Finite Group Databasebeta

← #223115

Subgroups — #223115

Order 512.

835 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36175C211yesmaximal, characteristic
2#21270C211yesmaximal, characteristic
3#20494C211yesmaximal, characteristic
4#9890C211yesmaximal, characteristic, complemented
5#9890C211yesmaximal, characteristic, complemented
6#9126C212yesmaximal, complemented
7#9122C212yesmaximal, complemented
8#8720C212yesmaximal, complemented
9#8580C211yesmaximal, characteristic, complemented
10C4×#330C211yesmaximal, characteristic, complemented
11#8035C212yesmaximal, complemented
12C2×#436—21no—
13C2×#432C2212yes—
14C2×C4×M16C2212yes—
15#2862C2211yescharacteristic
16#2851C2212yes—
17#2563C411yescharacteristic
18#2547C411yescharacteristic
19#2527—22no—
20#2523—22no—
21#2514—21no—
22#2495C412yes—
23#2494C412yes—
24#2447—21no—
25#2369C411yescharacteristic
26#2367—21no—
27#2365C411yescharacteristic
28#2356—21no—
29C4×#125C2211yescharacteristic
30#2283C2212yes—
31C42×D4C2211yescharacteristic
32#1954—21no—
33#1939—21no—
34#1924—22no—
35C8×C4⋊C4—21no—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1753—22no—
39#1749C2212yes—
40#1749C2212yes—
41C4×#97C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×#330C2211yescharacteristic
44C2×#330C2211yescharacteristic
45C2×#330C2212yes—
46#1295C2211yescharacteristic
47#1295C2212yes—
48#1295C2212yes—
49#1295C2212yes—
50#1295C2211yescharacteristic
51#1295C2212yes—
52#1295C2212yes—
53C22×M16—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2311yescharacteristic
57C2×#124—21no—
58C2×C4×Q8—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C22×C42C2312yes—
63#436—28no—
64#432—24no—
65#432—42no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C4×M16—24no—
73C4×M16—42no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97C2312yes—
77C2×#97C2312yes—
78C2×#97C2311yescharacteristic
79C2×#97C2311yescharacteristic
80C2×#97C2311yescharacteristic
81C2×#97C2311yescharacteristic
82C2×C4×C8—21no—
83C2×C4×C8—22no—
84C2×C4×C8—21no—
85#401—22no—
86#400—22no—
87#397—22no—
88#397—22no—
89#397—22no—
90#393—22no—
91#391—21no—
92#391—21no—
93#391—21no—
94#391—21no—
95#391—21no—
96#391—21no—
97#391—22no—
98#390—21no—
99#389—21no—
100#389C2×C411yescharacteristic
101#389C2×C411yescharacteristic
102#389C2×C412yes—
103#389C2×C412yes—
104#388—22no—
105#388—22no—
106#387—21no—
107#387—21no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#385—22no—
113#385—22no—
114#383—22no—
115#383—22no—
116#382—21no—
117#382C2×C412yes—
118#382C2311yescharacteristic
119#382C2312yes—
120#381—22no—
121#380C2×C412yes—
122#380C2×C412yes—
123#380C2×C412yes—
124#380C2×C412yes—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C22⋊C4—22no—
136C4×C22⋊C4C2×C412yes—
137C4×C22⋊C4D412yes—
138C4×C22⋊C4D412yes—
139#376—22no—
140C43C2311yescharacteristic
141#336—21no—
142#336—22no—
143#336—22no—
144#336—21no—
145#330—22no—
146#330—22no—
147#330—42no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×D4C22⋊C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C2×M16—28no—
155C2×M16—44no—
156C2×M16—44no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165#125—21no—
166#125—21no—
167#125—21no—
168#125—21no—
169#124—44no—
170C4×Q8—44no—
171C4×D4—24no—
172C4×D4—42no—
173C4×D4—42no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2411yescharacteristic
216C4⋊C8—24no—
217C4⋊C8—44no—
218C4⋊C8—42no—
219C22⋊C8—28no—
220C22⋊C8—44no—
221C22⋊C8—44no—
222#97—21no—
223#97—21no—
224#97—22no—
225#97—22no—
226#97—21no—
227#97—21no—
228#97—22no—
229#97—22no—
230#97—21no—
231#97—21no—
232#97—42no—
233#97—42no—
234#97#4112yes—
235#97#4112yes—
236C4×C8—24no—
237C4×C8—44no—
238C4×C8—42no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260C24C2×C22⋊C412yes—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298M16—416no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—82no—
337C4⋊C4#10414yes—
338C22⋊C4—24no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—42no—
358C42—44no—
359C42—42no—
360C42—82no—
361C42C2×#4111yescharacteristic
362C42C2×#4111yescharacteristic
363C42H(5,𝔽2)11yescharacteristic
364C422−1+411yescharacteristic
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23C22×C22⋊C411yescharacteristic
373Q8—84no—
374Q8—162no—
375D4—44no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4#41312yes—
424C2×C4C2×#10414yes—
425C2×C4C2×#10412yes—
426C2×C4#42112yes—
427C2×C4C4×#4111yescharacteristic
428C2×C4#52011yescharacteristic
429C2×C4#54611yescharacteristic
430C2×C4#54611yescharacteristic
431C2×C4#54711yescharacteristic
432C2×C4#54711yescharacteristic
433C8—24no—
434C8—42no—
435C8—42no—
436C8—42no—
437C8—44no—
438C8—82no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22#233911yescharacteristic
447C22C2×#41311yescharacteristic
448C22#288911yescharacteristic
449C22C22×#10411yescharacteristic
450C22C2×#42111yescharacteristic
451C22#290111yescharacteristic
452C22#290111yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—24no—
458C4—21no—
459C4—21no—
460C4—41no—
461C4—41no—
462C4—42no—
463C4—42no—
464C4—41no—
465C4—81no—
466C4—82no—
467C4—82no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4#152914yes—
472C2—82no—
473C2C2×#152911yescharacteristic
474C2#1765411yescharacteristic
475C2#1768211yescharacteristic
476C2#1769111yescharacteristic
477C2#2248411yescharacteristic
478C2#2250511yescharacteristic
479C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9890C2semidirect
5. #9890C2semidirect
6. #9126C2semidirect
7. #9122C2semidirect
8. #8720C2semidirect
9. #8580C2semidirect
10. C4×#330C2semidirect
11. #8035C2semidirect

Maximal subgroups

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

subgroupindex
#361752
#212702
#204942
#98902
#91262
#91222
#87202
#85802
C4×#3302
#80352