Finite Group Databasebeta

← #414251

Subgroups — #414251

Order 512.

928 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23617C212yesmaximal
2C2×#1944C211yesmaximal, characteristic, complemented
3#20384C211yesmaximal, characteristic
4#11330C211yesmaximal, characteristic, complemented
5#11270C211yesmaximal, characteristic, complemented
6#11204C212yesmaximal, complemented
7#11107C211yesmaximal, characteristic
8#11106C211yesmaximal, characteristic
9#10896C212yesmaximal, complemented
10#10845C211yesmaximal, characteristic, complemented
11#10774C212yesmaximal
12C22×#131C2211yescharacteristic
13C2×#428C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#423C2212yes—
17C2×#417C2212yes—
18C2×#417C2212yes—
19C2×#408C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C22×#97C2211yescharacteristic
22#2297C2211yescharacteristic
23C2×#389C2212yes—
24#2068—22no—
25#2067—22no—
26#2066—22no—
27#2065—22no—
28#1944—22no—
29#1944—22no—
30#1753—22no—
31#1752—22no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341C2211yescharacteristic
37C2×#341C2212yes—
38C2×#341C2211yescharacteristic
39#1736C2212yes—
40C2×#340C2212yes—
41#1729C2211yescharacteristic
42#1728C2211yescharacteristic
43C22×#41C2311yescharacteristic
44C2×#131—22no—
45C2×#131—42no—
46C2×#131—42no—
47C2×#131Q814yes—
48C22×M16—21no—
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2312yes—
53C23×C8—21no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#117C2311yescharacteristic
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42C2311yescharacteristic
61#428—24no—
62#428—42no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#107—42no—
66C2×#106—22no—
67C2×#106—24no—
68C2×#106—22no—
69#423—24no—
70#423—42no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73#417—24no—
74#417—24no—
75#417—42no—
76#417—42no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#102—22no—
80C2×#102—22no—
81#408—22no—
82#408Q814yes—
83C2×C22⋊C8—22no—
84#405—42no—
85C2×#97—21no—
86C2×#97—42no—
87C2×#97D412yes—
88C2×#97D412yes—
89C2×C4×C8—21no—
90#391D412yes—
91#390D412yes—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#386—21no—
97#386—21no—
98C4×C22⋊C4—22no—
99C2×#95—22no—
100#343—24no—
101#343—42no—
102#343—42no—
103#343—42no—
104#341—24no—
105#341—22no—
106#341—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×#41C2×Q812yes—
120C2×#41C2×Q812yes—
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—21no—
133C2×M16—21no—
134C2×M16—22no—
135C2×M16—24no—
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×M16C2×Q812yes—
153C2×M16C2×Q812yes—
154C2×M16#4114yes—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—24no—
163C22×C8—22no—
164C22×C8—42no—
165C22×C8—42no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D412yes—
169C22×C8C2×Q812yes—
170C22×C8C2×Q812yes—
171C22×C8#4112yes—
172C22×C8#4112yes—
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⋊C4C2×D412yes—
190C2×C4⋊C4#4114yes—
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#107—44no—
201#106—28no—
202#106—44no—
203C4⋊C8—44no—
204C4⋊C8—44no—
205#103—44no—
206#103—44no—
207#102—44no—
208#102—44no—
209C22⋊C8—28no—
210#97—42no—
211#97—42no—
212#97—42no—
213C4×C8—42no—
214#95—22no—
215#95—22no—
216#95—42no—
217#95—42no—
218C24—21no—
219C24C2×#4111yescharacteristic
220#41—24no—
221#41—24no—
222#41—24no—
223#41—24no—
224#41—42no—
225#41—42no—
226#41—42no—
227#41—42no—
228C2×Q8—21no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8C4⋊Q812yes—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4C4⋊Q812yes—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4#12112yes—
263C22×C4#12112yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×Q811yescharacteristic
267C22×C4C22×Q811yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4H(5,𝔽2)12yes—
273C22×C4H(5,𝔽2)12yes—
274M16—24no—
275M16—24no—
276M16—24no—
277M16—24no—
278M16—48no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16—42no—
285M16—42no—
286M16—42no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—48no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8C22⋊Q814yes—
311C2×C8C22⋊Q812yes—
312C2×C8C22⋊Q812yes—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—84no—
324C4⋊C4—84no—
325C22⋊C4—42no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—41no—
333C42—41no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—41no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23C2×#12111yescharacteristic
342C23#53311yescharacteristic
343C23#53311yescharacteristic
344C23#55312yes—
345C23#55312yes—
346Q8—24no—
347Q8—42no—
348D4—24no—
349D4—44no—
350D4—44no—
351D4—42no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—162no—
379C2×C4—162no—
380C2×C4C2×C22⋊Q811yescharacteristic
381C2×C4C2×C22⋊Q811yescharacteristic
382C2×C4#52511yescharacteristic
383C2×C4#52912yes—
384C2×C4C2×C4⋊Q811yescharacteristic
385C2×C4#53211yescharacteristic
386C2×C4#53211yescharacteristic
387C2×C4#53312yes—
388C2×C4#53411yescharacteristic
389C2×C4#53511yescharacteristic
390C2×C4#53812yes—
391C2×C4#54312yes—
392C2×C4#54612yes—
393C2×C4#54612yes—
394C2×C4#55312yes—
395C2×C4#55312yes—
396C2×C4#55412yes—
397C2×C4#55412yes—
398C8—24no—
399C8—22no—
400C8—22no—
401C8—42no—
402C8—42no—
403C8—44no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—42no—
412C22—44no—
413C22—82no—
414C22—81no—
415C22—81no—
416C22#253511yescharacteristic
417C22#254311yescharacteristic
418C22#258712yes—
419C22#260812yes—
420C22#285711yescharacteristic
421C22#306812yes—
422C22#306912yes—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—82no—
427C4—84no—
428C4—162no—
429C4#239611yescharacteristic
430C4#254912yes—
431C4#255912yes—
432C4#256611yescharacteristic
433C4#257911yescharacteristic
434C4#262611yescharacteristic
435C2—22no—
436C2—22no—
437C2—82no—
438C2#1516511yescharacteristic
439C2#2817812yes—
440C2#2859411yescharacteristic
441C2#2859511yescharacteristic
442C2#3322312yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C23×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
4. #11330C2semidirect
5. #11270C2semidirect
6. #11204C2semidirect
9. #10896C2semidirect
10. #10845C2semidirect

Maximal subgroups

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

subgroupindex
#236172
C2×#19442
#203842
#113302
#112702
#112042
#111072
#111062
#108962
#108452
#107742