Finite Group Databasebeta

← #302849

Subgroups — #302849

Order 512.

1091 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59817C211yesmaximal, characteristic
2#29208C211yesmaximal, characteristic, complemented
3#29139C211yesmaximal, characteristic
4#28687C211yesmaximal, characteristic, complemented
5#10575C214yesmaximal, complemented
6#9865C211yesmaximal, characteristic, complemented
7#9865C211yesmaximal, characteristic, complemented
8#8921C214yesmaximal
9C4×#331C211yesmaximal, characteristic
10C2×D4×Q8—21no—
11C2×D4×Q8—21no—
12C2×#498—21no—
13#2868C2211yescharacteristic
14#2683—22no—
15#2551C2212yes—
16#2538—21no—
17#2538—21no—
18#2452—22no—
19#2452—22no—
20#2409—22no—
21#2366C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C4×#127C2211yescharacteristic
25#2084—22no—
26#2057—22no—
27#2023—22no—
28#1990—21no—
29#1990—21no—
30#1957C2212yes—
31#1948C2214yes—
32#1947C2214yes—
33#1845—21no—
34#1834C2212yes—
35#1802—22no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#331C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—22no—
44D4×Q8—44no—
45D4×Q8—44no—
46D4×Q8—44no—
47D4×Q8—44no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8D411yescharacteristic
54C2×C4⋊Q8D411yescharacteristic
55C2×#127C2311yescharacteristic
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4D412yes—
67#498—44no—
68C2×#107—24no—
69C2×#106—24no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×#102—24no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8C2312yes—
77#396—24no—
78#396—24no—
79#395—24no—
80#393—24no—
81#391—21no—
82#391—21no—
83#390—22no—
84#390—22no—
85#390—21no—
86#390—21no—
87#390D412yes—
88#389—22no—
89#389—22no—
90#389—21no—
91#389—21no—
92#389D412yes—
93#389D412yes—
94#386—24no—
95#385—24no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384D411yescharacteristic
100#384D411yescharacteristic
101#384C2312yes—
102#384C2311yescharacteristic
103#380—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2312yes—
107C4×C4⋊C4C2312yes—
108C43C2311yescharacteristic
109#340—24no—
110#340—24no—
111#340—24no—
112#336—21no—
113#336—21no—
114#336—21no—
115#331Q818yescomplemented
116C22×Q8—24no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—48no—
120C22×Q8—48no—
121C22×Q8—41no—
122C22×Q8—84no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4C2×D412yes—
126C23×C4—24no—
127C23×C4—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—82no—
140C4⋊Q8—82no—
141C4⋊Q8D814yes—
142C4⋊Q8SD1614yes—
143#127—22no—
144#127C2×Q814yes—
145C22⋊Q8—416no—
146C22⋊Q8—416no—
147C22⋊Q8—48no—
148C22⋊Q8—48no—
149C4×Q8—42no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×D4—44no—
154C4×D4—44no—
155C4×D4—44no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4—42no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—42no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2411yescharacteristic
200#107—84no—
201#106—48no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—42no—
205C4⋊C8C2×Q814yescomplemented
206C4⋊C8C2×Q814yescomplemented
207#102—84no—
208C4×C8—24no—
209C4×C8—42no—
210C4×C8—42no—
211#95—24no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216C24—22no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—48no—
220C2×Q8—48no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8—84no—
228C2×Q8—832no—
229C2×Q8—82no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4C4⋊Q814yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—84no—
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—44no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4C2×D814yes—
297C4⋊C4C2×SD1614yes—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—82no—
303C22⋊C4—84no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—41no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42C2×D812yes—
313C42C2×SD1612yes—
314C42AGL(1,ℤ/8)12yes—
315C42AGL(1,ℤ/8)12yes—
316C42C22×Q812yes—
317C422−1+412yes—
318C23—24no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23C2×#12011yescharacteristic
326Q8—48no—
327Q8—48no—
328Q8—816no—
329Q8—164no—
330D4—44no—
331D4—44no—
332D4—44no—
333D4—44no—
334D4—44no—
335D4—82no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—816no—
361C2×C4—88no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4#44712yes—
366C2×C4#44712yes—
367C2×C4#45012yes—
368C2×C4#45012yes—
369C2×C4#45512yes—
370C2×C4C2×C4⋊Q811yescharacteristic
371C2×C4#53611yescharacteristic
372C2×C4D4×Q812yes—
373C2×C4D4×Q812yes—
374C2×C4C22×D811yescharacteristic
375C2×C4C22×SD1611yescharacteristic
376C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
377C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
378C2×C4#57712yes—
379C2×C4#57812yes—
380C8—44no—
381C8—44no—
382C8—82no—
383C22—44no—
384C22—44no—
385C22—44no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22#243411yescharacteristic
391C22C2×#44711yescharacteristic
392C22C2×#45011yescharacteristic
393C22#301011yescharacteristic
394C22#301111yescharacteristic
395C22C2×#45511yescharacteristic
396C22#302111yescharacteristic
397C4—21no—
398C4—21no—
399C4—22no—
400C4—22no—
401C4—41no—
402C4—44no—
403C4—44no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—162no—
408C4#163914yes—
409C4Q8×D812yes—
410C4Q8×SD1612yes—
411C4#338412yes—
412C4#338612yes—
413C2—84no—
414C2C2×#163911yescharacteristic
415C2#1872511yescharacteristic
416C2#1891211yescharacteristic
417C2#1891311yescharacteristic
418C2#2582311yescharacteristic
419C2#2583411yescharacteristic
420C2#2583811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29208C2semidirect
4. #28687C2semidirect
5. #10575C2semidirect
6. #9865C2semidirect
7. #9865C2semidirect
115. #331Q8semidirect
205. C4⋊C8C2×Q8semidirect
206. C4⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598172
#292082
#291392
#286872
#105752
#98652
#89212
C4×#3312