Finite Group Databasebeta

← #403313

Subgroups — #403313

Order 512.

883 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2058C211yesmaximal, characteristic
2#21313C211yesmaximal, characteristic
3#21034C211yesmaximal, characteristic
4#20430C211yesmaximal, characteristic
5#11382C211yesmaximal, characteristic
6#11326C211yesmaximal, characteristic
7#11242C212yesmaximal
8#11238C212yesmaximal
9#11204C211yesmaximal, characteristic
10#10896C212yesmaximal
11#10869C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2211yescharacteristic
15C22×#106C2212yes—
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C2×#405C2212yes—
19C2×#389C2211yescharacteristic
20C4×#117C2211yescharacteristic
21#2058—21no—
22#2058—21no—
23#2058—22no—
24#1988—21no—
25#1988—21no—
26#1987—21no—
27#1987—21no—
28#1921—21no—
29#1920—21no—
30#1919—21no—
31#1918—21no—
32#1768—22no—
33#1767—21no—
34C4×C4⋊C8—21no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2212yes—
40#1736C2211yescharacteristic
41#1736C2211yescharacteristic
42#1736C2211yescharacteristic
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2212yes—
46#1728C2211yescharacteristic
47#1728C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C23×Q8—21no—
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—42no—
61C2×#117C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C42C2311yescharacteristic
69C2×#108—24no—
70C2×#108—42no—
71C2×#108—42no—
72#428—22no—
73#428Q814yescomplemented
74C2×#107—21no—
75C2×#106—21no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—24no—
79C2×#106—42no—
80#424—24no—
81#424Q818yescomplemented
82#423—42no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87#405—24no—
88#405—42no—
89C2×#97—22no—
90C2×C4×C8—22no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95C4×C4⋊C4D411yescharacteristic
96C4×C4⋊C4D411yescharacteristic
97C4×C4⋊C4D411yescharacteristic
98C4×C4⋊C4D411yescharacteristic
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101#376—21no—
102C2×#95—21no—
103C43—21no—
104#343—24no—
105#343—42no—
106#343—42no—
107#343—42no—
108#341—24no—
109#340—22no—
110#340—22no—
111#340—21no—
112#340—21no—
113#340—21no—
114#340—21no—
115#340—21no—
116#340—21no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#336—21no—
122#336—21no—
123#336—21no—
124#336—21no—
125#336D411yescharacteristic
126#336D411yescharacteristic
127#336D412yes—
128C22×Q8—44no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8—82no—
132C22×Q8—82no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4#4111yescharacteristic
137C23×C4#4111yescharacteristic
138C23×C4C2411yescharacteristic
139C2×M16—22no—
140C2×M16—21no—
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—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—21no—
157C2×M16C2×Q814yes—
158C2×M16C2×Q812yes—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×Q814yes—
170C22×C8#4114yes—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—44no—
174C4⋊Q8—44no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177#117—22no—
178#117—21no—
179#117—21no—
180#117C2×Q812yes—
181#117C2×Q814yes—
182#117#4112yes—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×Q812yes—
208C2×C4⋊C4#4114yes—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223#108—48no—
224#107—24no—
225#106—24no—
226#106—44no—
227#106—44no—
228C4⋊C8—28no—
229C4⋊C8—28no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232#97—44no—
233C4×C8—44no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—41no—
239#95—41no—
240C24C2×#4111yescharacteristic
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8—816no—
247C2×Q8—82no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×Q812yes—
279C22×C4C22×Q811yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4112yes—
282C22×C4C2×#4112yes—
283C22×C4H(5,𝔽2)12yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C42−1+411yescharacteristic
286M16—24no—
287M16—24no—
288M16—24no—
289M16—42no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
295M16—42no—
296M16—42no—
297M16—42no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8C22⋊Q814yes—
319C2×C8C22⋊Q814yes—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4C22⋊Q812yes—
340C4⋊C4C22⋊Q812yes—
341C4⋊C4C4⋊Q812yes—
342C4⋊C4C4⋊Q812yes—
343C22⋊C4—24no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C42—24no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—42no—
356C42—42no—
357C42—81no—
358C42—81no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23C2×#12111yescharacteristic
364C23#53312yes—
365C23#55112yes—
366C23#55211yescharacteristic
367C23#55411yescharacteristic
368Q8—88no—
369Q8—88no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—84no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—84no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—162no—
408C2×C4#47112yes—
409C2×C4#47112yes—
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×C22⋊Q812yes—
412C2×C4#52511yescharacteristic
413C2×C4#52912yes—
414C2×C4C2×C4⋊Q811yescharacteristic
415C2×C4#53211yescharacteristic
416C2×C4#53312yes—
417C2×C4#53511yescharacteristic
418C2×C4#53711yescharacteristic
419C2×C4#53812yes—
420C2×C4#54212yes—
421C2×C4#54611yescharacteristic
422C2×C4#54612yes—
423C2×C4D4×Q811yescharacteristic
424C2×C4#55412yes—
425C2×C4#55412yes—
426C2×C4#55712yes—
427C2×C4Q8212yes—
428C8—24no—
429C8—24no—
430C8—42no—
431C8—44no—
432C8—82no—
433C8—81no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22#241111yescharacteristic
443C22#241711yescharacteristic
444C22#252012yes—
445C22#258612yes—
446C22#285711yescharacteristic
447C22C2×#47111yescharacteristic
448C22C2×#47111yescharacteristic
449C22#306811yescharacteristic
450C22#306911yescharacteristic
451C4—22no—
452C4—22no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—41no—
458C4—41no—
459C4—82no—
460C4—84no—
461C4—161no—
462C4#239611yescharacteristic
463C4#245012yes—
464C4#245112yes—
465C4#256711yescharacteristic
466C4#258411yescharacteristic
467C4#266211yescharacteristic
468C2—22no—
469C2—22no—
470C2#1487211yescharacteristic
471C2#2452111yescharacteristic
472C2#2452311yescharacteristic
473C2#2485611yescharacteristic
474C2#2485811yescharacteristic
475C2#3322011yescharacteristic
476C2#3322311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
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
73. #428Q8semidirect
81. #424Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
#213132
#210342
#204302
#113822
#113262
#112422
#112382
#112042
#108962
#108692