Finite Group Databasebeta

← #224183

Subgroups — #224183

Order 512.

835 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36562C211yesmaximal, characteristic
2#21345C212yesmaximal, complemented
3#9895C212yesmaximal, complemented
4#9894C212yesmaximal, complemented
5#9891C212yesmaximal, complemented
6#8583C211yesmaximal, characteristic
7#8581C212yesmaximal
8#8226C212yesmaximal
9#8224C211yesmaximal, characteristic
10C2×#557C411yescharacteristic
11C2×#554C411yescharacteristic
12#2862C2211yescharacteristic
13#2859C2212yes—
14#2567C411yescharacteristic
15#2566C412yes—
16#2565C412yes—
17#2556—22no—
18#2556—22no—
19#2542C411yescharacteristic
20#2429—22no—
21#2360—22no—
22C4×#124C2211yescharacteristic
23C4×#124C2212yes—
24C42×Q8C2211yescharacteristic
25#1984—22no—
26#1939—22no—
27#1917—22no—
28#1917—22no—
29#1767C2212yes—
30#1767C2212yes—
31#1767C2212yes—
32#1745C2212yes—
33#1745C2212yes—
34C4×#97C2212yes—
35C2×#329C2211yescharacteristic
36C2×#329C2212yes—
37C2×#329C2211yescharacteristic
38#1295C2212yes—
39#1295C2212yes—
40#1295C2212yes—
41#1295C2212yes—
42#1295C2212yes—
43#1295C2211yescharacteristic
44#1295C2211yescharacteristic
45#557—44no—
46#554—44no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125—22no—
51C2×#125C2×C412yes—
52C2×#125C2×C412yes—
53C2×#124C2311yescharacteristic
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D412yes—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C411yescharacteristic
64C22×C4⋊C4—21no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×#97—22no—
69C2×#97C2312yes—
70C2×#97C2312yes—
71C2×#97C2312yes—
72C2×#97C2312yes—
73C2×C4×C8—22no—
74#399—21no—
75#399—21no—
76#398—22no—
77#398—22no—
78#396—22no—
79#396—22no—
80#395—21no—
81#395—21no—
82#391—22no—
83#391—22no—
84#391C2×C411yescharacteristic
85#391C2×C412yes—
86#390C2×C411yescharacteristic
87#390C2×C412yes—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#385—22no—
100#384C2×C412yes—
101#384C2×C412yes—
102#382C2×C412yes—
103#382C2311yescharacteristic
104#382C2312yes—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C43C2312yes—
114C43C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#329—22no—
123#329—24no—
124#329Q814yescomplemented
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22×C411yescharacteristic
129C22×D4C22×C411yescharacteristic
130C23×C4—21no—
131C23×C4—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—44no—
139#125—44no—
140#125—44no—
141#125—48no—
142#124—22no—
143#124C2×Q812yes—
144#124#4112yes—
145C22⋊Q8—48no—
146C22⋊Q8—48no—
147C22⋊Q8—44no—
148C4×Q8—24no—
149C4×Q8—44no—
150C4×Q8—44no—
151C4×Q8—42no—
152C4×Q8—44no—
153C4×D4—42no—
154C4×D4—44no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42C22×C412yes—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2411yescharacteristic
196C4⋊C8—44no—
197C4⋊C8—44no—
198C4⋊C8—44no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—22no—
204#97—22no—
205#97—22no—
206#97—44no—
207#97C2×Q814yes—
208C4×C8—44no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95C22⋊C412yes—
218#95C22⋊C412yes—
219#95C22⋊C411yescharacteristic
220#95C22⋊C411yescharacteristic
221C24—21no—
222C2×Q8—21no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—82no—
230C2×Q8#10014yes—
231C2×Q8C4×Q812yes—
232C2×D4—21no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4C4×Q812yes—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D412yes—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—82no—
295C42—24no—
296C42—22no—
297C42—24no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—41no—
303C42—42no—
304C42—42no—
305C42—44no—
306C42—42no—
307C42C22×Q811yescharacteristic
308C42C2×#4111yescharacteristic
309C422−1+412yes—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—42no—
314C23—42no—
315C23C22×C22⋊C411yescharacteristic
316Q8—44no—
317Q8—42no—
318Q8—42no—
319Q8—84no—
320Q8—162no—
321D4—42no—
322D4—42no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—84no—
357C2×C4—164no—
358C2×C4C2×#10012yes—
359C2×C4C2×#10014yes—
360C2×C4#42114yes—
361C2×C4C2×C4×Q811yescharacteristic
362C2×C4#51911yescharacteristic
363C2×C4#54812yes—
364C2×C4D4×Q812yes—
365C8—44no—
366C8—44no—
367C8—82no—
368C8—82no—
369C22—44no—
370C22—42no—
371C22—81no—
372C22—82no—
373C22—81no—
374C22Q8×C22⋊C411yescharacteristic
375C22C22×#10011yescharacteristic
376C22#288911yescharacteristic
377C22C2×#42112yes—
378C22#290112yes—
379C4—22no—
380C4—22no—
381C4—22no—
382C4—24no—
383C4—24no—
384C4—21no—
385C4—21no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—41no—
390C4—42no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4—82no—
395C2—82no—
396C2#1767511yescharacteristic
397C2#1768012yes—
398C2#1768811yescharacteristic
399C2Q8×#10011yescharacteristic
400C2#2252912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21345C2semidirect
3. #9895C2semidirect
4. #9894C2semidirect
5. #9891C2semidirect
124. #329Q8semidirect

Maximal subgroups

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

subgroupindex
#365622
#213452
#98952
#98942
#98912
#85832
#85812
#82262
#82242