Finite Group Databasebeta

← #216356

Subgroups — #216356

Order 512.

858 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34621C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic, complemented
3#21349C211yesmaximal, characteristic, complemented
4#9866C212yesmaximal, complemented
5#9674C212yesmaximal, complemented
6#9074C212yesmaximal, complemented
7#9072C212yesmaximal, complemented
8#8569C212yesmaximal, complemented
9#8567C212yesmaximal, complemented
10C2×#431C2211yescharacteristic
11C22×C4×C8C2211yescharacteristic
12#2651—22no—
13#2648—22no—
14#2647C411yescharacteristic
15#2644C412yes—
16#2641C411yescharacteristic
17#2629C411yescharacteristic
18#2624C412yes—
19#2619C411yescharacteristic
20#2395—22no—
21#2392—22no—
22C4×#127C2211yescharacteristic
23C4×#125C2211yescharacteristic
24C4×#121C2212yes—
25#2293C2212yes—
26C4×#117C2211yescharacteristic
27#1939—22no—
28#1923—22no—
29#1916—22no—
30#1850—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34C4×C4⋊C8C2212yes—
35C4×C4⋊C8C2212yes—
36C4×C4⋊C8C2212yes—
37C2×D4⋊C8C2212yes—
38C2×D4⋊C8C2212yes—
39C2×D4⋊C8C2212yes—
40C2×D4⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C23×C8—21no—
46C2×#127C2×C411yescharacteristic
47C2×#125C2×C411yescharacteristic
48C2×#121C2×C412yes—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2312yes—
52C2×#117—22no—
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C22×C42C2311yescharacteristic
56#431—41no—
57#431—41no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—22no—
66C2×C4×C8—21no—
67C2×C4×C8—22no—
68C2×C4×C8—41no—
69C2×C4×C8—41no—
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72#400—22no—
73#399—22no—
74#398—22no—
75#398—22no—
76#397—22no—
77#397—22no—
78#397—22no—
79#397—22no—
80#396—22no—
81#396—22no—
82#394—22no—
83#394—22no—
84#394—22no—
85#394—22no—
86#394—22no—
87#392—22no—
88#390—22no—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#388—22no—
98#388—22no—
99#388—22no—
100#387—22no—
101#386—22no—
102#384C2×C411yescharacteristic
103#382C2×C411yescharacteristic
104#380C2×C412yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4C2312yes—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4D412yes—
112C4×C22⋊C4D412yes—
113#376C2×C411yescharacteristic
114#376C2×C411yescharacteristic
115C43C2311yescharacteristic
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122D4⋊C8—42no—
123D4⋊C8—42no—
124D4⋊C8—42no—
125D4⋊C8—42no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4C22×C412yes—
129C23×C4—22no—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—44no—
138C22×C8—42no—
139#127—24no—
140#125—24no—
141#121—28no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—82no—
147C4×D4—82no—
148C4×D4—82no—
149C4×D4—82no—
150#117—24no—
151#117—24no—
152#117—82no—
153#117—82no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2411yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—42no—
200C4⋊C8—44no—
201C4⋊C8—44no—
202C4⋊C8—82no—
203C22⋊C8—42no—
204C22⋊C8—82no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—42no—
210C4×C8—82no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222C24—22no—
223C24C2×C22⋊C411yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—82no—
231C2×D4—82no—
232C2×D4—82no—
233C2×D4—82no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4#13114yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C2×C8—24no—
265C2×C8—42no—
266C2×C8—48no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—44no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—44no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42#13114yes—
321C42H(5,𝔽2)11yescharacteristic
322C42H(5,𝔽2)11yescharacteristic
323C422−1+411yescharacteristic
324C422−1+411yescharacteristic
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23#40812yes—
337C23C22×C22⋊C411yescharacteristic
338D4—44no—
339D4—44no—
340D4—88no—
341D4—88no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4#40814yes—
381C2×C4#40812yes—
382C2×C4#52011yescharacteristic
383C2×C4#52011yescharacteristic
384C2×C4#53511yescharacteristic
385C2×C4#53611yescharacteristic
386C2×C4#53711yescharacteristic
387C2×C4#53711yescharacteristic
388C2×C4C2×#13112yes—
389C2×C4C2×#13112yes—
390C2×C4#57612yes—
391C2×C4#57712yes—
392C8—44no—
393C8—82no—
394C8—82no—
395C8—82no—
396C22—24no—
397C22—22no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22#231111yescharacteristic
409C22C2×#40811yescharacteristic
410C22C2×#40811yescharacteristic
411C22#289712yes—
412C22#290112yes—
413C4—24no—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—41no—
421C4—41no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4#299412yes—
426C4#299412yes—
427C4#325712yes—
428C4#325911yescharacteristic
429C4#326011yescharacteristic
430C2—42no—
431C2—162no—
432C2#1746812yes—
433C2#1748312yes—
434C2#2184811yescharacteristic
435C2#2189711yescharacteristic
436C2#2190111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2619C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21922C2semidirect
3. #21349C2semidirect
4. #9866C2semidirect
5. #9674C2semidirect
6. #9074C2semidirect
7. #9072C2semidirect
8. #8569C2semidirect
9. #8567C2semidirect

Maximal subgroups

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

subgroupindex
#346212
#219222
#213492
#98662
#96742
#90742
#90722
#85692
#85672