Finite Group Databasebeta

← #116271

Subgroups — #116271

Order 512.

706 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1986C211yesmaximal, characteristic
2C2×#1515C211yesmaximal, characteristic
3C2×#1506C211yesmaximal, characteristic
4#13952C211yesmaximal, characteristic
5C2×#1300C211yesmaximal, characteristic
6C2×#1288C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×#383C2211yescharacteristic
13C2×C4×C4⋊C4C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1986—22no—
16#1986—22no—
17C2×#339—22no—
18#1515—22no—
19#1515—22no—
20#1506—22no—
21#1506C414yescomplemented
22C2×#328—21no—
23C2×#328—21no—
24C2×#328—21no—
25C2×#328—21no—
26#1444—21no—
27#1442—21no—
28#1436—22no—
29C4⋊#95—21no—
30#1434—21no—
31#1300—22no—
32#1288—22no—
33#1285—22no—
34C22×M16—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533D412yes—
43C2×C4⋊Q8Q811yescharacteristic
44C2×C4⋊Q8Q811yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C42—21no—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#432D412yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16D412yes—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8—21no—
76#383—21no—
77#383—21no—
78#383—21no—
79#383—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84#376—21no—
85#376—21no—
86#376C2×C412yes—
87#376D412yes—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95D411yescharacteristic
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99#339—28no—
100#328—24no—
101#328—24no—
102#328—24no—
103#328—24no—
104C22×Q8—21no—
105C22×Q8—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8C4⋊C412yescomplemented
130C4⋊Q8C4⋊C412yescomplemented
131#125—22no—
132#125—22no—
133#125—41no—
134#125—41no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—41no—
151#117—22no—
152#117—22no—
153#117—41no—
154#117—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—82no—
167C2×C22⋊C4—41no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—44no—
173C2×C42—41no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—41no—
200#97—41no—
201#97—41no—
202C4×C8—22no—
203C4×C8—41no—
204#95—22no—
205#95—22no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—44no—
213#95—44no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—82no—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223C24#12011yescharacteristic
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—82no—
229C2×Q8—82no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4#12411yescharacteristic
280C22×C4#12411yescharacteristic
281C22×C4#12411yescharacteristic
282C22×C4#12711yescharacteristic
283M16—24no—
284M16—42no—
285M16—42no—
286M16—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—82no—
293C2×C8—82no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—88no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C42—44no—
312C42—44no—
313C42—44no—
314C42—42no—
315C42—42no—
316C42—82no—
317C42C2×C4⋊C411yescharacteristic
318C42#11711yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42#12111yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C42#12311yescharacteristic
330C42#12311yescharacteristic
331C42#12611yescharacteristic
332C42#12611yescharacteristic
333C23—22no—
334C23—22no—
335C23#38611yescharacteristic
336C23#39211yescharacteristic
337C23#39311yescharacteristic
338C23#45511yescharacteristic
339C23#45511yescharacteristic
340C23#45611yescharacteristic
341C23#45611yescharacteristic
342Q8—82no—
343Q8—82no—
344Q8—82no—
345Q8—82no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4#38011yescharacteristic
402C2×C4#38511yescharacteristic
403C2×C4#38511yescharacteristic
404C2×C4#38811yescharacteristic
405C2×C4#38811yescharacteristic
406C2×C4#39011yescharacteristic
407C2×C4#39411yescharacteristic
408C2×C4#39411yescharacteristic
409C2×C4#39611yescharacteristic
410C2×C4#39611yescharacteristic
411C2×C4#39711yescharacteristic
412C2×C4#39911yescharacteristic
413C8—44no—
414C8—82no—
415C8—82no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22#143911yescharacteristic
425C22#188211yescharacteristic
426C22#188211yescharacteristic
427C22#200411yescharacteristic
428C22#200511yescharacteristic
429C22#202811yescharacteristic
430C22#202811yescharacteristic
431C22#203112yes—
432C22#203112yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—44no—
446C4—84no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4—161no—
451C4—161no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2#1037912yes—
457C2#1038112yes—
458C2#1106411yescharacteristic
459C2#1108111yescharacteristic
460C2#1109111yescharacteristic
subgroupG/H
center, socleC23#393
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
21. #1506C4semidirect
129. C4⋊Q8C4⋊C4semidirect
130. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19862
C2×#15152
C2×#15062
#139522
C2×#13002
C2×#12882
C2×#12852