Finite Group Databasebeta

← #192993

Subgroups — #192993

Order 512.

843 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#21970C211yesmaximal, characteristic
3#17815C211yesmaximal, characteristic
4#17449C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8909C214yesmaximal, complemented
7#8907C214yesmaximal, complemented
8#8078C211yesmaximal, characteristic
9#7886C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11#2984—22no—
12#2984—22no—
13#2983—22no—
14#2983—21no—
15#2982—21no—
16Q8×M16—22no—
17Q8×M16C2214yes—
18#2920—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22#2920C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25#1950—22no—
26#1950C2214yes—
27#1811—22no—
28#1572—22no—
29#1572—22no—
30#1567C2212yes—
31#1566C2212yes—
32#1491—21no—
33#1491—21no—
34#1487C412yes—
35#1487C414yes—
36#1487C412yes—
37#1487C2214yes—
38#1479—22no—
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41#1301C2214yes—
42#1278C2212yes—
43#1278C2212yes—
44C23×Q8Q811yescharacteristic
45C22×M16—21no—
46C2×C4×Q8—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×C412yes—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8Q812yes—
59C2×C4×Q8C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4Q811yescharacteristic
62C22×C42—21no—
63C22×C42C2311yescharacteristic
64#446—28no—
65#446—24no—
66#446—24no—
67#446—28no—
68C8×Q8—24no—
69C8×Q8—24no—
70#432—24no—
71#432—24no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432D412yes—
80#431—24no—
81#431—24no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431C2×C412yes—
91#431C2×C412yes—
92#431D412yes—
93#431C2312yes—
94C2×#108—24no—
95#423—24no—
96#423—21no—
97#423—21no—
98#423—21no—
99#423C2×C412yes—
100#423C2×C412yes—
101#423D412yes—
102#423C2312yes—
103#423C2312yes—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2312yes—
107#407—21no—
108#407—21no—
109#407—21no—
110#407D412yes—
111C2×C22⋊C8—21no—
112C4×M16—24no—
113C4×M16—21no—
114C4×M16—21no—
115C4×M16—21no—
116C4×M16C2312yes—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#341—24no—
120#335—24no—
121#334—24no—
122Q8⋊C8—24no—
123Q8⋊C8—24no—
124Q8⋊C8C2×C414yes—
125Q8⋊C8C2×C414yes—
126Q8⋊C8C2×C414yes—
127Q8⋊C8C2×C414yes—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—44no—
132C22×Q8C4⋊C411yescharacteristic
133C22×Q8C4⋊C412yes—
134C22×Q8C4⋊C411yescharacteristic
135C22×Q8C2×Q812yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×Q811yescharacteristic
140C23×C4#4111yescharacteristic
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—41no—
146C2×M16—82no—
147C2×M16C2×D414yes—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C4×Q8—21no—
152C4×Q8—22no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—22no—
156C4×Q8—21no—
157C4×Q8—21no—
158C4×Q8—22no—
159C4×Q8—21no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8C4⋊C412yes—
168C4×Q8C4⋊C412yes—
169C4×Q8C4⋊C412yes—
170C4×Q8C4⋊C412yes—
171C4×Q8C22×C412yes—
172C4×Q8C22×C412yes—
173C4×Q8C2×D412yes—
174C4×Q8C2×D412yes—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C4⋊C412yes—
187C2×C4⋊C4C4⋊C411yescharacteristic
188C2×C4⋊C4C4⋊C411yescharacteristic
189C2×C4⋊C4C2×Q812yes—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q811yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42C2411yescharacteristic
210#108—84no—
211C4⋊C8—28no—
212C4⋊C8—28no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—41no—
225C4⋊C8C22×C414yes—
226C4⋊C8C2×D414yes—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—41no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237#97—28no—
238#97—22no—
239#97—22no—
240#97—41no—
241#97—41no—
242C4×C8—28no—
243C4×C8—22no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—22no—
247C4×C8—22no—
248C4×C8—41no—
249C4×C8C22×C412yes—
250C4×C8C22×C412yes—
251C24C22⋊Q811yescharacteristic
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—21no—
255C2×Q8—22no—
256C2×Q8—21no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8C2×C4⋊C412yes—
266C2×Q8C2×C4⋊C412yes—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—81no—
280C22×C4—82no—
281C22×C4C2×C4⋊C411yescharacteristic
282C22×C4C2×C4⋊C411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4C22⋊Q811yescharacteristic
290C22×C4C22⋊Q811yescharacteristic
291C22×C4C22⋊Q811yescharacteristic
292C22×C4C22⋊Q811yescharacteristic
293C22×C4C22⋊Q811yescharacteristic
294C22×C4#12311yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×Q811yescharacteristic
298C22×C4C2×#4111yescharacteristic
299M16—24no—
300M16—24no—
301M16—42no—
302M16—42no—
303M16—42no—
304M16—42no—
305M16—82no—
306M16—82no—
307M16—82no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—44no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—81no—
317C2×C8—81no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4C2×C4⋊C412yes—
331C4⋊C4C2×C4⋊C412yes—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42C2×C4⋊C411yescharacteristic
341C42C2×C4⋊C411yescharacteristic
342C42C4×D411yescharacteristic
343C42C4×D411yescharacteristic
344C42C4×D411yescharacteristic
345C42C4×D411yescharacteristic
346C42#12011yescharacteristic
347C42#12011yescharacteristic
348C42C22⋊Q811yescharacteristic
349C42C22⋊Q811yescharacteristic
350C42#12311yescharacteristic
351C42#12311yescharacteristic
352C42C23×C411yescharacteristic
353C42C22×D411yescharacteristic
354C42C22×D411yescharacteristic
355C42C2×#4111yescharacteristic
356C23—22no—
357C23—41no—
358C23—41no—
359C23#38511yescharacteristic
360C23#38511yescharacteristic
361C23C2×C22⋊Q811yescharacteristic
362Q8—24no—
363Q8—22no—
364Q8—22no—
365Q8—22no—
366Q8—44no—
367Q8—44no—
368Q8—42no—
369Q8#43014yes—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4#38511yescharacteristic
402C2×C4#38511yescharacteristic
403C2×C4#38511yescharacteristic
404C2×C4#38511yescharacteristic
405C2×C4#38511yescharacteristic
406C2×C4#38511yescharacteristic
407C2×C4C22×C4⋊C411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×C4×D411yescharacteristic
410C2×C4C2×#12011yescharacteristic
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×#12311yescharacteristic
413C2×C4#57712yes—
414C2×C4#57812yes—
415C8—42no—
416C8—42no—
417C8—44no—
418C8—44no—
419C8—44no—
420C8—82no—
421C8—82no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—44no—
431C22C2×#38511yescharacteristic
432C22#308212yes—
433C4—24no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4C2×#43012yes—
447C4#294911yescharacteristic
448C4#295011yescharacteristic
449C4#301312yes—
450C2—22no—
451C2—41no—
452C2—41no—
453C2#2085111yescharacteristic
454C2#2086711yescharacteristic
455C2#2088011yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C22⋊Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8909C2semidirect
7. #8907C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#219702
#178152
#174492
C2×#14872
#89092
#89072
#80782
#78862