Finite Group Databasebeta

← #497607

Subgroups — #497607

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39612C211yesmaximal, characteristic
2#22517C212yesmaximal
3#13303C212yesmaximal
4#13303C212yesmaximal
5#13302C212yesmaximal
6#12804C211yesmaximal, characteristic
7#12804C211yesmaximal, characteristic
8#12802C211yesmaximal, characteristic
9#12631C211yesmaximal, characteristic
10#12631C211yesmaximal, characteristic
11C4×#356C211yesmaximal, characteristic
12C2×#557C411yescharacteristic
13C2×#557C2211yescharacteristic
14Q8×M16—22no—
15#2859C411yescharacteristic
16#2734—22no—
17#2726—22no—
18#2660C411yescharacteristic
19#2660C411yescharacteristic
20#2567C411yescharacteristic
21#2567C2211yescharacteristic
22#2565C411yescharacteristic
23#2565C411yescharacteristic
24#2565C2211yescharacteristic
25#2565C2211yescharacteristic
26#2462—22no—
27#2462—22no—
28#2327C2211yescharacteristic
29#2327C2211yescharacteristic
30C4×C4⋊Q8C2211yescharacteristic
31#2295C411yescharacteristic
32C2×#356C2211yescharacteristic
33C2×#356C2211yescharacteristic
34C2×#356C2211yescharacteristic
35C2×#356C2211yescharacteristic
36#1998—22no—
37#1998—22no—
38#1997—22no—
39#1801C2212yes—
40#1801C2212yes—
41#1801C2212yes—
42C4×#101C2212yes—
43C4×#101C2212yes—
44C4×#101C2212yes—
45#1394C2212yes—
46#1394C2212yes—
47#1394C2212yes—
48#1394C2212yes—
49#1394C2212yes—
50#1394C2212yes—
51#557—42no—
52#557—42no—
53#557—42no—
54#557—42no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8D412yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×#125—21no—
61C2×#125—21no—
62C2×#125C2×C411yescharacteristic
63C2×#125C2×C411yescharacteristic
64C2×#125C2×C411yescharacteristic
65C2×#125C2×C411yescharacteristic
66C2×#125D412yes—
67C2×#125D412yes—
68C2×C4×Q8—21no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8D412yes—
74C2×C4×Q8C2311yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77#446—42no—
78#446—42no—
79#446—42no—
80C8×Q8—42no—
81#423—22no—
82#423—22no—
83#423—22no—
84C2×#101C2312yes—
85C2×#101C2312yes—
86C2×#101C2312yes—
87C2×#101C2312yes—
88C4×M16—22no—
89C4×M16—22no—
90C4×M16—22no—
91#398—22no—
92#398—22no—
93#398—22no—
94#398—22no—
95#398—22no—
96#398—22no—
97#395—22no—
98#395—22no—
99#395—22no—
100#395—22no—
101#391—21no—
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#391C2311yescharacteristic
106#391C2311yescharacteristic
107#391C2311yescharacteristic
108#389—22no—
109#389—22no—
110#389—22no—
111#389—21no—
112#389—22no—
113#389—22no—
114#387—22no—
115#387—22no—
116#387—21no—
117#387—21no—
118#387—22no—
119#387—22no—
120#387—22no—
121#387—22no—
122#387—22no—
123#387—22no—
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#384C2×C411yescharacteristic
127#384C2×C411yescharacteristic
128#384C2×C411yescharacteristic
129#382C2×C411yescharacteristic
130#382C2×C411yescharacteristic
131#382C2×C411yescharacteristic
132#382C2×C411yescharacteristic
133C4×C4⋊C4—22no—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137#376C2×C411yescharacteristic
138#376C2×C411yescharacteristic
139C43C2×C411yescharacteristic
140#356—22no—
141#356—22no—
142#356—22no—
143#356Q814yescomplemented
144#343—22no—
145#343—22no—
146#343—22no—
147#343—22no—
148#343—22no—
149#343—22no—
150C22×Q8—21no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22⋊C412yes—
153C22×Q8C22×C411yescharacteristic
154C22×Q8C2411yescharacteristic
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C4⋊Q8—22no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—44no—
165C4⋊Q8C2×Q812yes—
166C4⋊Q8#4112yes—
167#125—41no—
168#125—41no—
169#125—42no—
170#125—42no—
171#125—42no—
172#125—42no—
173#125—42no—
174#125—41no—
175#125—41no—
176#125—42no—
177#125—44no—
178#125—42no—
179#125—42no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—41no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—41no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—42no—
195C4×Q8—42no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251#101—22no—
252#101—22no—
253#101—22no—
254#101—22no—
255#101—22no—
256#101—22no—
257#101—22no—
258#101C2×Q814yes—
259#97—42no—
260#97—42no—
261#97—42no—
262C4×C8—42no—
263C4×C8—42no—
264C4×C8—42no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—82no—
292C2×Q8—81no—
293C2×Q8—82no—
294C2×Q8C4×Q812yes—
295C2×Q8C22×Q811yescharacteristic
296C2×Q8C2×#4111yescharacteristic
297C2×Q82−1+412yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D412yes—
317M16—24no—
318M16—42no—
319M16—42no—
320M16—42no—
321M16—42no—
322M16—42no—
323M16—42no—
324M16—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—81no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4#9912yes—
368C4⋊C4#9912yes—
369C42—21no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—21no—
377C42—21no—
378C42—22no—
379C42—22no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—41no—
387C42—41no—
388C42—81no—
389C42C4×Q812yes—
390C23C22×C22⋊C411yescharacteristic
391Q8—41no—
392Q8—41no—
393Q8—42no—
394Q8—42no—
395Q8—42no—
396Q8—44no—
397Q8—82no—
398Q8—82no—
399Q8—84no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
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—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4C2×#9911yescharacteristic
441C2×C4C2×#9911yescharacteristic
442C2×C4C2×#9912yes—
443C2×C4C2×#9912yes—
444C2×C4C2×C4×Q811yescharacteristic
445C2×C4#51911yescharacteristic
446C2×C4#54812yes—
447C2×C4D4×Q812yes—
448C8—44no—
449C8—82no—
450C8—82no—
451C8—82no—
452C22—21no—
453C22—21no—
454C22Q8×C22⋊C411yescharacteristic
455C22C22×#9911yescharacteristic
456C22#288411yescharacteristic
457C4—22no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—42no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C2—21no—
480C2—21no—
481C2Q8×#9911yescharacteristic
482C2#3336911yescharacteristic
483C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC23C22×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#557 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
143. #356Q8semidirect

Maximal subgroups

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

subgroupindex
#396122
#225172
#133032
#133022
#128042
#128022
#126312
C4×#3562