Finite Group Databasebeta

← #97663

Subgroups — #97663

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#557C211yesmaximal, characteristic
2#9870C211yesmaximal, characteristic
3#8564C211yesmaximal, characteristic
4#8564C211yesmaximal, characteristic
5C4×Q8⋊C8C211yesmaximal, characteristic
6C4×Q8⋊C8C211yesmaximal, characteristic
7#7570C211yesmaximal, characteristic
8C2×#557C411yescharacteristic
9#2370—22no—
10#2370—21no—
11#2365—21no—
12#2365C411yescharacteristic
13#2355—21no—
14#2355C411yescharacteristic
15#2351—21no—
16Q8×C4⋊C4C411yescharacteristic
17C4×C4⋊Q8—21no—
18C4×C4⋊Q8C2211yescharacteristic
19C4×#125—22no—
20C4×#125—21no—
21C42×Q8—21no—
22C42×Q8—21no—
23C42×Q8C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1842C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C42×C8C2211yescharacteristic
29C2×Q8⋊C8—21no—
30C2×Q8⋊C8—21no—
31C2×Q8⋊C8—21no—
32C2×Q8⋊C8—21no—
33#1279—21no—
34#1279—21no—
35#1279—22no—
36#1279—21no—
37#1279—21no—
38#1271—22no—
39#557—22no—
40#557—22no—
41#557—22no—
42#557C814yescomplemented
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—22no—
46C2×#125—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4×C8—21no—
64C2×C4×C8—21no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68#391—21no—
69#391—21no—
70#391—21no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—21no—
75#389—22no—
76#389—21no—
77#389—21no—
78#389D412yes—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#384—21no—
86#384C2×C411yescharacteristic
87#382—22no—
88#382—21no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4D411yescharacteristic
114C43—22no—
115C43—21no—
116C43—21no—
117C43C2311yescharacteristic
118#336—21no—
119#336—21no—
120Q8⋊C8—24no—
121Q8⋊C8—24no—
122Q8⋊C8—42no—
123Q8⋊C8—42no—
124C22×Q8—21no—
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22⋊C411yescharacteristic
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C4⋊Q8—24no—
132C4⋊Q8—24no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8C2×C812yes—
136C4⋊Q8M1612yes—
137#125—28no—
138#125—24no—
139#125—28no—
140#125—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—21no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×Q8C2×C812yes—
168C4×Q8C2×C812yes—
169C4×Q8M1612yes—
170C4×Q8M1612yes—
171C4×Q8SD1612yes—
172C4×Q8Q1612yes—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C412yes—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—42no—
226C4⋊C8—41no—
227C4⋊C8—41no—
228C4⋊C8—41no—
229C4⋊C8—41no—
230C4×C8—22no—
231C4×C8—22no—
232C4×C8—22no—
233C4×C8—41no—
234C4×C8—41no—
235C4×C8—41no—
236C4×C8—41no—
237C4×C8—41no—
238C4×C8—41no—
239C4×C8—41no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—22no—
254C2×Q8—22no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8C22⋊C812yes—
269C2×Q8C22⋊C812yes—
270C2×Q8#10312yes—
271C2×Q8#10412yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—81no—
295C2×C8—81no—
296C2×C8—81no—
297C2×C8—81no—
298C2×C8—81no—
299C2×C8—81no—
300C2×C8—81no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—82no—
335C4⋊C4C22⋊C812yes—
336C4⋊C4C22⋊C812yes—
337C4⋊C4C22⋊C814yes—
338C4⋊C4#10312yes—
339C4⋊C4#10412yes—
340C42—28no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—22no—
350C42—22no—
351C42—21no—
352C42—21no—
353C42—41no—
354C42—42no—
355C42—42no—
356C42—44no—
357C42—44no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—42no—
364C42—41no—
365C42—42no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—81no—
371C42—81no—
372C42—82no—
373C42C22×C811yescharacteristic
374C42C2×M1611yescharacteristic
375C42C2×M1611yescharacteristic
376C42C2×M1611yescharacteristic
377C42C2×SD1611yescharacteristic
378C42C2×Q1611yescharacteristic
379C42#13511yescharacteristic
380C42#13511yescharacteristic
381C42AGL(1,ℤ/8)11yescharacteristic
382C42#13711yescharacteristic
383C42#13711yescharacteristic
384C42#13711yescharacteristic
385C23#37911yescharacteristic
386Q8—22no—
387Q8—22no—
388Q8—22no—
389Q8—22no—
390Q8—22no—
391Q8—42no—
392Q8—42no—
393Q8—41no—
394Q8—42no—
395Q8—41no—
396Q8—42no—
397Q8Q8⋊C814yescomplemented
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
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—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4C2×C22⋊C811yescharacteristic
438C2×C4C2×C22⋊C811yescharacteristic
439C2×C4C2×C22⋊C811yescharacteristic
440C2×C4#40711yescharacteristic
441C2×C4#40711yescharacteristic
442C2×C4#40711yescharacteristic
443C2×C4C2×#10311yescharacteristic
444C2×C4#41611yescharacteristic
445C2×C4#41711yescharacteristic
446C2×C4#41911yescharacteristic
447C2×C4C2×#10411yescharacteristic
448C2×C4C2×#10411yescharacteristic
449C2×C4#42111yescharacteristic
450C2×C4#42111yescharacteristic
451C2×C4#44811yescharacteristic
452C2×C4#44811yescharacteristic
453C2×C4#44911yescharacteristic
454C2×C4#44911yescharacteristic
455C2×C4#45111yescharacteristic
456C2×C4#45111yescharacteristic
457C2×C4#45211yescharacteristic
458C2×C4#45211yescharacteristic
459C2×C4#45411yescharacteristic
460C2×C4#45611yescharacteristic
461C8—44no—
462C8—82no—
463C8—82no—
464C8—161no—
465C22#177911yescharacteristic
466C22#178711yescharacteristic
467C22#178811yescharacteristic
468C22#178911yescharacteristic
469C22#178911yescharacteristic
470C22#179311yescharacteristic
471C22#179411yescharacteristic
472C4—24no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—42no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—41no—
485C4—42no—
486C4—82no—
487C4—81no—
488C4—81no—
489C4C2×Q8⋊C811yescharacteristic
490C4#147611yescharacteristic
491C4#147711yescharacteristic
492C4#147911yescharacteristic
493C4#148711yescharacteristic
494C4#148811yescharacteristic
495C4#148911yescharacteristic
496C4#149111yescharacteristic
497C4#162611yescharacteristic
498C4#163611yescharacteristic
499C4#164811yescharacteristic
500C4#165311yescharacteristic
501C2#831111yescharacteristic
502C2#831211yescharacteristic
503C2#831911yescharacteristic
504C2#843711yescharacteristic
505C2#843911yescharacteristic
506C2#844211yescharacteristic
507C2#844811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C22×C8
Frattini, cosocleC43C23
socle, prime radicalC23#379
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
42. #557C8semidirect
397. Q8Q8⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5572
#98702
#85642
C4×Q8⋊C82
#75702