Finite Group Databasebeta

← #379129

Subgroups — #379129

Order 512.

960 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2079C211yesmaximal, characteristic
2C2×#1944C211yesmaximal, characteristic, complemented
3#21093C212yesmaximal
4#20384C211yesmaximal, characteristic
5#11366C211yesmaximal, characteristic, complemented
6#11270C211yesmaximal, characteristic, complemented
7#11107C211yesmaximal, characteristic
8#10913C212yesmaximal, complemented
9#10827C212yesmaximal, complemented
10#10774C212yesmaximal
11#10705C211yesmaximal, characteristic, complemented
12C22×#131C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C2×#417C2212yes—
16C2×#417C2212yes—
17C2×#408C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C2×C4×M16C2212yes—
20C22×#97C2211yescharacteristic
21#2297C2211yescharacteristic
22C2×#389C2212yes—
23#2079C418yescomplemented
24#1944—22no—
25#1944—22no—
26#1936—22no—
27#1935—22no—
28#1934—22no—
29#1933—22no—
30#1753—22no—
31#1752—22no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341C2212yes—
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39#1736C2212yes—
40#1736C2212yes—
41C2×#340C2212yes—
42#1729C2211yescharacteristic
43#1728C2211yescharacteristic
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⋊C4—22no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C42C2311yescharacteristic
62C2×#107—22no—
63C2×#107—42no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106C2×C414yes—
67#417—24no—
68#417—24no—
69#417—42no—
70#417C2×C418yes—
71C2×#103—22no—
72C2×#103—22no—
73C2×#102—22no—
74C2×#102—22no—
75#408—22no—
76#408C2×C414yes—
77C2×C22⋊C8—22no—
78#405—24no—
79C4×M16—24no—
80C4×M16—42no—
81C2×#97—21no—
82C2×#97—22no—
83C2×#97—42no—
84C2×#97D412yes—
85C2×#97D412yes—
86C2×C4×C8—21no—
87C2×C4×C8—22no—
88#391D412yes—
89#390D412yes—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#386—21no—
95#386—21no—
96C4×C22⋊C4—22no—
97C2×#95—22no—
98#343—24no—
99#343—24no—
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#340—22no—
110#340—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—42no—
150C2×M16—42no—
151C2×M16C22×C414yes—
152C2×M16C22×C412yes—
153C2×M16#4112yes—
154C2×M16#4114yes—
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⋊C4—22no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—84no—
189C2×C4⋊C4C22×C414yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199#107—44no—
200#106—28no—
201#103—28no—
202#103—44no—
203#102—28no—
204#102—44no—
205C22⋊C8—28no—
206#97—42no—
207#97—42no—
208#97—44no—
209#97—42no—
210C4×C8—44no—
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—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
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—24no—
277M16—24no—
278M16—48no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16—42no—
285C2×C8—24no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—22no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—48no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8C4×D414yes—
309C2×C8#12412yes—
310C2×C8#12412yes—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
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—42no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—41no—
330C42—41no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23C4×#4111yescharacteristic
339C23C2×#12111yescharacteristic
340C23#52911yescharacteristic
341C23#55312yes—
342C23#55612yes—
343Q8—24no—
344Q8—42no—
345D4—24no—
346D4—44no—
347D4—44no—
348D4—42no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—162no—
376C2×C4—162no—
377C2×C4C2×C4×D411yescharacteristic
378C2×C4C2×C4×D411yescharacteristic
379C2×C4C4×#4112yes—
380C2×C4#52012yes—
381C2×C4#52511yescharacteristic
382C2×C4C2×#12411yescharacteristic
383C2×C4#52912yes—
384C2×C4#53211yescharacteristic
385C2×C4#53211yescharacteristic
386C2×C4#53411yescharacteristic
387C2×C4#53611yescharacteristic
388C2×C4#53912yes—
389C2×C4#54612yes—
390C2×C4#54712yes—
391C2×C4#55112yes—
392C2×C4#55112yes—
393C2×C4#55212yes—
394C2×C4#55212yes—
395C8—22no—
396C8—22no—
397C8—24no—
398C8—42no—
399C8—44no—
400C8—44no—
401C8—42no—
402C22—22no—
403C22—22no—
404C22—24no—
405C22—22no—
406C22—22no—
407C22—44no—
408C22—42no—
409C22—82no—
410C22—81no—
411C22—81no—
412C22C4×#12111yescharacteristic
413C22#237312yes—
414C22#254311yescharacteristic
415C22#254711yescharacteristic
416C22#257312yes—
417C22#306812yes—
418C22#306912yes—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—84no—
423C4—82no—
424C4—162no—
425C4C4×#12411yescharacteristic
426C4#232211yescharacteristic
427C4#235912yes—
428C4#255312yes—
429C4#256111yescharacteristic
430C4#261111yescharacteristic
431C2—22no—
432C2—22no—
433C2—82no—
434C2#1422911yescharacteristic
435C2#2155712yes—
436C2#2859411yescharacteristic
437C2#2859511yescharacteristic
438C2#2873512yes—
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×#1944C2semidirect
5. #11366C2semidirect
6. #11270C2semidirect
8. #10913C2semidirect
9. #10827C2semidirect
11. #10705C2semidirect
23. #2079C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20792
C2×#19442
#210932
#203842
#113662
#112702
#111072
#109132
#108272
#107742
#107052