Finite Group Databasebeta

← #223113

Subgroups — #223113

Order 512.

869 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36165C211yesmaximal, characteristic
2#21270C211yesmaximal, characteristic
3#20494C211yesmaximal, characteristic
4#9891C211yesmaximal, characteristic, complemented
5#9891C211yesmaximal, characteristic, complemented
6#9125C212yesmaximal, complemented
7#9122C212yesmaximal, complemented
8#8718C212yesmaximal, complemented
9#8580C211yesmaximal, characteristic, complemented
10C4×#330C211yesmaximal, characteristic, complemented
11#8033C212yesmaximal, complemented
12C2×#436—21no—
13C2×#432C2212yes—
14C2×C4×M16C2212yes—
15#2859C2211yescharacteristic
16#2851C2212yes—
17#2571C411yescharacteristic
18#2571C411yescharacteristic
19#2508—21no—
20#2508—21no—
21#2505—22no—
22#2505—22no—
23#2442C412yes—
24#2442C412yes—
25#2349—21no—
26#2349—21no—
27D4×C4⋊C4C411yescharacteristic
28D4×C4⋊C4C411yescharacteristic
29C4×#125C2211yescharacteristic
30C2×#382C2212yes—
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×#125C2×C412yes—
55C2×#125C2311yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C42C2312yes—
64#436—28no—
65#432—24no—
66#432—42no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C4×M16—24no—
74C4×M16—42no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97C2312yes—
78C2×#97C2312yes—
79C2×#97C2311yescharacteristic
80C2×#97C2311yescharacteristic
81C2×#97C2311yescharacteristic
82C2×#97C2311yescharacteristic
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86#400—22no—
87#400—22no—
88#400—22no—
89#400—22no—
90#398—22no—
91#398—22no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389C2×C411yescharacteristic
101#389C2×C411yescharacteristic
102#389C2×C412yes—
103#389C2×C412yes—
104#387—21no—
105#387—21no—
106#387—22no—
107#387—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#382—21no—
117#382—21no—
118#382—22no—
119#382—22no—
120#382C2×C412yes—
121#382C2×C412yes—
122#382C2311yescharacteristic
123#382C2312yes—
124#380—22no—
125#380—22no—
126#380C2×C412yes—
127#380C2×C412yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C22⋊C4C2×C412yes—
136C4×C22⋊C4D412yes—
137C4×C22⋊C4D412yes—
138C2×#95—22no—
139C2×#95—22no—
140C43C2311yescharacteristic
141#336—21no—
142#336—22no—
143#336—22no—
144#336—21no—
145#330—22no—
146#330—22no—
147#330—42no—
148C22×D4C22⋊C411yescharacteristic
149C23×C4—22no—
150C23×C4—22no—
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—
163#125—21no—
164#125—21no—
165#125—21no—
166#125—21no—
167#125—82no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—42no—
171C4×D4—44no—
172C4×D4—44no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2411yescharacteristic
219C4⋊C8—24no—
220C4⋊C8—44no—
221C4⋊C8—42no—
222C22⋊C8—28no—
223C22⋊C8—44no—
224C22⋊C8—44no—
225#97—21no—
226#97—21no—
227#97—22no—
228#97—22no—
229#97—21no—
230#97—21no—
231#97—22no—
232#97—22no—
233#97—21no—
234#97—21no—
235#97—42no—
236#97—42no—
237#97#4112yes—
238#97#4112yes—
239C4×C8—24no—
240C4×C8—44no—
241C4×C8—42no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261C24C2×C22⋊C412yes—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4#10414yes—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299M16—416no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4#10114yes—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—42no—
363C42—42no—
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C42H(5,𝔽2)11yescharacteristic
367C422−1+411yescharacteristic
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23C2×#10414yes—
375C23C22×C22⋊C411yescharacteristic
376D4—28no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—164no—
425C2×C4—164no—
426C2×C4C2×#10114yes—
427C2×C4C2×#10112yes—
428C2×C4C2×#10412yes—
429C2×C4#42112yes—
430C2×C4C4×#4111yescharacteristic
431C2×C4#52011yescharacteristic
432C2×C4#54611yescharacteristic
433C2×C4#54611yescharacteristic
434C2×C4#54711yescharacteristic
435C2×C4#54711yescharacteristic
436C8—24no—
437C8—42no—
438C8—42no—
439C8—42no—
440C8—44no—
441C8—82no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22#233911yescharacteristic
450C22C22×#10111yescharacteristic
451C22#288811yescharacteristic
452C22C22×#10411yescharacteristic
453C22C2×#42111yescharacteristic
454C22#290111yescharacteristic
455C22#290111yescharacteristic
456C4—21no—
457C4—21no—
458C4—24no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—41no—
464C4—41no—
465C4—42no—
466C4—42no—
467C4—41no—
468C4—82no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—82no—
473C4—81no—
474C4#152814yes—
475C2—44no—
476C2C2×#152811yescharacteristic
477C2#1765211yescharacteristic
478C2#1768111yescharacteristic
479C2#1769211yescharacteristic
480C2#2248111yescharacteristic
481C2#2250511yescharacteristic
482C2#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. #9891C2semidirect
5. #9891C2semidirect
6. #9125C2semidirect
7. #9122C2semidirect
8. #8718C2semidirect
9. #8580C2semidirect
10. C4×#330C2semidirect
11. #8033C2semidirect

Maximal subgroups

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

subgroupindex
#361652
#212702
#204942
#98912
#91252
#91222
#87182
#85802
C4×#3302
#80332