Finite Group Databasebeta

← #190328

Subgroups — #190328

Order 512.

897 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34128C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×#1467C211yesmaximal, characteristic
4#8775C214yesmaximal, complemented
5#8774C214yesmaximal, complemented
6#8071C212yesmaximal
7#7978C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10C2×#432C2212yes—
11#2356C412yes—
12#2356C2212yes—
13#2356C2212yes—
14#2351C412yes—
15#2344—21no—
16#2344—22no—
17Q8×C22⋊C4—21no—
18#2295—22no—
19#2295—21no—
20#2283C412yes—
21#2283C2211yescharacteristic
22C42×Q8—21no—
23C4×#117C412yes—
24C4×#117C2211yescharacteristic
25#1821—21no—
26#1801—22no—
27#1784—21no—
28C4×#101C2214yes—
29#1467—21no—
30#1467—21no—
31#1467C2214yes—
32C2×#324—22no—
33#1300C2218yes—
34#1288C2214yes—
35#1288C2214yes—
36C23×Q8D411yescharacteristic
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×Q8C2×C411yescharacteristic
44C2×C4×Q8D412yes—
45C2×C4×Q8D411yescharacteristic
46C2×C4×Q8D411yescharacteristic
47C2×C4×Q8C2312yes—
48C2×#117—22no—
49C2×#117—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D412yes—
52C22×C42C2×C412yes—
53C22×C42C2311yescharacteristic
54#432—22no—
55#432—22no—
56#432—22no—
57#432—22no—
58#432—22no—
59#432C2314yes—
60#431—22no—
61#431—22no—
62#423—22no—
63C2×C4⋊C8—22no—
64C2×#101—24no—
65#407—22no—
66C2×C22⋊C8—22no—
67C4×M16C2314yes—
68C2×#97—22no—
69#391—22no—
70#391—22no—
71#391—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#383C2×C412yes—
82#383C2×C412yes—
83#383C2311yescharacteristic
84#383C2311yescharacteristic
85#382C2×C412yes—
86#382C2×C412yes—
87#382C2×C412yes—
88#381—22no—
89#381—21no—
90#380—22no—
91#380—21no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4C2×C411yescharacteristic
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4C2×C411yescharacteristic
107#376—22no—
108#376—21no—
109#376C2×C412yes—
110#376C2×C412yes—
111#376C2312yes—
112C43—22no—
113C43—21no—
114#343—24no—
115#324—24no—
116#324—24no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—44no—
120C22×Q8C22⋊C411yescharacteristic
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C2×D412yes—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×M16—24no—
129C22×C8—42no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—24no—
142C4×Q8—48no—
143#117—44no—
144#117—48no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—21no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42C22⋊C412yes—
179C2×C42C22⋊C412yes—
180C2×C42C22×C412yes—
181C2×C42C22×C412yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4112yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189C2×C42C2411yescharacteristic
190C4⋊C8—24no—
191C4⋊C8—42no—
192C4⋊C8—42no—
193C4⋊C8—42no—
194#101—48no—
195C22⋊C8—24no—
196C22⋊C8—42no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199#97—24no—
200#97—24no—
201#97—42no—
202C4×C8—24no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95—21no—
216#95—21no—
217#95C22×C412yes—
218#95C22×C412yes—
219#95C22×C412yes—
220#95C22×C412yes—
221C24C2×C22⋊C411yescharacteristic
222C2×Q8—24no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—48no—
230C2×Q8—48no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4#11711yescharacteristic
253C22×C4#11711yescharacteristic
254C22×C4#11711yescharacteristic
255C22×C4#11711yescharacteristic
256C22×C4#12312yes—
257C22×C4#12312yes—
258C22×C4#12312yes—
259C22×C4#12312yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263M16—48no—
264C2×C8—44no—
265C2×C8—44no—
266C2×C8—84no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C22⋊C4—24no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—42no—
292C42—44no—
293C42—42no—
294C42—44no—
295C42—44no—
296C42—44no—
297C42—44no—
298C42—44no—
299C42—44no—
300C42#11712yes—
301C42#11712yes—
302C42#12312yes—
303C42#12312yes—
304C42#12312yes—
305C42#12312yes—
306C42C2×#4112yes—
307C42C2×#4112yes—
308C23—22no—
309C23—22no—
310C23—41no—
311C23#38111yescharacteristic
312C23#38111yescharacteristic
313C23C22×C22⋊C411yescharacteristic
314Q8—28no—
315Q8—48no—
316Q8—44no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4#38111yescharacteristic
357C2×C4#38112yes—
358C2×C4#38112yes—
359C2×C4#38111yescharacteristic
360C2×C4#42114yes—
361C2×C4C2×#11711yescharacteristic
362C2×C4C2×#11711yescharacteristic
363C2×C4C2×#12312yes—
364C2×C4C2×#12312yes—
365C8—84no—
366C8—84no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—44no—
375C22#182712yes—
376C22C2×#38111yescharacteristic
377C22C2×#42112yes—
378C4—22no—
379C4—22no—
380C4—24no—
381C4—22no—
382C4—22no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—44no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—82no—
392C4—82no—
393C4#183012yes—
394C2—22no—
395C2—22no—
396C2—41no—
397C2#2070511yescharacteristic
398C2C2×#182711yescharacteristic
399C2C2×#183011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ 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
#207272
C2×#14672
#87752
#87742
#80712
#79782