Finite Group Databasebeta

← #107604

Subgroups — #107604

Order 512.

935 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34236C211yesmaximal, characteristic
2#17731C211yesmaximal, characteristic
3#17589C211yesmaximal, characteristic
4#8098C211yesmaximal, characteristic, complemented
5#8050C211yesmaximal, characteristic, complemented
6#8021C211yesmaximal, characteristic, complemented
7#7978C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12#2369—21no—
13#2366—21no—
14#2361—22no—
15#2357C412yescomplemented
16#2349C412yescomplemented
17#2339—22no—
18#2337—22no—
19#2319—21no—
20C4×#121—22no—
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#2275—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1557C414yescomplemented
26#1556—21no—
27#1555—21no—
28#1540—22no—
29#1517C414yescomplemented
30#1510—22no—
31#1505—22no—
32#1303—21no—
33#1303—21no—
34#1297—22no—
35#1294—22no—
36#1288—22no—
37C22×#41—21no—
38#532—21no—
39#532—21no—
40#532—21no—
41#532—21no—
42#532C2×C412yes—
43#532C2×C412yes—
44#532D412yes—
45#532Q812yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×#127D411yescharacteristic
48C2×#124—21no—
49C2×#121—21no—
50C2×#121D412yes—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4—21no—
55C2×C4×D4C2×C412yes—
56C22×C4⋊C4—21no—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432C2×C412yescomplemented
67#431—21no—
68#431—21no—
69#431—21no—
70#431C2×C412yescomplemented
71#423—21no—
72#423C2×C412yescomplemented
73C2×C4⋊C8—21no—
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C2×#97—21no—
80#391—24no—
81#390—22no—
82#390—22no—
83#389—22no—
84#389—22no—
85#388—24no—
86#388—24no—
87#386—24no—
88#386—24no—
89#385—24no—
90#383—21no—
91#383D412yes—
92#382C2×C411yescharacteristic
93#382C2×C411yescharacteristic
94#382C2×C412yescomplemented
95#381—21no—
96#381—22no—
97#381—21no—
98#380—21no—
99#380—22no—
100#380—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4D411yescharacteristic
109C4×C4⋊C4D411yescharacteristic
110C4×C22⋊C4—24no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4D412yes—
114#376—21no—
115#376—21no—
116C2×#95—22no—
117C2×#95D412yes—
118#332—24no—
119#331—24no—
120#327—28no—
121C2×#41—22no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—84no—
126C22×Q8—21no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4C22⋊C412yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—22no—
138C2×M16—41no—
139C22×C8—41no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C22⋊C412yes—
143C4⋊Q8C4⋊C412yes—
144#127—21no—
145#127—21no—
146#127C22⋊C412yes—
147#127C4⋊C412yes—
148#124—22no—
149#124—22no—
150#124—22no—
151#124—22no—
152#121—22no—
153#121—22no—
154#121—22no—
155#121—22no—
156#121—22no—
157#121—22no—
158#121C22⋊C414yes—
159#121C4⋊C414yes—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—48no—
167C4×D4—84no—
168C4×D4—88no—
169C4×D4C4214yes—
170C4×D4C4⋊C414yes—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—41no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—41no—
194C2×C42—82no—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×Q811yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219#97—22no—
220#97—41no—
221#97—41no—
222#97—41no—
223C4×C8—22no—
224C4×C8—41no—
225#95—24no—
226#95—24no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—41no—
235#95—41no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24—22no—
241C24#12011yescharacteristic
242#41—84no—
243#41—84no—
244#41—84no—
245#41—84no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—82no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4—48no—
260C2×D4—82no—
261C2×D4#9514yes—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—48no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—82no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—84no—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C4⋊C411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4C22⋊Q811yescharacteristic
296C22×C4C22⋊Q811yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4#12311yescharacteristic
299M16—81no—
300M16—81no—
301M16—82no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—82no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—82no—
315C4⋊C4—84no—
316C4⋊C4#9514yes—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—48no—
322C22⋊C4—44no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C22⋊C4—81no—
326C22⋊C4—81no—
327C42—44no—
328C42—44no—
329C42—81no—
330C42—81no—
331C42—82no—
332C42—84no—
333C42C2×C4211yescharacteristic
334C42C2×C22⋊C411yescharacteristic
335C42C2×C4⋊C411yescharacteristic
336C42C2×C4⋊C411yescharacteristic
337C42#11711yescharacteristic
338C42#11711yescharacteristic
339C42C4×D411yescharacteristic
340C42C4×D411yescharacteristic
341C42C4×D411yescharacteristic
342C42C4×D411yescharacteristic
343C42#12111yescharacteristic
344C42#12111yescharacteristic
345C42C22⋊Q811yescharacteristic
346C42C22⋊Q811yescharacteristic
347C42#12311yescharacteristic
348C42#12311yescharacteristic
349C23—21no—
350C23—21no—
351C23—22no—
352C23—24no—
353C23—41no—
354C23—44no—
355C23—44no—
356C23—42no—
357C23—42no—
358C23#37911yescharacteristic
359C23#38511yescharacteristic
360C23#38611yescharacteristic
361Q8—82no—
362Q8—82no—
363D4—44no—
364D4—44no—
365D4—44no—
366D4—48no—
367D4—48no—
368D4—44no—
369D4—82no—
370D4—82no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—88no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—168no—
412C2×C4C2×#9511yescharacteristic
413C2×C4C4×C22⋊C411yescharacteristic
414C2×C4C4×C22⋊C411yescharacteristic
415C2×C4#38011yescharacteristic
416C2×C4#38011yescharacteristic
417C2×C4#38111yescharacteristic
418C2×C4#38511yescharacteristic
419C2×C4#38511yescharacteristic
420C2×C4#38511yescharacteristic
421C2×C4#38611yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C8—82no—
425C8—82no—
426C8—82no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—42no—
435C22—42no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—84no—
440C22—84no—
441C22C22⋊#9511yescharacteristic
442C22#179511yescharacteristic
443C22#179611yescharacteristic
444C22#190412yes—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—44no—
457C4—44no—
458C4—82no—
459C4—82no—
460C4—82no—
461C4—162no—
462C4#190612yes—
463C2—21no—
464C2—21no—
465C2—22no—
466C2—41no—
467C2—84no—
468C2#1076511yescharacteristic
469C2#1077211yescharacteristic
470C2#1078511yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8098C2semidirect
5. #8050C2semidirect
6. #8021C2semidirect
7. #7978C2semidirect
15. #2357C4semidirect
16. #2349C4semidirect
25. #1557C4semidirect
29. #1517C4semidirect
66. #432C2×C4semidirect
70. #431C2×C4semidirect
72. #423C2×C4semidirect
94. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342362
#177312
#175892
#80982
#80502
#80212
#79782