Finite Group Databasebeta

← #190314

Subgroups — #190314

Order 512.

965 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34128C211yesmaximal, characteristic
2C2×C4×#101C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#8775C214yesmaximal, complemented
5#8774C214yesmaximal, complemented
6#8068C212yesmaximal
7#7974C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10C22×#101—21no—
11C2×C4×M16C2212yes—
12#2356C411yescharacteristic
13#2356C411yescharacteristic
14#2356C2212yes—
15#2356C2212yes—
16#2351C411yescharacteristic
17#2351C411yescharacteristic
18#2344—22no—
19#2344—21no—
20Q8×C22⋊C4—21no—
21#2295—22no—
22#2295—21no—
23#2283C412yes—
24#2283C2211yescharacteristic
25C42×Q8—21no—
26C4×#117C412yes—
27C4×#117C2211yescharacteristic
28C4×#101—22no—
29C4×#101C2214yes—
30C2×#343—21no—
31#1467C2214yes—
32#1463—22no—
33#1461—21no—
34#1461—21no—
35#1300C2214yes—
36#1300C2214yes—
37#1288C2218yes—
38C23×Q8D411yescharacteristic
39C22×M16—22no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D412yes—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×Q8C2312yes—
52C2×#117—22no—
53C2×#117—21no—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4D411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42C2311yescharacteristic
58#432—22no—
59#432C2314yes—
60#431—22no—
61#431—22no—
62#423—22no—
63C2×#101—28no—
64C2×#101—24no—
65#407—22no—
66C4×M16—22no—
67C4×M16—22no—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16C2314yes—
71C2×#97—22no—
72C2×C4×C8—22no—
73#391—21no—
74#391—21no—
75#391—22no—
76#391—22no—
77#389—21no—
78#389—21no—
79#389—22no—
80#389—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#383C2×C412yes—
88#383C2×C412yes—
89#383C2312yes—
90#382C2×C411yescharacteristic
91#382C2×C411yescharacteristic
92#382C2×C412yes—
93#382C2×C412yes—
94#381—22no—
95#381—21no—
96#380—22no—
97#380—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4C2×C411yescharacteristic
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4C2×C411yescharacteristic
115#376—22no—
116#376—21no—
117#376C2×C412yes—
118#376C2×C412yes—
119#376C2311yescharacteristic
120#376C2311yescharacteristic
121C43—22no—
122C43—21no—
123#343—24no—
124#324—24no—
125#324—24no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—42no—
130C22×Q8C22⋊C411yescharacteristic
131C22×Q8C22⋊C411yescharacteristic
132C22×Q8C2×D412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—28no—
139C2×M16—24no—
140C2×M16—44no—
141C2×M16—42no—
142C2×M16—42no—
143C22×C8—42no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—24no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—44no—
157C4×Q8—44no—
158#117—44no—
159#117—48no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C412yes—
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42C2411yescharacteristic
209C4⋊C8—24no—
210C4⋊C8—42no—
211#101—216no—
212C22⋊C8—24no—
213C22⋊C8—42no—
214#97—24no—
215#97—24no—
216#97—42no—
217C4×C8—24no—
218C4×C8—42no—
219C4×C8—42no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—22no—
234#95C22×C411yescharacteristic
235#95C22×C411yescharacteristic
236#95C22×C412yes—
237#95C22×C412yes—
238#95C22×C411yescharacteristic
239#95C22×C411yescharacteristic
240C24C2×C22⋊C411yescharacteristic
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—42no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8#10414yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4#11711yescharacteristic
277C22×C4#11711yescharacteristic
278C22×C4#11711yescharacteristic
279C22×C4#11711yescharacteristic
280C22×C4#12312yes—
281C22×C4#12312yes—
282C22×C4#12312yes—
283C22×C4#12312yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—28no—
288M16—44no—
289M16—48no—
290M16—84no—
291C2×C8—48no—
292C2×C8—44no—
293C2×C8—44no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C22⋊C4—24no—
310C22⋊C4—44no—
311C22⋊C4—42no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—42no—
321C42—44no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—44no—
326C42—44no—
327C42—44no—
328C42—44no—
329C42—42no—
330C42—42no—
331C42#11712yes—
332C42#11712yes—
333C42#12312yes—
334C42#12312yes—
335C42#12312yes—
336C42#12312yes—
337C42C2×#4112yes—
338C42C2×#4112yes—
339C23—22no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23#38111yescharacteristic
344C23#38111yescharacteristic
345C23C22×C22⋊C411yescharacteristic
346Q8—28no—
347Q8—48no—
348Q8—44no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4#38111yescharacteristic
395C2×C4#38112yes—
396C2×C4#38112yes—
397C2×C4#38111yescharacteristic
398C2×C4C2×#10414yes—
399C2×C4C2×#10412yes—
400C2×C4#42112yes—
401C2×C4C2×#11711yescharacteristic
402C2×C4C2×#11711yescharacteristic
403C2×C4C2×#12312yes—
404C2×C4C2×#12312yes—
405C8—48no—
406C8—84no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22#182712yes—
417C22C2×#38111yescharacteristic
418C22C22×#10411yescharacteristic
419C22C2×#42111yescharacteristic
420C4—22no—
421C4—24no—
422C4—22no—
423C4—22no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—81no—
433C4—82no—
434C4—82no—
435C4—81no—
436C4#182614yes—
437C4#183012yes—
438C2—21no—
439C2—21no—
440C2—22no—
441C2—41no—
442C2C2×#182611yescharacteristic
443C2C2×#182711yescharacteristic
444C2C2×#183011yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#381
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ 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
4. #8775C2semidirect
5. #8774C2semidirect

Maximal subgroups

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

subgroupindex
#341282
C2×C4×#1012
#173652
#87752
#87742
#80682
#79742