Finite Group Databasebeta

← #370282

Subgroups — #370282

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C211yesmaximal, characteristic
2C2×#1951C211yesmaximal, characteristic
3#21159C212yesmaximal
4C2×#1943C211yesmaximal, characteristic
5C2×#1778C211yesmaximal, characteristic
6#20436C211yesmaximal, characteristic
7#11290C212yesmaximal
8#11216C212yesmaximal
9#10913C212yesmaximal
10#10870C211yesmaximal, characteristic
11#10869C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#428C2212yes—
15C2×#428C2212yes—
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C2×C4×M16C2212yes—
22C22×#97C2211yescharacteristic
23#2296C2211yescharacteristic
24C2×#389C2212yes—
25#1952C418yescomplemented
26#1951—24no—
27#1944—22no—
28#1943—21no—
29#1943—21no—
30#1943—21no—
31#1943—21no—
32#1943—22no—
33#1942—22no—
34#1941—22no—
35#1778—24no—
36#1771—21no—
37#1771—21no—
38#1770—21no—
39#1770—21no—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#341C2212yes—
45#1736C2212yes—
46#1736C2212yes—
47C2×#340C2212yes—
48#1729C2211yescharacteristic
49#1728C2211yescharacteristic
50C22×M16C2311yescharacteristic
51C22×M16C2312yes—
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C42C2311yescharacteristic
62C2×#108—22no—
63C2×#108—22no—
64C2×#108C2×C414yes—
65C2×#108D414yes—
66C2×#108Q814yes—
67C2×#108Q814yes—
68#428—24no—
69#428—24no—
70#428—42no—
71#428C2×C418yes—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107C2×C414yes—
78C2×#106—22no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—22no—
84#424—22no—
85#424—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88#405—22no—
89#405—22no—
90#405—22no—
91#405C2×C414yes—
92C4×M16—24no—
93C4×M16—42no—
94C2×#97—22no—
95C2×#97—21no—
96C2×#97—22no—
97C2×#97—21no—
98C2×#97—22no—
99C2×#97D411yescharacteristic
100C2×#97D411yescharacteristic
101C2×#97D411yescharacteristic
102C2×#97D411yescharacteristic
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389D412yes—
111#389D412yes—
112#385—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C2×#95—22no—
116#343—24no—
117#343—24no—
118#343—42no—
119#343C2×C418yes—
120#341—24no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#336—21no—
128#336—21no—
129#336—21no—
130#336—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4#4112yes—
134C23×C4C2411yescharacteristic
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—21no—
143C2×M16—22no—
144C2×M16—21no—
145C2×M16—22no—
146C2×M16—21no—
147C2×M16—22no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16C22×C412yes—
154C2×M16C22×C414yes—
155C2×M16C2×D414yes—
156C2×M16C2×Q814yes—
157C2×M16C2×Q814yes—
158C2×M16#4112yes—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C22×C412yes—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D412yes—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×Q812yes—
176C22×C8C2×Q812yes—
177C22×C8#4112yes—
178C22×C8#4112yes—
179#117—21no—
180#117—21no—
181#117—21no—
182#117—21no—
183#117—21no—
184#117—21no—
185#117C22×C412yes—
186#117#4112yes—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4C22×C414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218#108—28no—
219#108C4⋊C4116yescomplemented
220#107—24no—
221#107—24no—
222#107—28no—
223#107—44no—
224#106—28no—
225#106—44no—
226#106—42no—
227#106—42no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230#97—24no—
231#97—22no—
232#97—22no—
233#97—22no—
234#97—22no—
235#97—44no—
236#97—42no—
237C4×C8—24no—
238C4×C8—44no—
239C4×C8—42no—
240#95—22no—
241#95—22no—
242#95—42no—
243#95—42no—
244C24C2×#4111yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—42no—
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×C4C23×C411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C22×Q811yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C42−1+411yescharacteristic
276C22×C42−1+411yescharacteristic
277C22×C42−1+411yescharacteristic
278M16—24no—
279M16—24no—
280M16—24no—
281M16—22no—
282M16—22no—
283M16—24no—
284M16—22no—
285M16—22no—
286M16—24no—
287M16—41no—
288M16—41no—
289M16—42no—
290M16—41no—
291M16—42no—
292M16—41no—
293M16C2×C4⋊C418yes—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—24no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8C2×C4⋊C414yes—
317C2×C8C4×D412yes—
318C2×C8C4×D412yes—
319C2×C8C4×Q814yes—
320C2×C8#12112yes—
321C2×C8#12112yes—
322C2×C8C22⋊Q814yes—
323C2×C8#12514yes—
324C2×C8C4⋊Q812yes—
325C2×C8C4⋊Q812yes—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C4⋊C4C4×D412yes—
343C4⋊C4C4×D412yes—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—22no—
349C42—21no—
350C42—21no—
351C42—22no—
352C42—21no—
353C42—21no—
354C42—22no—
355C42—22no—
356C42—41no—
357C42—41no—
358C23—22no—
359C23—22no—
360C23C4×#4111yescharacteristic
361C23C2×#12111yescharacteristic
362C23#52911yescharacteristic
363C23#54811yescharacteristic
364C23#55311yescharacteristic
365C23#55711yescharacteristic
366C23#55711yescharacteristic
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—162no—
391C2×C4—162no—
392C2×C4C22×C4⋊C411yescharacteristic
393C2×C4C2×C4×D411yescharacteristic
394C2×C4C2×C4×D411yescharacteristic
395C2×C4C2×C4×Q811yescharacteristic
396C2×C4#51911yescharacteristic
397C2×C4#52011yescharacteristic
398C2×C4C2×#12111yescharacteristic
399C2×C4C2×C22⋊Q811yescharacteristic
400C2×C4#52511yescharacteristic
401C2×C4C2×#12511yescharacteristic
402C2×C4C2×C4⋊Q811yescharacteristic
403C2×C4#53211yescharacteristic
404C2×C4#53611yescharacteristic
405C2×C4#53711yescharacteristic
406C2×C4#54111yescharacteristic
407C2×C4#54211yescharacteristic
408C2×C4D4211yescharacteristic
409C2×C4#54711yescharacteristic
410C2×C4#54711yescharacteristic
411C2×C4#54811yescharacteristic
412C2×C4D4×Q811yescharacteristic
413C2×C4D4×Q811yescharacteristic
414C2×C4#55211yescharacteristic
415C2×C4#55411yescharacteristic
416C2×C4#55411yescharacteristic
417C2×C4#55611yescharacteristic
418C2×C4#55711yescharacteristic
419C2×C4#55711yescharacteristic
420C8—28no—
421C8—42no—
422C8—41no—
423C8—41no—
424C8—42no—
425C8—44no—
426C8#38914yes—
427C8#38914yes—
428C22—22no—
429C22—22no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—22no—
436C22C4×#12111yescharacteristic
437C22#235111yescharacteristic
438C22#237011yescharacteristic
439C22#241611yescharacteristic
440C22#245311yescharacteristic
441C22#250911yescharacteristic
442C22#254611yescharacteristic
443C22#306812yes—
444C22#306912yes—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—84no—
450C4—82no—
451C4—162no—
452C4C2×#38911yescharacteristic
453C4#232311yescharacteristic
454C4D4×C4⋊C411yescharacteristic
455C4#236511yescharacteristic
456C4#241511yescharacteristic
457C4#245211yescharacteristic
458C4#250611yescharacteristic
459C4#258411yescharacteristic
460C2—21no—
461C2—21no—
462C2—21no—
463C2—21no—
464C2#1404611yescharacteristic
465C2#2155511yescharacteristic
466C2#2155611yescharacteristic
467C2#2480512yes—
468C2#2870312yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
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
25. #1952C4semidirect
219. #108C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
C2×#19512
#211592
C2×#19432
C2×#17782
#204362
#112902
#112162
#109132
#108702
#108692