Finite Group Databasebeta

← #192108

Subgroups — #192108

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34377C211yesmaximal, characteristic
2#21363C211yesmaximal, characteristic, complemented
3#21358C211yesmaximal, characteristic, complemented
4C2×#1469C211yesmaximal, characteristic
5#14288C211yesmaximal, characteristic
6#8858C214yesmaximal, complemented
7#8853C214yesmaximal, complemented
8#8025C211yesmaximal, characteristic, complemented
9#8018C211yesmaximal, characteristic, complemented
10C22×#124C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2379—22no—
14#2367—22no—
15#2342—22no—
16Q8×C22⋊C4—21no—
17D4×C22⋊C4—21no—
18#2337—22no—
19#2326—22no—
20#2315C2212yes—
21#2312C2212yes—
22#2285C2211yescharacteristic
23#2283C2211yescharacteristic
24C2×#383C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1949—22no—
27#1947—22no—
28#1853C2212yes—
29#1852C2214yes—
30#1852C2212yes—
31#1776—22no—
32C4×#107—22no—
33#1469C418yescomplemented
34C2×#323C2214yes—
35#1446—21no—
36#1439C2214yes—
37#1438—21no—
38#1437—22no—
39#1294C2214yes—
40#1294C2212yes—
41#1294C2212yes—
42C23×Q8D411yescharacteristic
43C23×D4D411yescharacteristic
44C2×#124—21no—
45C2×#124—21no—
46C2×#124—21no—
47C2×#124—21no—
48C2×#124—22no—
49C2×#124—22no—
50C2×#124C2×C414yes—
51C2×#124Q814yes—
52C2×C4×Q8—22no—
53C2×C4×D4—22no—
54C2×#117—21no—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—24no—
58C22×C22⋊C4D412yes—
59C22×C22⋊C4C2312yes—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#432—22no—
63#432C2×C414yescomplemented
64#431—22no—
65#431C2×C414yescomplemented
66C2×#107—24no—
67C2×#107—22no—
68C2×#106—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C22⋊C8C2312yes—
72C2×C22⋊C8C2312yes—
73C2×#97—21no—
74C2×C4×C8—21no—
75#391—22no—
76#391—24no—
77#390—22no—
78#388—24no—
79#388—24no—
80#388—24no—
81#388—24no—
82#386—24no—
83#385—24no—
84#385—24no—
85#385—24no—
86#384D412yes—
87#383—21no—
88#383—22no—
89#383—22no—
90#382—21no—
91#382D412yes—
92#381—21no—
93#380—22no—
94#380—21no—
95#380C2311yescharacteristic
96#380C2312yes—
97#380C2311yescharacteristic
98#379—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—21no—
109#376—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—24no—
114C2×#95C2312yes—
115#340—24no—
116#340—22no—
117#340—22no—
118#323C2×C4116yescomplemented
119C25C2×D412yes—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—82no—
124C22×Q8C4⋊C414yes—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—82no—
129C22×D4C4⋊C414yes—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139#124—24no—
140#124—24no—
141#124—24no—
142#124—24no—
143#124—24no—
144#124—24no—
145#124—24no—
146#124—24no—
147C4×Q8—84no—
148C4×D4—84no—
149#117—82no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D414yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—48no—
173C2×C22⋊C4—84no—
174C2×C22⋊C4—84no—
175C2×C22⋊C4—84no—
176C2×C22⋊C4C4⋊C418yes—
177C2×C22⋊C4C22×C414yes—
178C2×C22⋊C4C2×Q814yes—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—82no—
185C2×C42C22×C412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q812yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193#107—44no—
194#107—48no—
195#106—44no—
196C4⋊C8—24no—
197C4⋊C8—24no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8C22×C414yes—
201C22⋊C8C22×C414yes—
202#97—24no—
203C4×C8—24no—
204#95—24no—
205#95—24no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—21no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—41no—
216#95—41no—
217#95—42no—
218#95—42no—
219#95—84no—
220C24—22no—
221C24—22no—
222C24—24no—
223C24—24no—
224C24—44no—
225C24—44no—
226C24C2×C4⋊C414yes—
227C24C22×D411yescharacteristic
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—44no—
231C2×Q8—82no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—44no—
237C2×D4—82no—
238C2×D4—84no—
239C2×D4—84no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4C2×C4⋊C412yes—
264C22×C4C2×C4⋊C414yes—
265C22×C4C2×C4⋊C412yes—
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12312yes—
271C22×C4#12312yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×Q811yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—44no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—28no—
298C22⋊C4—28no—
299C22⋊C4—42no—
300C22⋊C4—44no—
301C22⋊C4—42no—
302C22⋊C4—84no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C22⋊C4—816no—
306C42—44no—
307C42—82no—
308C42—82no—
309C42—82no—
310C42—84no—
311C42C4×D412yes—
312C42C4×D412yes—
313C42C4×D412yes—
314C42C4×D412yes—
315C42C22⋊Q812yes—
316C42C22⋊Q812yes—
317C42C22⋊Q812yes—
318C42C22⋊Q812yes—
319C23—22no—
320C23—22no—
321C23—24no—
322C23—28no—
323C23—44no—
324C23—44no—
325C23—42no—
326C23—42no—
327C23—44no—
328C23—44no—
329C23—44no—
330C23—82no—
331C23—82no—
332C23—84no—
333C23—84no—
334C23—88no—
335C23C22×C4⋊C411yescharacteristic
336C23C2×#12011yescharacteristic
337C23C2×#12311yescharacteristic
338C23#57612yes—
339C23#57812yes—
340Q8—48no—
341Q8—84no—
342D4—48no—
343D4—84no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—84no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—168no—
384C2×C4#38512yes—
385C2×C4#38512yes—
386C2×C4#38512yes—
387C2×C4#38512yes—
388C2×C4C2×C4×D411yescharacteristic
389C2×C4C2×C4×D411yescharacteristic
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4C2×C22⋊Q811yescharacteristic
392C8—44no—
393C8—44no—
394C22—24no—
395C22—24no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—88no—
404C22—84no—
405C22—84no—
406C22—164no—
407C22C2×#38511yescharacteristic
408C22#291312yes—
409C22#300811yescharacteristic
410C22#301111yescharacteristic
411C22#301912yes—
412C22#302512yes—
413C22#309812yes—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—44no—
423C4—84no—
424C4—84no—
425C4—82no—
426C4—82no—
427C4—162no—
428C2—22no—
429C2—22no—
430C2—84no—
431C2#1824012yes—
432C2#1824312yes—
433C2#2084711yescharacteristic
434C2#2085611yescharacteristic
435C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21363C2semidirect
3. #21358C2semidirect
6. #8858C2semidirect
7. #8853C2semidirect
8. #8025C2semidirect
9. #8018C2semidirect
33. #1469C4semidirect
63. #432C2×C4semidirect
65. #431C2×C4semidirect
118. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343772
#213632
#213582
C2×#14692
#142882
#88582
#88532
#80252
#80182