Finite Group Databasebeta

← #122154

Subgroups — #122154

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37624C211yesmaximal, characteristic
2#8046C211yesmaximal, characteristic
3#8032C211yesmaximal, characteristic
4#8025C211yesmaximal, characteristic
5#8018C211yesmaximal, characteristic
6#8018C211yesmaximal, characteristic
7#7900C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#2841—22no—
12#2666—22no—
13#2660—21no—
14#2647—21no—
15#2629—21no—
16#2628—21no—
17#2571C411yescharacteristic, complemented
18#2570—21no—
19#2552C411yescharacteristic, complemented
20#2552C411yescharacteristic, complemented
21#2551C411yescharacteristic, complemented
22#2550—21no—
23#2441—21no—
24#2439—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2283C2211yescharacteristic
28C2×#383C2211yescharacteristic
29#1296—21no—
30#1296—21no—
31#1295—21no—
32#1295—21no—
33#1294—22no—
34#1294—21no—
35#1294—21no—
36#1294—21no—
37#1294—21no—
38#1279—22no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×#125—21no—
42C2×#125—21no—
43C2×#117—21no—
44C2×#117—21no—
45C22×C42C2311yescharacteristic
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432Q812yescomplemented
53#432Q812yescomplemented
54#431—21no—
55#431—21no—
56#431—21no—
57#431Q812yescomplemented
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×#97—21no—
65C2×#97—21no—
66C2×C4×C8—21no—
67#400—22no—
68#400—22no—
69#400—22no—
70#399—22no—
71#398—22no—
72#398—22no—
73#398—22no—
74#398—22no—
75#398—22no—
76#398—22no—
77#396—22no—
78#396—22no—
79#395—22no—
80#395—22no—
81#395—22no—
82#393—22no—
83#391—22no—
84#389—22no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#387—22no—
94#387—22no—
95#384—21no—
96#384—21no—
97#384C2×C411yescharacteristic
98#384C2×C411yescharacteristic
99#384D411yescharacteristic
100#384D411yescharacteristic
101#383—21no—
102#383—21no—
103#383—21no—
104#383—21no—
105#383Q812yescomplemented
106#382—21no—
107#382—21no—
108#382—21no—
109#382—21no—
110#382D411yescharacteristic
111#382D411yescharacteristic
112#381—21no—
113#380—21no—
114#380—21no—
115#380D412yes—
116#380D411yescharacteristic
117#380D411yescharacteristic
118#380D411yescharacteristic
119#380D411yescharacteristic
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—21no—
128#376—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C22×Q8—22no—
133C23×C4—41no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C4⋊Q8—82no—
141C4⋊Q8—82no—
142#125—82no—
143#125—82no—
144#117—82no—
145#117—82no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×Q811yescharacteristic
188C2×C42C2×Q811yescharacteristic
189C2×C42C2×Q811yescharacteristic
190C2×C42C2×Q811yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207#97—22no—
208#97—22no—
209#97—41no—
210#97—41no—
211C4×C8—22no—
212C4×C8—41no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—41no—
237#95—41no—
238#95—41no—
239C24#12011yescharacteristic
240C2×Q8—82no—
241C2×Q8—82no—
242C2×Q8—82no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4#12411yescharacteristic
279C22×C4#12411yescharacteristic
280C22×C4#12411yescharacteristic
281C22×C4C4⋊Q811yescharacteristic
282C22×C4C4⋊Q811yescharacteristic
283C22×C4C4⋊Q811yescharacteristic
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—81no—
288C2×C8—81no—
289C2×C8—81no—
290C2×C8—81no—
291C2×C8—81no—
292C2×C8—81no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C22⋊C4—82no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—82no—
330C42C4×Q811yescharacteristic
331C42C4×Q811yescharacteristic
332C42C4×Q811yescharacteristic
333C42C4×Q811yescharacteristic
334C42C22⋊Q811yescharacteristic
335C42C22⋊Q811yescharacteristic
336C42C22⋊Q811yescharacteristic
337C42C22⋊Q811yescharacteristic
338C42C22⋊Q811yescharacteristic
339C42C22⋊Q811yescharacteristic
340C42#12411yescharacteristic
341C42#12411yescharacteristic
342C42#12411yescharacteristic
343C42C4⋊Q811yescharacteristic
344C42C4⋊Q811yescharacteristic
345C42C4⋊Q811yescharacteristic
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23#37911yescharacteristic
353C23#39311yescharacteristic
354C23#39511yescharacteristic
355Q8—164no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—161no—
414C2×C4—161no—
415C2×C4—161no—
416C2×C4—161no—
417C2×C4#39111yescharacteristic
418C2×C4#39111yescharacteristic
419C2×C4#39111yescharacteristic
420C2×C4#39111yescharacteristic
421C2×C4#39111yescharacteristic
422C2×C4#39111yescharacteristic
423C2×C4#39311yescharacteristic
424C2×C4#39311yescharacteristic
425C2×C4#39311yescharacteristic
426C2×C4#39511yescharacteristic
427C2×C4#39511yescharacteristic
428C2×C4#39511yescharacteristic
429C8—82no—
430C8—82no—
431C8—82no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22#144611yescharacteristic
443C22#179411yescharacteristic
444C22#179411yescharacteristic
445C22#179511yescharacteristic
446C22#179511yescharacteristic
447C22#179611yescharacteristic
448C22#179611yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—81no—
462C4—82no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—82no—
471C4—162no—
472C2—41no—
473C2—41no—
474C2#850011yescharacteristic
475C2#850111yescharacteristic
476C2#850211yescharacteristic
477C2#850311yescharacteristic
478C2#1138811yescharacteristic
479C2#1138811yescharacteristic
480C2#1139311yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2571C4semidirect
19. #2552C4semidirect
20. #2552C4semidirect
21. #2551C4semidirect
52. #432Q8semidirect
53. #432Q8semidirect
57. #431Q8semidirect
105. #383Q8semidirect

Maximal subgroups

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

subgroupindex
#376242
#80462
#80322
#80252
#80182
#79002