharacteristic
349C2×C4#48611yescharacteristic
350C2×C4#48911yescharacteristic
351C2×C4#48911yescharacteristic
352C2×C4#49211yescharacteristic
353C2×C4#49611yescharacteristic
354C2×C4#49711yescharacteristic
355C8—24no—
356C8#38512yes—
357C8#38812yes—
358C8#39412yes—
359C8#39612yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—84no—
368C22—82no—
369C22—162no—
370C22—162no—
371C22#180211yescharacteristic
372C22#187411yescharacteristic
373C22#195711yescharacteristic
374C22#201111yescharacteristic
375C22#202311yescharacteristic
376C22#207111yescharacteristic
377C22#209311yescharacteristic
378C4—44no—
379C4—82no—
380C4—164no—
381C4—162no—
382C4#143911yescharacteristic
383C4#180211yescharacteristic
384C4#187411yescharacteristic
385C4#195911yescharacteristic
386C4#201311yescharacteristic
387C4#202311yescharacteristic
388C4#207111yescharacteristic
389C4#209411yescharacteristic
390C2—21no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—21no—
396C2—162no—
397C2#1104911yescharacteristic
398C2#1300812yes—
subgroupG/H
centerC2×C4#481
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
socleC22#1874
prime radicalC22×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
3. #20799C2semidirect
4. C2×#1384C2semidirect
5. C2×#1377C2semidirect
19. #1972C4semidirect
20. #1971C4semidirect
108. C2×Q16C4⋊C4semidirect
116. C2×D8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21452
#212282
#207992
C2×#13842
C2×#13772