Finite Group Databasebeta

← #376811

Subgroups — #376811

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60193C211yesmaximal, characteristic
2#23560C211yesmaximal, characteristic
3#21164C212yesmaximal
4#21025C211yesmaximal, characteristic
5#11393C211yesmaximal, characteristic
6#11343C211yesmaximal, characteristic
7#11212C211yesmaximal, characteristic
8#10901C212yesmaximal
9#10900C212yesmaximal
10#10887C212yesmaximal
11#10711C211yesmaximal, characteristic
12#3458C2211yescharacteristic
13#2962—24no—
14#2962—24no—
15#2961—22no—
16#2960—22no—
17C2×#430C2212yes—
18#2911C2212yes—
19#2905C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#424C2212yes—
22#2873C2212yes—
23C2×#405C2212yes—
24C2×#405C2211yescharacteristic
25#2285C2211yescharacteristic
26#2061C412yescomplemented
27#2061C412yescomplemented
28#2060C412yescomplemented
29#2060C412yescomplemented
30#1946—24no—
31#1945—24no—
32#1917—22no—
33#1916—22no—
34C2×#344C2212yes—
35#1742C2211yescharacteristic
36#1740C2211yescharacteristic
37C2×#341C2212yes—
38#1735C2212yes—
39#1733C2212yes—
40#1733C2212yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43#1729C2211yescharacteristic
44#1729C2211yescharacteristic
45#1728C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C22×M16C2311yescharacteristic
48C22×M16C2312yes—
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53#544—22no—
54#544—22no—
55#544C2×C412yes—
56#544C2×C412yes—
57#543—22no—
58#533—24no—
59#533—22no—
60C2×#117C2312yes—
61C2×#117C2311yescharacteristic
62C2×#117C2312yes—
63C2×#117C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#446—24no—
66#446—24no—
67#446—28no—
68#446—28no—
69C8×Q8—24no—
70C8×Q8—24no—
71#430—24no—
72#430—42no—
73C2×#108—24no—
74#428—22no—
75#428—22no—
76#428C2×C414yes—
77#428C2×C414yes—
78#427—24no—
79#424—24no—
80#424—22no—
81#424—22no—
82#424—22no—
83#424—22no—
84#424D414yes—
85#424D414yes—
86#423—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91#405—22no—
92#405—22no—
93#405—24no—
94#405—24no—
95#405—24no—
96#405—42no—
97C4×M16—24no—
98C2×#97—22no—
99C2×#97D412yes—
100C2×C4×C8—22no—
101C2×C4×C8D412yes—
102#384C2×C411yescharacteristic
103#384C2×C411yescharacteristic
104#382C2×C411yescharacteristic
105#382C2×C411yescharacteristic
106#380—22no—
107C4×C4⋊C4—22no—
108#344Q818yescomplemented
109#343C2×C412yes—
110#343C2×C412yes—
111#342—22no—
112#341—24no—
113#340—22no—
114#340—22no—
115#339—24no—
116#339—24no—
117#339—42no—
118#339C2×C414yes—
119#339C2×C414yes—
120#336—21no—
121#336—22no—
122#336—22no—
123#336—21no—
124#336—22no—
125C23×C4—21no—
126C23×C4C2411yescharacteristic
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16C22×C414yes—
143C2×M16C2×D414yes—
144C2×M16C2×Q814yes—
145C2×M16C2×Q814yes—
146C2×M16#4114yes—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167#125—24no—
168#125—24no—
169#125—24no—
170C22⋊Q8—28no—
171C22⋊Q8—28no—
172C22⋊Q8—24no—
173C22⋊Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117C22×C414yes—
186#117C2×D414yes—
187#117C2×Q814yes—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4#4112yes—
196C2×C4⋊C4#4112yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4#4112yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212#108—48no—
213#107—24no—
214#107—24no—
215#106—24no—
216#106—24no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—24no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—24no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—42no—
226#97—28no—
227#97—24no—
228#97—44no—
229C4×C8—28no—
230C4×C8—22no—
231C4×C8—22no—
232C4×C8—44no—
233#95—22no—
234#95—22no—
235C24C2×#4111yescharacteristic
236C2×Q8—24no—
237C2×Q8—24no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—81no—
249C22×C4C4×D412yes—
250C22×C4C4×D412yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D412yes—
253C22×C4C22×D411yescharacteristic
254C22×C4C22×Q812yes—
255C22×C4C2×#4112yes—
256C22×C4C2×#4112yes—
257C22×C4H(5,𝔽2)11yescharacteristic
258C22×C42−1+411yescharacteristic
259C22×C42−1+411yescharacteristic
260C22×C42−1+411yescharacteristic
261M16—24no—
262M16—24no—
263M16—24no—
264M16—24no—
265M16—24no—
266M16—24no—
267M16—24no—
268M16—42no—
269M16—42no—
270M16—42no—
271M16—42no—
272M16C4×Q818yes—
273C2×C8—24no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—22no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8#12011yescharacteristic
298C2×C8#12011yescharacteristic
299C2×C8#12012yes—
300C2×C8C4⋊Q814yes—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—82no—
315C4⋊C4C4×D414yes—
316C4⋊C4C4×D414yes—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—24no—
324C42—24no—
325C42—22no—
326C42—22no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—24no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—42no—
337C23—41no—
338C23—41no—
339C23C2×C4×D411yescharacteristic
340C23#52512yes—
341C23#53312yes—
342C23#54111yescharacteristic
343C23#54211yescharacteristic
344Q8—84no—
345Q8—84no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4C2×C4×D412yes—
370C2×C4C2×C4×Q812yes—
371C2×C4#51911yescharacteristic
372C2×C4#52011yescharacteristic
373C2×C4C2×#12011yescharacteristic
374C2×C4#52511yescharacteristic
375C2×C4C2×C4⋊Q811yescharacteristic
376C2×C4#53211yescharacteristic
377C2×C4#53611yescharacteristic
378C2×C4#53711yescharacteristic
379C2×C4#54612yes—
380C2×C4#54712yes—
381C2×C4D4×Q812yes—
382C2×C4D4×Q812yes—
383C2×C4#55212yes—
384C2×C4#55412yes—
385C2×C4#55612yes—
386C2×C4#55712yes—
387C8—28no—
388C8—21no—
389C8—22no—
390C8—21no—
391C8—24no—
392C8—44no—
393C8—81no—
394C8—81no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22C4×#12011yescharacteristic
403C22C4×C4⋊Q811yescharacteristic
404C22#232311yescharacteristic
405C22#246212yes—
406C22#246512yes—
407C4—42no—
408C4—42no—
409C4—44no—
410C4—42no—
411C4—44no—
412C4—81no—
413C4—84no—
414C4—81no—
415C4—81no—
416C4#236012yes—
417C4#236512yes—
418C4#243011yescharacteristic
419C4#243411yescharacteristic
420C4#258411yescharacteristic
421C4#260411yescharacteristic
422C2—21no—
423C2—22no—
424C2—21no—
425C2—22no—
426C2#1418611yescharacteristic
427C2#2147711yescharacteristic
428C2#2147811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C4×#120
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #2061C4semidirect
27. #2061C4semidirect
28. #2060C4semidirect
29. #2060C4semidirect
108. #344Q8semidirect

Maximal subgroups

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

subgroupindex
#601932
#235602
#211642
#210252
#113932
#113432
#112122
#109012
#109002
#108872
#107112