Finite Group Databasebeta

← #378895

Subgroups — #378895

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2079C211yesmaximal, characteristic
2#21059C212yesmaximal
3C2×C4×#106C211yesmaximal, characteristic, complemented
4#20383C211yesmaximal, characteristic
5#11366C211yesmaximal, characteristic, complemented
6#11330C211yesmaximal, characteristic, complemented
7#11106C211yesmaximal, characteristic
8#10896C212yesmaximal, complemented
9#10822C211yesmaximal, characteristic, complemented
10#10802C212yesmaximal, complemented
11#10774C212yesmaximal
12C22×#131C2211yescharacteristic
13C2×#428C2212yes—
14C22×#106C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#417C2212yes—
17C2×#417C2212yes—
18C2×#408C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2297C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23#2079C418yescomplemented
24#1928—22no—
25#1927—22no—
26#1926—22no—
27#1925—22no—
28C4×#106—22no—
29C4×#106—22no—
30#1753—22no—
31C8×C22⋊C4—22no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341C2212yes—
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39#1736C2212yes—
40C2×#340C2212yes—
41#1729C2211yescharacteristic
42#1728C2211yescharacteristic
43#1728C2212yes—
44C22×#41C2311yescharacteristic
45C2×#131—22no—
46C2×#131—22no—
47C2×#131—24no—
48C2×#131—42no—
49C22×M16—21no—
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C23×C8—21no—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42—22no—
61C22×C42C2311yescharacteristic
62#428—24no—
63#428—42no—
64C2×#107—22no—
65C2×#106—22no—
66C2×#106—22no—
67C2×#106—22no—
68C2×#106—24no—
69C2×#106—22no—
70C2×#106C2×C414yes—
71#417—24no—
72#417—24no—
73#417—42no—
74#417C2×C418yes—
75C2×#103—22no—
76C2×#103—22no—
77C2×#102—22no—
78C2×#102—22no—
79#408—22no—
80#408C2×C414yes—
81C2×C22⋊C8—22no—
82#405—42no—
83C2×#97—21no—
84C2×C4×C8—24no—
85C2×C4×C8—21no—
86C2×C4×C8D412yes—
87C2×C4×C8D412yes—
88#391D412yes—
89#390D412yes—
90#386—21no—
91#386—21no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—22no—
97C2×#95—22no—
98#343—24no—
99#343—42no—
100#343—42no—
101#343C2×C418yes—
102#341—22no—
103#341—24no—
104#341C2×C414yes—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113C2×#41—21no—
114C2×#41—21no—
115C2×#41—22no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41C22×C412yes—
120C2×#41#4112yes—
121C22×Q8C2×D411yescharacteristic
122C22×D4—21no—
123C22×D4C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4#4111yescharacteristic
128C23×C4#4111yescharacteristic
129C23×C4C2411yescharacteristic
130#131—48no—
131#131—48no—
132C2×M16—24no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—42no—
151C2×M16—42no—
152C2×M16C22×C414yes—
153C2×M16C22×C412yes—
154C2×M16#4112yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—24no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8C22×C412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8#4112yes—
172C22×C8#4114yes—
173#117—22no—
174#117—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22×C414yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4#4114yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—84no—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199#107—44no—
200#106—28no—
201#106—28no—
202#106—44no—
203#103—28no—
204#103—44no—
205#102—28no—
206#102—44no—
207C22⋊C8—28no—
208#97—42no—
209C4×C8—24no—
210C4×C8—24no—
211C4×C8—42no—
212#95—22no—
213#95—22no—
214#95—42no—
215#95—42no—
216C24—21no—
217C24C2×#4111yescharacteristic
218#41—24no—
219#41—24no—
220#41—24no—
221#41—24no—
222#41—42no—
223#41—42no—
224#41—42no—
225#41—42no—
226C2×Q8—21no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8C4×D412yes—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—21no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4C4×D412yes—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4#12112yes—
261C22×C4#12112yes—
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)12yes—
271C22×C42−1+412yes—
272M16—24no—
273M16—24no—
274M16—24no—
275M16—24no—
276M16—48no—
277M16—42no—
278M16—42no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16—42no—
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—24no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—48no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8C4×D414yes—
309C2×C8#12312yes—
310C2×C8#12312yes—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C22⋊C4—44no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C42—22no—
325C42—22no—
326C42—24no—
327C42—41no—
328C42—41no—
329C42—84no—
330C42—84no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23C4×#4111yescharacteristic
339C23C2×#12111yescharacteristic
340C23#52911yescharacteristic
341C23#55212yes—
342C23#55312yes—
343Q8—24no—
344Q8—42no—
345D4—24no—
346D4—44no—
347D4—44no—
348D4—42no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4#47114yes—
377C2×C4C2×C4×D411yescharacteristic
378C2×C4C2×C4×D411yescharacteristic
379C2×C4C4×#4112yes—
380C2×C4#52012yes—
381C2×C4C2×#12311yescharacteristic
382C2×C4#52511yescharacteristic
383C2×C4#52511yescharacteristic
384C2×C4#52912yes—
385C2×C4#53211yescharacteristic
386C2×C4#53511yescharacteristic
387C2×C4#53611yescharacteristic
388C2×C4#54212yes—
389C2×C4#54612yes—
390C2×C4#54712yes—
391C2×C4#55112yes—
392C2×C4#55112yes—
393C2×C4#55212yes—
394C2×C4#55612yes—
395C8—22no—
396C8—22no—
397C8—24no—
398C8—42no—
399C8—44no—
400C8—42no—
401C8—82no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—44no—
409C22—42no—
410C22—82no—
411C22—81no—
412C22—81no—
413C22C4×#12111yescharacteristic
414C22#237312yes—
415C22#248611yescharacteristic
416C22#251512yes—
417C22#253511yescharacteristic
418C22C2×#47112yes—
419C22#306712yes—
420C4—24no—
421C4—42no—
422C4—42no—
423C4—82no—
424C4—84no—
425C4—84no—
426C4C4×#12311yescharacteristic
427C4#232611yescharacteristic
428C4#235912yes—
429C4#248211yescharacteristic
430C4#252112yes—
431C4#259911yescharacteristic
432C2—22no—
433C2—22no—
434C2—82no—
435C2#1422511yescharacteristic
436C2C4×#47112yes—
437C2#2767211yescharacteristic
438C2#2767411yescharacteristic
439C2#2817512yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#106C2semidirect
5. #11366C2semidirect
6. #11330C2semidirect
8. #10896C2semidirect
9. #10822C2semidirect
10. #10802C2semidirect
23. #2079C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20792
#210592
C2×C4×#1062
#203832
#113662
#113302
#111062
#108962
#108222
#108022
#107742