Finite Group Databasebeta

← #192589

Subgroups — #192589

Order 512.

818 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2#21352C211yesmaximal, characteristic
3#17822C211yesmaximal, characteristic
4#17562C211yesmaximal, characteristic
5#17496C211yesmaximal, characteristic
6#8881C214yesmaximal, complemented
7#8879C214yesmaximal, complemented
8#8093C211yesmaximal, characteristic
9#7920C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11Q8×M16—22no—
12Q8×M16—22no—
13#2962—22no—
14#2962—22no—
15#2961C2214yes—
16#2960—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#432C2211yescharacteristic
20#2905—21no—
21C2×#423C2211yescharacteristic
22#2873—21no—
23C2×C4×M16C2211yescharacteristic
24#2283C2211yescharacteristic
25#1940C2212yes—
26#1940C2212yes—
27#1851—21no—
28#1851—21no—
29#1768—22no—
30#1573—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1569—22no—
34#1515C414yes—
35#1511C414yes—
36#1503C2214yes—
37#1497—22no—
38#1491—22no—
39#1481C2212yes—
40#1481C2212yes—
41#1303C2214yes—
42#1281C2214yes—
43C22×M16—21no—
44C22×M16D411yescharacteristic
45C22×M16D411yescharacteristic
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533C2×C412yes—
51#533D412yes—
52#533Q812yes—
53#533C2312yes—
54C2×C4⋊Q8Q811yescharacteristic
55C2×C4⋊Q8Q811yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8C2×C412yes—
59C2×C4×Q8—21no—
60C2×C4×Q8D412yes—
61C2×#117—21no—
62C22×C42C2311yescharacteristic
63#446—24no—
64#446—24no—
65#446—28no—
66#446—28no—
67C8×Q8—24no—
68C8×Q8—24no—
69#432—21no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432C2×C412yes—
77#431—21no—
78#431—21no—
79#431C2312yes—
80#431C2312yes—
81#424—24no—
82#424—24no—
83#423—24no—
84#423—21no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423D412yes—
89#423C2312yes—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8D412yes—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96#407—21no—
97#407C2×C412yes—
98C2×C22⋊C8—21no—
99#405—24no—
100#405—24no—
101C4×M16—24no—
102C4×M16—21no—
103C4×M16—21no—
104C4×M16—21no—
105C4×M16—21no—
106C4×M16—21no—
107C4×M16—21no—
108C2×#97—21no—
109C2×#97C2312yes—
110C2×C4×C8D412yes—
111#383C2×C412yes—
112#382—21no—
113#381C2×C412yes—
114#380—21no—
115C4×C4⋊C4C2312yes—
116C4×C22⋊C4—21no—
117#376—21no—
118#336—22no—
119#336—22no—
120#335—24no—
121#334—24no—
122#328C2×C4116yes—
123Q8⋊C8—24no—
124Q8⋊C8—24no—
125C22×Q8—21no—
126C22×Q8C4⋊C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—24no—
138C2×M16—42no—
139C2×M16—44no—
140C2×M16—41no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8C4⋊C412yes—
151C4⋊Q8C4⋊C412yes—
152C4⋊Q8C2×Q812yes—
153C4⋊Q8C2×Q812yes—
154#125—22no—
155#125—22no—
156#125—22no—
157#125—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8C4⋊C414yes—
165C22⋊Q8C22×C414yes—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8C4⋊C414yes—
173C4×Q8C2×D414yes—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C4⋊C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×Q811yescharacteristic
196C2×C42C2×Q811yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42C2411yescharacteristic
201C4⋊C8—28no—
202C4⋊C8—28no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—44no—
214C4⋊C8—44no—
215C4⋊C8—41no—
216C4⋊C8C2×D414yes—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—41no—
220C22⋊C8—41no—
221C22⋊C8—41no—
222C22⋊C8C22×C414yes—
223#97—28no—
224#97—22no—
225#97—22no—
226#97—22no—
227#97—41no—
228#97—41no—
229#97—41no—
230C4×C8—28no—
231C4×C8—22no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95C22×C414yes—
239C24C4×D411yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8C2×C4⋊C414yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4111yescharacteristic
271M16—24no—
272M16—24no—
273M16—42no—
274M16—42no—
275M16—44no—
276M16—44no—
277M16—82no—
278M16—82no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—44no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8#13714yes—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—82no—
305C4⋊C4C2×C4⋊C414yes—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—82no—
309C42—24no—
310C42—24no—
311C42—42no—
312C42—42no—
313C42C2×C4⋊C411yescharacteristic
314C42C2×C4⋊C411yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42#12011yescharacteristic
318C42#12011yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42C22⋊Q811yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42#12311yescharacteristic
324C42#12311yescharacteristic
325C42C22×D411yescharacteristic
326C42C22×D411yescharacteristic
327C42C22×Q811yescharacteristic
328C42C2×#4111yescharacteristic
329C23—22no—
330C23—41no—
331C23—41no—
332C23#38511yescharacteristic
333C23#38511yescharacteristic
334C23C2×C4×D411yescharacteristic
335Q8—44no—
336Q8—44no—
337Q8—44no—
338Q8—82no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4#38511yescharacteristic
366C2×C4#38511yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4#38511yescharacteristic
369C2×C4#38511yescharacteristic
370C2×C4#38511yescharacteristic
371C2×C4C22×C4⋊C411yescharacteristic
372C2×C4C2×C4×D411yescharacteristic
373C2×C4C2×#12011yescharacteristic
374C2×C4C2×C22⋊Q811yescharacteristic
375C2×C4C2×C22⋊Q811yescharacteristic
376C2×C4C2×#12311yescharacteristic
377C2×C4C2×#13712yes—
378C2×C4#57512yes—
379C8—24no—
380C8—44no—
381C8—42no—
382C8—44no—
383C8—82no—
384C8—81no—
385C8—81no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—82no—
395C22C2×#38511yescharacteristic
396C22C4×#13712yes—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—44no—
408C4—44no—
409C4—84no—
410C4#300411yescharacteristic
411C4#300511yescharacteristic
412C4#307812yes—
413C2—22no—
414C2—41no—
415C2—41no—
416C2#2086711yescharacteristic
417C2#2087411yescharacteristic
418C2#2089211yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C4×D4
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
6. #8881C2semidirect
7. #8879C2semidirect

Maximal subgroups

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

subgroupindex
#601922
#213522
#178222
#175622
#174962
#88812
#88792
#80932
#79202