Finite Group Databasebeta

← #481139

Subgroups — #481139

Order 512.

736 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2960C211yesmaximal, characteristic
2#33418C211yesmaximal, characteristic
3#33418C211yesmaximal, characteristic
4#11640C214yesmaximal, complemented
5#11638C214yesmaximal, complemented
6#11607C214yesmaximal, complemented
7C2×#533C411yescharacteristic
8#2960—21no—
9#2960—21no—
10#2960—21no—
11#2960—21no—
12#2960C412yes—
13#2960C412yes—
14#2960C2212yes—
15#2960C2212yes—
16C2×#446—21no—
17C2×#446C412yes—
18C2×C8×Q8—21no—
19C2×C8×Q8C2212yes—
20C2×#424—21no—
21C2×#423C411yescharacteristic
22C2×#405—21no—
23C22×C4×C8C2211yescharacteristic
24#2163—21no—
25#2163—21no—
26#2163—21no—
27#2163C2212yes—
28#2162—21no—
29#2162C2212yes—
30#2162C2212yes—
31#2162C2212yes—
32#2150—21no—
33#2150C2212yes—
34#2112—21no—
35#2112—21no—
36C4×M32—21no—
37C4×M32C2212yes—
38Q8⋊C16C22116yescomplemented
39C22×M16—21no—
40C23×C8D411yescharacteristic
41C23×C8D411yescharacteristic
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533C812yes—
47#533C812yes—
48#533C2×C412yes—
49#533C2×C412yes—
50C2×C4⋊Q8C811yescharacteristic
51C2×C4⋊Q8C811yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C812yes—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C412yes—
57C2×#117—21no—
58C22×C42C2×C411yescharacteristic
59C2×M32—22no—
60C2×M32—22no—
61#446—22no—
62#446—22no—
63#446—22no—
64#446—22no—
65#446—22no—
66#446—22no—
67#446C2×C414yes—
68#446C2×C414yes—
69C8×Q8—22no—
70C8×Q8—22no—
71C8×Q8—22no—
72C8×Q8—22no—
73C8×Q8—22no—
74C8×Q8—22no—
75C8×Q8C2×C414yes—
76C8×Q8C2314yes—
77#424—22no—
78#424—22no—
79#424—22no—
80#424—22no—
81#423—21no—
82#423—21no—
83#423C2×C412yes—
84#423C2×C412yes—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2×C412yes—
89#405—22no—
90#405—22no—
91#405—22no—
92#405—22no—
93C2×#97—22no—
94C2×C4×C8—22no—
95C2×C4×C8C2×C411yescharacteristic
96C2×C4×C8C2×C411yescharacteristic
97C2×C4×C8C2×C411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8D411yescharacteristic
100C2×C4×C8D411yescharacteristic
101C2×C4×C8D411yescharacteristic
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C4⋊C16—22no—
108C4⋊C16—22no—
109C4⋊C16—22no—
110C4⋊C16C2314yes—
111C22⋊C16—22no—
112C22⋊C16—22no—
113C22⋊C16—22no—
114C22⋊C16—22no—
115#346—22no—
116#346—22no—
117C4×C16—22no—
118C4×C16C2314yes—
119C22×Q8—21no—
120C22×Q8C2×C812yes—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C2×C811yescharacteristic
123C23×C4M1611yescharacteristic
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—44no—
127C22×C8—24no—
128C22×C8—22no—
129C22×C8—42no—
130C22×C8C22⋊C411yescharacteristic
131C22×C8C22⋊C411yescharacteristic
132C22×C8C22⋊C411yescharacteristic
133C22×C8C22⋊C411yescharacteristic
134C22×C8C22⋊C411yescharacteristic
135C22×C8C22⋊C411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D411yescharacteristic
141C22×C8C2×D411yescharacteristic
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8C2×C812yes—
147C4⋊Q8C2×C812yes—
148C4⋊Q8C2×C812yes—
149C4⋊Q8C2×C812yes—
150#125—22no—
151#125—22no—
152#125—22no—
153#125—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8C2×C814yes—
161C22⋊Q8C2×C814yes—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8C2×C814yes—
169C4×Q8C22×C414yes—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C2×C812yes—
180C2×C22⋊C4—22no—
181C2×C42—22no—
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C2×C811yescharacteristic
186C2×C42C2×C811yescharacteristic
187C2×C42C2×C811yescharacteristic
188C2×C42M1611yescharacteristic
189C2×C42M1611yescharacteristic
190C2×C42M1611yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194M32—44no—
195M32—44no—
196C2×C16—24no—
197C2×C16—24no—
198C4⋊C8—24no—
199C4⋊C8—24no—
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8C22×C414yes—
206#97—24no—
207#97—24no—
208C4×C8—24no—
209C4×C8—24no—
210C4×C8C22⋊C411yescharacteristic
211C4×C8C22⋊C411yescharacteristic
212C4×C8C22⋊C411yescharacteristic
213C4×C8C22⋊C411yescharacteristic
214C4×C8C22⋊C411yescharacteristic
215C4×C8C22⋊C411yescharacteristic
216C4×C8C22×C411yescharacteristic
217C4×C8C22×C411yescharacteristic
218C4×C8C22×C411yescharacteristic
219C4×C8C2×D411yescharacteristic
220C4×C8C2×D411yescharacteristic
221C4×C8C2×D411yescharacteristic
222C4×C8C2×D411yescharacteristic
223C4×C8C2×D411yescharacteristic
224C4×C8C2×D411yescharacteristic
225C4×C8C2411yescharacteristic
226C24C22⋊C811yescharacteristic
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8C22×C814yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊C811yescharacteristic
240C22×C4C22⋊C811yescharacteristic
241C22×C4C22⋊C811yescharacteristic
242C22×C4C22⋊C811yescharacteristic
243C22×C4C22⋊C811yescharacteristic
244C22×C4C22⋊C811yescharacteristic
245C22×C4C22⋊C811yescharacteristic
246C22×C4C22⋊C811yescharacteristic
247C22×C4C22⋊C811yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C22×C811yescharacteristic
252C22×C4C22×C811yescharacteristic
253C22×C4C22×C811yescharacteristic
254C22×C4C2×M1611yescharacteristic
255C22×C4C2×M1611yescharacteristic
256C22×C4C2×M1611yescharacteristic
257M16—44no—
258M16—44no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—24no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—44no—
272C2×C8C2×C22⋊C411yescharacteristic
273C2×C8C2×C22⋊C411yescharacteristic
274C2×C8C2×C22⋊C411yescharacteristic
275C2×C8C2×C22⋊C411yescharacteristic
276C2×C8C2×C22⋊C411yescharacteristic
277C2×C8C2×C22⋊C411yescharacteristic
278C2×C8#13714yes—
279C2×C8C22×D411yescharacteristic
280C2×C8C22×D411yescharacteristic
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4C22×C814yes—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C42—24no—
296C42—24no—
297C42C22⋊C811yescharacteristic
298C42C22⋊C811yescharacteristic
299C42C22⋊C811yescharacteristic
300C42C22⋊C811yescharacteristic
301C42C22⋊C811yescharacteristic
302C42C22⋊C811yescharacteristic
303C42C2×C22⋊C411yescharacteristic
304C42C2×C22⋊C411yescharacteristic
305C42C2×C22⋊C411yescharacteristic
306C42C22×C811yescharacteristic
307C42C22×C811yescharacteristic
308C42C22×C811yescharacteristic
309C42C2×M1611yescharacteristic
310C42C2×M1611yescharacteristic
311C42C2×M1611yescharacteristic
312C42C23×C411yescharacteristic
313C16—44no—
314C16—44no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23C2×C22⋊C811yescharacteristic
319C23C2×C22⋊C811yescharacteristic
320C23C2×C22⋊C811yescharacteristic
321Q8—28no—
322Q8—44no—
323Q8—44no—
324Q8—44no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4C2×C22⋊C811yescharacteristic
349C2×C4C2×C22⋊C811yescharacteristic
350C2×C4C2×C22⋊C811yescharacteristic
351C2×C4C2×C22⋊C811yescharacteristic
352C2×C4C2×C22⋊C811yescharacteristic
353C2×C4C2×C22⋊C811yescharacteristic
354C2×C4C2×C22⋊C811yescharacteristic
355C2×C4C2×C22⋊C811yescharacteristic
356C2×C4C2×C22⋊C811yescharacteristic
357C2×C4#41912yes—
358C2×C4#42112yes—
359C2×C4C22×C22⋊C411yescharacteristic
360C2×C4C23×C811yescharacteristic
361C2×C4C22×M1611yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C8—44no—
369C8#41912yes—
370C8C2×#13712yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—44no—
380C22#147912yes—
381C22C22×C22⋊C811yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—44no—
391C4—44no—
392C4#147912yes—
393C4#211712yes—
394C4C2×#41911yescharacteristic
395C4C2×#42111yescharacteristic
396C4#340712yes—
397C2—22no—
398C2—41no—
399C2—41no—
400C2C2×#147911yescharacteristic
401C2#3333311yescharacteristic
402C2C2×#211711yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC4×C8C24
socleC22C22×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11640C2semidirect
5. #11638C2semidirect
6. #11607C2semidirect
38. Q8⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29602
#334182
#116402
#116382
#116072