Finite Group Databasebeta

← #224904

Subgroups — #224904

Order 512.

765 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36591C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic, complemented
3#21345C211yesmaximal, characteristic, complemented
4#9688C212yesmaximal, complemented
5#9688C212yesmaximal, complemented
6#9685C211yesmaximal, characteristic
7#9685C211yesmaximal, characteristic
8#8581C212yesmaximal, complemented
9#8580C211yesmaximal, characteristic, complemented
10#8224C212yesmaximal, complemented
11C4×#330C211yesmaximal, characteristic, complemented
12#2861C2212yes—
13#2858C2211yescharacteristic
14#2595—22no—
15#2568C411yescharacteristic
16#2548C411yescharacteristic
17#2516—22no—
18#2513—21no—
19#2448—21no—
20#2448C412yes—
21#2447C412yes—
22#2367—21no—
23#2366C411yescharacteristic
24#2365C411yescharacteristic
25#2357—21no—
26C4×#125C2211yescharacteristic
27C4×#124C2212yes—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1917—22no—
34#1917—22no—
35#1767C2211yescharacteristic
36#1767C2211yescharacteristic
37#1767C2212yes—
38#1767C2212yes—
39#1745C2212yes—
40#1745C2212yes—
41C4×#97C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×#330C2211yescharacteristic
44C2×#330C2211yescharacteristic
45C2×#330C2212yes—
46C2×#329C2212yes—
47C2×#329C2212yes—
48C2×#329C2212yes—
49C2×#329C2212yes—
50#1295C2211yescharacteristic
51#1295C2211yescharacteristic
52#1295C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#125C2311yescharacteristic
56C2×#124—21no—
57C2×#124C2×C412yes—
58C2×#124C2312yes—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×#97—21no—
70C2×#97—21no—
71C2×#97C2311yescharacteristic
72C2×#97C2311yescharacteristic
73C2×#97C2312yes—
74C2×#97C2312yes—
75C2×#97C2311yescharacteristic
76C2×#97C2311yescharacteristic
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#400—22no—
80#399—22no—
81#397—22no—
82#397—22no—
83#397—22no—
84#394—22no—
85#391—21no—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2×C412yes—
89#390—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390C2×C412yes—
96#390C2×C412yes—
97#390C2×C412yes—
98#389—21no—
99#389C2×C411yescharacteristic
100#389C2×C411yescharacteristic
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#387—21no—
110#387—21no—
111#387—22no—
112#386—22no—
113#385—22no—
114#382—21no—
115#382C2311yescharacteristic
116#381—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#330—22no—
141#330—22no—
142#330—42no—
143#329—24no—
144#329—24no—
145#329—42no—
146#329—42no—
147C22×Q8C22⋊C411yescharacteristic
148C22×Q8C22×C412yes—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22×C412yes—
152C23×C4—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—42no—
160#127—42no—
161#125—21no—
162#125—21no—
163#125—21no—
164#125—21no—
165#124—22no—
166#124—22no—
167#124—22no—
168#124—22no—
169#124—44no—
170#124—82no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—82no—
175C4×D4—44no—
176C4×D4—82no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C412yes—
210C2×C42C22×C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2411yescharacteristic
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—44no—
224C4⋊C8—44no—
225#97—22no—
226#97—22no—
227#97—21no—
228#97—21no—
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—42no—
240C4×C8—42no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—81no—
273C2×D4—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—44no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—81no—
330C4⋊C4—81no—
331C4⋊C4—82no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—41no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—41no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—82no—
360C42C2×#4111yescharacteristic
361C42C2×#4111yescharacteristic
362C42H(5,𝔽2)11yescharacteristic
363C422−1+411yescharacteristic
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23C22×C22⋊C411yescharacteristic
372Q8—44no—
373Q8—82no—
374Q8—82no—
375D4—82no—
376D4—82no—
377D4—84no—
378D4—162no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—164no—
422C2×C4#41312yes—
423C2×C4#42112yes—
424C2×C4#42112yes—
425C2×C4C4×#4111yescharacteristic
426C2×C4#52011yescharacteristic
427C2×C4#54811yescharacteristic
428C2×C4#54811yescharacteristic
429C2×C4#55011yescharacteristic
430C2×C4#55011yescharacteristic
431C8—42no—
432C8—42no—
433C8—42no—
434C8—42no—
435C8—82no—
436C8—82no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22#234111yescharacteristic
445C22C2×#41311yescharacteristic
446C22#288811yescharacteristic
447C22C2×#42111yescharacteristic
448C22C2×#42111yescharacteristic
449C22#290111yescharacteristic
450C22#290111yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—41no—
458C4—42no—
459C4—41no—
460C4—42no—
461C4—42no—
462C4—41no—
463C4—42no—
464C4—82no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4#153514yes—
470C2—162no—
471C2C2×#153511yescharacteristic
472C2#1768111yescharacteristic
473C2#1768411yescharacteristic
474C2#1769111yescharacteristic
475C2#2253911yescharacteristic
476C2#2255111yescharacteristic
477C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21345C2semidirect
3. #21345C2semidirect
4. #9688C2semidirect
5. #9688C2semidirect
8. #8581C2semidirect
9. #8580C2semidirect
10. #8224C2semidirect
11. C4×#330C2semidirect

Maximal subgroups

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

subgroupindex
#365912
#213452
#96882
#96852
#85812
#85802
#82242
C4×#3302