Finite Group Databasebeta

← #223147

Subgroups — #223147

Order 512.

767 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36175C211yesmaximal, characteristic
2#21270C211yesmaximal, characteristic
3#21270C211yesmaximal, characteristic
4#9890C211yesmaximal, characteristic, complemented
5#9890C211yesmaximal, characteristic, complemented
6#9126C212yesmaximal, complemented
7#9125C212yesmaximal, complemented
8#8720C212yesmaximal, complemented
9#8580C211yesmaximal, characteristic, complemented
10C4×#330C211yesmaximal, characteristic, complemented
11#8032C212yesmaximal, complemented
12C2×#432C2212yes—
13C2×#432C2212yes—
14#2862C2211yescharacteristic
15#2851C2212yes—
16#2563C411yescharacteristic
17#2547C411yescharacteristic
18#2527—22no—
19#2523—22no—
20#2514—21no—
21#2495C412yes—
22#2494C412yes—
23#2447—21no—
24#2369C411yescharacteristic
25#2367—21no—
26#2365C411yescharacteristic
27#2356—21no—
28C4×#125C2211yescharacteristic
29#2283C2212yes—
30C42×D4C2211yescharacteristic
31#1954—21no—
32#1954—21no—
33#1939—21no—
34#1939—21no—
35#1924—22no—
36#1924—22no—
37#1767C2211yescharacteristic
38#1767C2211yescharacteristic
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—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125C2311yescharacteristic
56C2×#124—21no—
57C2×C4×Q8—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C22×C42C2312yes—
62#432—24no—
63#432—24no—
64#432—42no—
65#432—42no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
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—21no—
84#401—22no—
85#400—22no—
86#397—22no—
87#397—22no—
88#397—22no—
89#393—22no—
90#391—21no—
91#391—21no—
92#391—21no—
93#391—21no—
94#391—21no—
95#391—21no—
96#391—22no—
97#390—21no—
98#389—21no—
99#389C2×C411yescharacteristic
100#389C2×C411yescharacteristic
101#389C2×C412yes—
102#389C2×C412yes—
103#388—22no—
104#388—22no—
105#387—21no—
106#387—21no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#385—22no—
112#385—22no—
113#383—22no—
114#383—22no—
115#382—21no—
116#382C2×C412yes—
117#382C2311yescharacteristic
118#382C2312yes—
119#381—22no—
120#380C2×C412yes—
121#380C2×C412yes—
122#380C2×C412yes—
123#380C2×C412yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C22⋊C4—22no—
135C4×C22⋊C4C2×C412yes—
136C4×C22⋊C4D412yes—
137C4×C22⋊C4D412yes—
138#376—22no—
139C43C2311yescharacteristic
140#336—21no—
141#336—21no—
142#336—22no—
143#336—22no—
144#330—22no—
145#330—22no—
146#330—42no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×D4C22⋊C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161#125—21no—
162#125—21no—
163#125—21no—
164#125—21no—
165#124—44no—
166C4×Q8—44no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2411yescharacteristic
212C4⋊C8—44no—
213C4⋊C8—44no—
214C4⋊C8—42no—
215C4⋊C8—42no—
216C22⋊C8—44no—
217C22⋊C8—44no—
218C22⋊C8—44no—
219C22⋊C8—44no—
220#97—21no—
221#97—21no—
222#97—22no—
223#97—22no—
224#97—21no—
225#97—21no—
226#97—22no—
227#97—22no—
228#97—21no—
229#97—21no—
230#97—42no—
231#97—42no—
232#97#4112yes—
233#97#4112yes—
234C4×C8—42no—
235C4×C8—42no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—22no—
252#95—21no—
253#95—22no—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257C24C2×C22⋊C412yes—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—81no—
263C2×Q8—81no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—81no—
332C4⋊C4—81no—
333C4⋊C4—82no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—44no—
354C42—42no—
355C42—42no—
356C42—82no—
357C42C2×#4111yescharacteristic
358C42C2×#4111yescharacteristic
359C42H(5,𝔽2)11yescharacteristic
360C422−1+411yescharacteristic
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23C22×C22⋊C411yescharacteristic
369Q8—84no—
370Q8—162no—
371D4—44no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4#41312yes—
421C2×C4#42112yes—
422C2×C4#42112yes—
423C2×C4C4×#4111yescharacteristic
424C2×C4#52011yescharacteristic
425C2×C4#54611yescharacteristic
426C2×C4#54611yescharacteristic
427C2×C4#54711yescharacteristic
428C2×C4#54711yescharacteristic
429C8—42no—
430C8—42no—
431C8—42no—
432C8—42no—
433C8—82no—
434C8—82no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22#233911yescharacteristic
443C22C2×#41311yescharacteristic
444C22#288911yescharacteristic
445C22C2×#42111yescharacteristic
446C22C2×#42111yescharacteristic
447C22#290111yescharacteristic
448C22#290111yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—81no—
462C4—81no—
463C4—82no—
464C4—81no—
465C4—81no—
466C4—82no—
467C4#153514yes—
468C2—82no—
469C2C2×#153511yescharacteristic
470C2#1768211yescharacteristic
471C2#1768311yescharacteristic
472C2#1769111yescharacteristic
473C2#2248411yescharacteristic
474C2#2250711yescharacteristic
475C2#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. #9125C2semidirect
8. #8720C2semidirect
9. #8580C2semidirect
10. C4×#330C2semidirect
11. #8032C2semidirect

Maximal subgroups

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

subgroupindex
#361752
#212702
#98902
#91262
#91252
#87202
#85802
C4×#3302
#80322