Finite Group Databasebeta

← #164396

Subgroups — #164396

Order 512.

881 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36446C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic, complemented
3#20573C211yesmaximal, characteristic, complemented
4#7702C212yesmaximal, complemented
5#7682C211yesmaximal, characteristic, complemented
6#7682C211yesmaximal, characteristic, complemented
7#7662C212yesmaximal, complemented
8#7635C211yesmaximal, characteristic
9#7618C212yesmaximal
10#7584C212yesmaximal
11#7567C211yesmaximal, characteristic
12C2×#557C411yescharacteristic
13C2×D4×Q8C411yescharacteristic
14C2×#446—21no—
15#2862C2211yescharacteristic
16#2862C2212yes—
17#2568C412yes—
18#2568C411yescharacteristic
19#2547C411yescharacteristic
20#2547C412yes—
21#2489—22no—
22#2489—21no—
23#2489—21no—
24#2489—22no—
25Q8×C22⋊C4—21no—
26Q8×C22⋊C4—21no—
27C4×#124C2211yescharacteristic
28C4×#124C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1771—22no—
36C8×C4⋊C4—21no—
37#1767C2211yescharacteristic
38#1767C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40#1745C2212yes—
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2211yescharacteristic
48#1272C2212yes—
49#1272C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C23×Q8—21no—
54#557—44no—
55D4×Q8—44no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2×C412yes—
60C2×#125—22no—
61C2×#125—21no—
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C412yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C412yes—
74#446—28no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×#97—22no—
80C2×#97—21no—
81C2×#97—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85#400—22no—
86#400—22no—
87#400—22no—
88#397—22no—
89#397—22no—
90#397—22no—
91#391—22no—
92#391—21no—
93#391—21no—
94#391—22no—
95#391—21no—
96#391—21no—
97#391—22no—
98#391—22no—
99#391C2×C412yes—
100#391C2×C412yes—
101#391C2×C412yes—
102#391C2312yes—
103#391C2311yescharacteristic
104#390C2311yescharacteristic
105#388—22no—
106#388—21no—
107#388—21no—
108#388—22no—
109#382—22no—
110#382—21no—
111#382—21no—
112#382—22no—
113#382C2×C411yescharacteristic
114#382C2×C412yes—
115#380—22no—
116#380—21no—
117#380—21no—
118#380—22no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C43C2312yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8—44no—
146C22×Q8—44no—
147C22×Q8—44no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22×C412yes—
150C22×Q8C22×C411yescharacteristic
151C22×D4C22×C411yescharacteristic
152C23×C4—22no—
153C23×C4—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C4⋊Q8—42no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163C4⋊Q8—42no—
164#125—48no—
165#125—44no—
166#124—22no—
167#124—24no—
168#124—41no—
169#124—42no—
170C22⋊Q8—48no—
171C22⋊Q8—44no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—42no—
175C4×Q8—44no—
176C4×Q8—42no—
177C4×Q8—44no—
178C4×D4—42no—
179C4×D4—44no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2411yescharacteristic
220C4⋊C8—24no—
221C4⋊C8—28no—
222C4⋊C8—42no—
223C4⋊C8—82no—
224#97—28no—
225#97—81no—
226#97—81no—
227C4×C8—24no—
228C4×C8—42no—
229C4×C8—82no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—22no—
242#95—21no—
243#95—22no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C24—21no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—21no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—816no—
265C2×Q8#10414yes—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4#13112yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4#10014yes—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—44no—
337C42—22no—
338C42—21no—
339C42—22no—
340C42—21no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—21no—
348C42—21no—
349C42—24no—
350C42—22no—
351C42—22no—
352C42—24no—
353C42—22no—
354C42—42no—
355C42—41no—
356C42—44no—
357C42—42no—
358C23—22no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C22×C22⋊C411yescharacteristic
366Q8—28no—
367Q8—82no—
368Q8—81no—
369Q8—82no—
370Q8—81no—
371Q8—88no—
372Q8—84no—
373D4—42no—
374D4—42no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—88no—
416C2×C4—84no—
417C2×C4C2×#10014yes—
418C2×C4C2×#10012yes—
419C2×C4C2×#10414yes—
420C2×C4C2×#10412yes—
421C2×C4#42112yes—
422C2×C4C2×#13111yescharacteristic
423C2×C4#56811yescharacteristic
424C2×C4#57711yescharacteristic
425C2×C4#57711yescharacteristic
426C2×C4#57811yescharacteristic
427C2×C4#57811yescharacteristic
428C8—42no—
429C8—44no—
430C8—42no—
431C8—81no—
432C8—82no—
433C8—81no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22#288011yescharacteristic
442C22#288411yescharacteristic
443C22C22×#10011yescharacteristic
444C22#289811yescharacteristic
445C22#289811yescharacteristic
446C22C22×#10411yescharacteristic
447C22C2×#42111yescharacteristic
448C4—21no—
449C4—21no—
450C4—24no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—24no—
456C4—41no—
457C4—42no—
458C4—41no—
459C4—42no—
460C4—41no—
461C4—82no—
462C4—81no—
463C4—81no—
464C4—82no—
465C4—81no—
466C4#152714yes—
467C2—82no—
468C2#1742511yescharacteristic
469C2#1751811yescharacteristic
470C2#1752311yescharacteristic
471C2#1754911yescharacteristic
472C2#1760111yescharacteristic
473C2C2×#152711yescharacteristic
474C2#1777911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22022C2semidirect
3. #20573C2semidirect
4. #7702C2semidirect
5. #7682C2semidirect
6. #7682C2semidirect
7. #7662C2semidirect

Maximal subgroups

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

subgroupindex
#364462
#220222
#205732
#77022
#76822
#76622
#76352
#76182
#75842
#75672