Finite Group Databasebeta

← #381972

Subgroups — #381972

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2079C211yesmaximal, characteristic
2C2×#1941C211yesmaximal, characteristic, complemented
3#20875C212yesmaximal
4C4×#408C211yesmaximal, characteristic
5#11366C211yesmaximal, characteristic, complemented
6#11330C211yesmaximal, characteristic, complemented
7#11290C212yesmaximal, complemented
8#11192C212yesmaximal
9#11106C211yesmaximal, characteristic
10#10913C212yesmaximal, complemented
11#10822C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#428C2212yes—
14C22×#106C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#417C2212yes—
17C2×#417C2212yes—
18C2×#408C2211yescharacteristic
19C2×#408C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#389C2212yes—
23#2079C418yescomplemented
24#1941—22no—
25#1941—22no—
26#1865—22no—
27#1864—22no—
28#1863—22no—
29#1862—22no—
30#1750—22no—
31C4×C22⋊C8—22no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341C2211yescharacteristic
37C2×#341C2212yes—
38C2×#341C2211yescharacteristic
39#1736C2212yes—
40C2×#340C2212yes—
41#1729C2211yescharacteristic
42#1729C2211yescharacteristic
43#1728C2212yes—
44C22×#41C2311yescharacteristic
45C22×M16C2312yes—
46C22×M16C2311yescharacteristic
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C4×#41—42no—
52C4×#41—42no—
53C2×C4×Q8D412yes—
54C2×C4×D4—22no—
55C2×C4×D4D412yes—
56C2×#117—21no—
57C2×#117C2311yescharacteristic
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C42—21no—
62C22×C42C2311yescharacteristic
63#428—42no—
64#428—42no—
65C2×#107—22no—
66C2×#106—22no—
67C2×#106—22no—
68C2×#106—22no—
69C2×#106—22no—
70C2×#106—42no—
71C2×#106C2×C414yescomplemented
72#417—24no—
73#417—42no—
74#417—42no—
75#417C2×C418yes—
76C2×#103—22no—
77C2×#103—22no—
78C2×#102—22no—
79C2×#102—22no—
80#408—22no—
81#408—22no—
82#408—22no—
83#408C2×C414yes—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C4×M16—42no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—42no—
90C2×C4×C8D412yes—
91C2×C4×C8D412yes—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96C2×#95—22no—
97#343—24no—
98#343—42no—
99#343—42no—
100#343C2×C418yescomplemented
101#341—22no—
102#341—42no—
103#341C2×C414yescomplemented
104#340—24no—
105#340—22no—
106#340—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111C2×#41—21no—
112C2×#41—21no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41—22no—
117C2×#41C22×C412yes—
118C2×#41#4112yes—
119C22×Q8C2×D411yescharacteristic
120C22×D4—21no—
121C22×D4C2×D411yescharacteristic
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4#4112yes—
127C23×C4C2411yescharacteristic
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—22no—
131C2×M16—22no—
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—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16C22×C414yes—
146C2×M16C22×C412yes—
147C2×M16#4112yes—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8C22×C412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8#4112yes—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—44no—
167C4×D4—44no—
168#117—22no—
169#117—22no—
170#117—42no—
171#117—42no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—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—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—84no—
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C2×D412yes—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200#107—44no—
201#106—28no—
202#106—44no—
203#106—44no—
204#103—28no—
205#103—44no—
206#102—28no—
207#102—44no—
208C22⋊C8—28no—
209C22⋊C8—28no—
210#97—42no—
211C4×C8—42no—
212C4×C8—42no—
213C4×C8—42no—
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—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4#12112yes—
264C22×C4#12112yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C22×C4H(5,𝔽2)12yes—
272C22×C4H(5,𝔽2)12yes—
273C22×C42−1+412yes—
274C22×C42−1+412yes—
275M16—24no—
276M16—24no—
277M16—24no—
278M16—24no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16—42no—
285M16—42no—
286M16—42no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—22no—
292C2×C8—24no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8C4×D414yes—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C22⋊C4—44no—
323C22⋊C4—42no—
324C42—21no—
325C42—21no—
326C42—22no—
327C42—22no—
328C42—21no—
329C42—21no—
330C42—44no—
331C42—41no—
332C42—41no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23C4×#4111yescharacteristic
341C23C2×#12111yescharacteristic
342C23#52911yescharacteristic
343C23#53812yes—
344C23#54112yes—
345Q8—24no—
346Q8—42no—
347D4—24no—
348D4—44no—
349D4—44no—
350D4—42no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—88no—
370C2×C4—84no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—162no—
378C2×C4—162no—
379C2×C4#47114yes—
380C2×C4C2×C4×D411yescharacteristic
381C2×C4C2×C4×D411yescharacteristic
382C2×C4#52012yes—
383C2×C4#52012yes—
384C2×C4#52511yescharacteristic
385C2×C4#53211yescharacteristic
386C2×C4#53411yescharacteristic
387C2×C4#53412yes—
388C2×C4#53511yescharacteristic
389C2×C4#53611yescharacteristic
390C2×C4#53612yes—
391C2×C4#53611yescharacteristic
392C2×C4#53912yes—
393C2×C4#53912yes—
394C2×C4#53912yes—
395C2×C4#53912yes—
396C2×C4#54212yes—
397C2×C4#54212yes—
398C8—24no—
399C8—44no—
400C8—42no—
401C8—42no—
402C8—42no—
403C8—82no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—44no—
411C22—42no—
412C22—82no—
413C22—81no—
414C22—81no—
415C22C4×#12111yescharacteristic
416C22#233412yes—
417C22#261711yescharacteristic
418C22#262111yescharacteristic
419C22#265212yes—
420C22C2×#47112yes—
421C22#306712yes—
422C4—24no—
423C4—42no—
424C4—42no—
425C4—84no—
426C4—82no—
427C4—162no—
428C4#232211yescharacteristic
429C4#232212yes—
430C4#232611yescharacteristic
431C4#263311yescharacteristic
432C4#263811yescharacteristic
433C4#266312yes—
434C2—22no—
435C2—22no—
436C2—82no—
437C2#1431311yescharacteristic
438C2C4×#47112yes—
439C2#2975012yes—
440C2#2988711yescharacteristic
441C2#2988811yescharacteristic
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
2. C2×#1941C2semidirect
5. #11366C2semidirect
6. #11330C2semidirect
7. #11290C2semidirect
10. #10913C2semidirect
11. #10822C2semidirect
23. #2079C4semidirect
71. C2×#106C2×C4semidirect
100. #343C2×C4semidirect
103. #341C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20792
C2×#19412
#208752
C4×#4082
#113662
#113302
#112902
#111922
#111062
#109132
#108222