Finite Group Databasebeta

← #224072

Subgroups — #224072

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2446C211yesmaximal, characteristic
2#22510C212yesmaximal
3#9882C212yesmaximal
4#9881C212yesmaximal
5#9880C212yesmaximal
6#8562C212yesmaximal
7#8560C211yesmaximal, characteristic
8#8154C212yesmaximal
9#8153C211yesmaximal, characteristic
10C22×C4⋊Q8C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#446—22no—
13#2446—22no—
14#2446—22no—
15#2446—22no—
16#2446—22no—
17C2×#400C412yes—
18C2×#398C412yes—
19C2×#391C411yescharacteristic
20C2×#391C411yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389C411yescharacteristic
23C2×#387C2212yes—
24C2×#387C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1985—22no—
27#1985—22no—
28#1984—22no—
29#1797C2212yes—
30#1797C2212yes—
31#1797C2212yes—
32#1749C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2212yes—
36C2×#324C2211yescharacteristic
37C2×#324C2211yescharacteristic
38#1280C2211yescharacteristic
39#1280C2211yescharacteristic
40#1280C2212yes—
41#1280C2212yes—
42#1280C2212yes—
43#1280C2212yes—
44#1280C2212yes—
45C23×Q8C2×C411yescharacteristic
46C23×Q8C2×C411yescharacteristic
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—44no—
51C2×C4×Q8—44no—
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68#446—48no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8C2312yes—
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75C2×C22⋊C8C2312yes—
76C2×#97—22no—
77C2×#97—22no—
78C2×C4×C8—22no—
79#400—22no—
80#400—22no—
81#400—22no—
82#400—22no—
83#398—22no—
84#398—22no—
85#398—22no—
86#398—22no—
87#391—22no—
88#391—21no—
89#391—21no—
90#391—22no—
91#391—21no—
92#391—21no—
93#391—22no—
94#391—22no—
95#389—22no—
96#389—21no—
97#389—21no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2311yescharacteristic
116C2×#95C2312yes—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123#324—24no—
124#324—22no—
125#324Q814yescomplemented
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8C22⋊C412yes—
137C22×Q8C22⋊C411yescharacteristic
138C22×Q8C22⋊C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C412yes—
142C23×C4C22×C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—48no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4×Q8—28no—
158C4×Q8—44no—
159C4×Q8—44no—
160C4×Q8—44no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C2×Q812yes—
189C2×C4⋊C4#4112yes—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—44no—
195C2×C42—42no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—42no—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C4⋊C8—44no—
206C4⋊C8—44no—
207C4⋊C8—44no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8C2×Q814yes—
216#97—44no—
217#97—44no—
218C4×C8—44no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235C24C23×C411yescharacteristic
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—48no—
245C2×Q8—48no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—84no—
253C2×Q8—84no—
254C2×Q8#9914yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C4×Q812yes—
310C22×C4C4×Q812yes—
311C22×C4#13112yes—
312C22×C4C22×D412yes—
313C22×C4C22×Q811yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C42−1+412yes—
316C2×C8—24no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23C22×C22⋊C411yescharacteristic
361C23C2×C4×Q811yescharacteristic
362C23#51911yescharacteristic
363C23C2×#13111yescharacteristic
364C23#56811yescharacteristic
365C23#56811yescharacteristic
366C23#56811yescharacteristic
367Q8—44no—
368Q8—48no—
369Q8—44no—
370Q8—84no—
371Q8—84no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4C2×#9912yes—
436C2×C4C2×#9914yes—
437C2×C4#41312yes—
438C2×C4#54812yes—
439C2×C4D4×Q812yes—
440C8—44no—
441C8—82no—
442C8—82no—
443C8—82no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22Q8×C22⋊C411yescharacteristic
457C22#287911yescharacteristic
458C22#288011yescharacteristic
459C22C22×#9911yescharacteristic
460C22#288411yescharacteristic
461C22C2×#41311yescharacteristic
462C22#288911yescharacteristic
463C22#296111yescharacteristic
464C22#296211yescharacteristic
465C22#298511yescharacteristic
466C22#298711yescharacteristic
467C4—22no—
468C4—22no—
469C4—24no—
470C4—22no—
471C4—22no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—82no—
481C4—82no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—21no—
486C2#1739211yescharacteristic
487C2#1741011yescharacteristic
488C2#1741411yescharacteristic
489C2#1742811yescharacteristic
490C2#2251111yescharacteristic
491C2Q8×#9911yescharacteristic
492C2#2251711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
125. #324Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24462
#225102
#98822
#98812
#98802
#85622
#85602
#81542
#81532