Finite Group Databasebeta

← D4×#356 #497313

Subgroups — D4×#356 #497313

Order 512.

1197 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2C22×#356C212yesmaximal, complemented
3D4×#101C212yesmaximal
4#13227C212yesmaximal, complemented
5#13094C214yesmaximal, complemented
6#12802C211yesmaximal, characteristic, complemented
7#12700C212yesmaximal, complemented
8C4×#356C211yesmaximal, characteristic, complemented
9C2×D4×Q8C411yescharacteristic
10C2×D4×Q8C2211yescharacteristic
11C22×C4⋊Q8C2212yes—
12D4×M16—22no—
13C22×#101C2214yes—
14#2868C411yescharacteristic
15#2857C412yes—
16#2551—22no—
17#2546C411yescharacteristic
18#2546C2211yescharacteristic
19#2440—24no—
20#2434C412yes—
21#2434C2212yes—
22D4×C4⋊C4—22no—
23C4×C4⋊Q8C2211yescharacteristic
24C42×D4C411yescharacteristic
25C2×#356—28no—
26C2×#356C2212yes—
27C2×#356C2214yes—
28C2×#356C2211yescharacteristic
29C2×#356C2214yes—
30C2×#356C2211yescharacteristic
31#1966—22no—
32#1901—24no—
33#1801C2212yes—
34#1784C2214yes—
35C4×#101C2212yes—
36#1394C2212yes—
37#1394C2212yes—
38C23×Q8C2×C412yes—
39C23×Q8C2312yes—
40C22×M16—24no—
41D4×Q8—41no—
42D4×Q8—42no—
43D4×Q8—41no—
44D4×Q8—42no—
45D4×Q8—41no—
46D4×Q8—41no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—44no—
52C2×C4⋊Q8C2×C412yes—
53C2×C4⋊Q8C2×C412yes—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8C2312yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2×C412yes—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8D414yes—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D412yes—
70C22×C4⋊C4—24no—
71C22×C42C2×C412yes—
72#436—42no—
73#435—44no—
74C8×D4—42no—
75#423—22no—
76C2×#101—28no—
77C2×#101—24no—
78C2×#101—24no—
79C2×#101—44no—
80C2×#101C2314yes—
81C2×#101C2312yes—
82C2×#101C2312yes—
83#407—24no—
84C4×M16—22no—
85#398—22no—
86#398—22no—
87#395—24no—
88#395—24no—
89#391C2×C411yescharacteristic
90#391C2×C412yes—
91#391C2×C412yes—
92#391C2311yescharacteristic
93#389—24no—
94#389—22no—
95#389—22no—
96#389—22no—
97#385—24no—
98#385—24no—
99#384—22no—
100#384—22no—
101#384C2×C411yescharacteristic
102#380—24no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C22⋊C4—22no—
107C4×C22⋊C4C2×C412yes—
108C43C2×C411yescharacteristic
109#356—24no—
110#356—22no—
111#356—28no—
112#356—44no—
113#356D414yesdirect product factor
114#343—22no—
115#343—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—82no—
127C22×Q8—82no—
128C22×Q8C22×C412yes—
129C22×Q8C22×C411yescharacteristic
130C22×Q8C22×C412yes—
131C22×Q8C2×D414yes—
132C22×Q8C2411yescharacteristic
133C22×D4C22⋊C411yescharacteristic
134C23×C4—22no—
135C23×C4—22no—
136C23×C4C22×C412yes—
137C23×C4C2×D414yes—
138C2×M16—22no—
139C2×M16—24no—
140C2×M16—22no—
141C2×M16—44no—
142C2×M16—48no—
143C2×M16—84no—
144C22×C8—44no—
145C4⋊Q8—22no—
146C4⋊Q8—24no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—48no—
160C4⋊Q8—48no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—82no—
164C4⋊Q8C2×D412yes—
165C4⋊Q8#4112yes—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C22⋊Q8—44no—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×D4—28no—
179C4×D4—22no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—22no—
183C4×D4—24no—
184C4×D4—22no—
185C4×D4—42no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C414yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C414yes—
212C2×C42—21no—
213C2×C42—24no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—42no—
218C2×C42C22×C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C4⋊C8—42no—
224C4⋊C8—42no—
225#101—22no—
226#101—24no—
227#101—22no—
228#101—22no—
229#101—24no—
230#101—48no—
231#101—84no—
232#101C2×D414yes—
233C22⋊C8—44no—
234C22⋊C8—44no—
235#97—42no—
236C4×C8—42no—
237#95—21no—
238#95—22no—
239#95—22no—
240#95—24no—
241#95—24no—
242#95—22no—
243C24C2×C22⋊C412yes—
244C2×Q8—22no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—44no—
254C2×Q8—42no—
255C2×Q8—44no—
256C2×Q8—42no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—82no—
264C2×Q8—82no—
265C2×Q8—84no—
266C2×Q8—82no—
267C2×Q8—84no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8—84no—
271C2×Q8—84no—
272C2×Q8—84no—
273C2×Q8—84no—
274C2×Q8C4×D412yes—
275C2×Q8C22×D411yescharacteristic
276C2×Q8C2×#4111yescharacteristic
277C2×Q8H(5,𝔽2)12yes—
278C2×D4—24no—
279C2×D4#9912yes—
280C2×D4#9911yescharacteristic
281C2×D4#9911yescharacteristic
282C22×C4—21no—
283C22×C4—28no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—82no—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C414yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D412yes—
312M16—24no—
313M16—42no—
314M16—42no—
315M16—44no—
316M16—44no—
317M16—42no—
318M16—88no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—44no—
322C2×C8—88no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4—41no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—84no—
346C22⋊C4—28no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C42—24no—
354C42—21no—
355C42—22no—
356C42—22no—
357C42—21no—
358C42—21no—
359C42—22no—
360C42—22no—
361C42—44no—
362C42—41no—
363C42—42no—
364C42—41no—
365C42—41no—
366C42C4×D412yes—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—42no—
371C23C2×#9912yes—
372C23C2×#9912yes—
373C23C22×C22⋊C411yescharacteristic
374Q8—41no—
375Q8—41no—
376Q8—42no—
377Q8—42no—
378Q8—42no—
379Q8—84no—
380Q8—84no—
381Q8—82no—
382Q8—82no—
383Q8—82no—
384Q8—82no—
385Q8—88no—
386D4—24no—
387D4—22no—
388D4#35614yesdirect product factor
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—416no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4C2×#9911yescharacteristic
433C2×C4C2×#9911yescharacteristic
434C2×C4C2×C4×D411yescharacteristic
435C2×C4#51811yescharacteristic
436C2×C4D4212yes—
437C2×C4#54612yes—
438C8—44no—
439C8—84no—
440C8—82no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—44no—
449C22—44no—
450C22C2×#35614yescomplemented
451C22D4×C22⋊C411yescharacteristic
452C22C22×#9911yescharacteristic
453C22#288311yescharacteristic
454C4—24no—
455C4—21no—
456C4—22no—
457C4—22no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—44no—
462C4—41no—
463C4—42no—
464C4—44no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—82no—
469C4C2×#35612yescomplemented
470C2—21no—
471C2—21no—
472C2—24no—
473C2—42no—
474C2D4×#9911yescharacteristic
475C2C22×#35611yescharacteristic
476C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×D4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#356C2semidirect
4. #13227C2semidirect
5. #13094C2semidirect
6. #12802C2semidirect
7. #12700C2semidirect
8. C4×#356C2semidirect
113. #356D4direct (N × H)
388. D4#356direct (N × H)
450. C22C2×#356semidirect
469. C4C2×#356semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
C22×#3562
D4×#1012
#132272
#130942
#128022
#127002
C4×#3562