Finite Group Databasebeta

← #190331

Subgroups — #190331

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34192C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×#1467C211yesmaximal, characteristic
4#8775C214yesmaximal, complemented
5#8774C214yesmaximal, complemented
6#8069C212yesmaximal
7#7977C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10C2×#432C2212yes—
11#2356C412yes—
12#2356C2212yes—
13#2356C2212yes—
14#2351C412yes—
15#2345—21no—
16#2345—22no—
17Q8×C22⋊C4—21no—
18#2320—22no—
19#2320—21no—
20#2283C412yes—
21#2283C412yes—
22#2283C2211yescharacteristic
23#2283C2211yescharacteristic
24#2276—21no—
25#1821—21no—
26#1801—22no—
27#1784—21no—
28C4×#101C2214yes—
29#1467—21no—
30#1467—21no—
31#1467C2214yes—
32C2×#324—22no—
33#1300C2214yes—
34#1300C2214yes—
35#1288C2218yes—
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—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D412yes—
50C22×C42C2×C412yes—
51C22×C42C2311yescharacteristic
52#432—22no—
53#432—22no—
54#432—22no—
55#432—22no—
56#432—22no—
57#432C2314yes—
58#431—22no—
59#431—22no—
60#423—22no—
61C2×C4⋊C8—22no—
62C2×#101—24no—
63#407—22no—
64C2×C22⋊C8—22no—
65C4×M16C2314yes—
66C2×#97—22no—
67#391—22no—
68#391—22no—
69#391—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#387—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#383—22no—
80#383—21no—
81#383C2×C412yes—
82#383C2×C412yes—
83#383C2312yes—
84#382—22no—
85#382—21no—
86#382C2×C412yes—
87#382C2×C412yes—
88#382C2×C412yes—
89#381C2×C411yescharacteristic
90#381C2×C411yescharacteristic
91#381C2×C412yes—
92#380—22no—
93#380—22no—
94#380—21no—
95#380—21no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C412yes—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108#376—22no—
109#376—21no—
110#376C2×C412yes—
111#376C2×C412yes—
112#376C2311yescharacteristic
113#376C2311yescharacteristic
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—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C22⋊C412yes—
175C2×C42C22⋊C412yes—
176C2×C42C22×C412yes—
177C2×C42C22×C412yes—
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42#4112yes—
182C2×C42#4112yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C2×C42C2411yescharacteristic
186C4⋊C8—24no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189C4⋊C8—42no—
190#101—48no—
191C22⋊C8—24no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195#97—24no—
196#97—24no—
197#97—42no—
198C4×C8—24no—
199#95—21no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—21no—
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#95C22×C412yes—
216#95C22×C412yes—
217#95C22×C412yes—
218#95C22×C412yes—
219C24C2×C22⋊C411yescharacteristic
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—48no—
228C2×Q8—48no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—81no—
242C22×C4—82no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4#11711yescharacteristic
251C22×C4#11711yescharacteristic
252C22×C4#11711yescharacteristic
253C22×C4#11711yescharacteristic
254C22×C4#12312yes—
255C22×C4#12312yes—
256C22×C4#12312yes—
257C22×C4#12312yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261M16—48no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8—84no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C22⋊C4—42no—
279C22⋊C4—44no—
280C22⋊C4—42no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—44no—
290C42—44no—
291C42—44no—
292C42—81no—
293C42—81no—
294C42—82no—
295C42—82no—
296C42#11712yes—
297C42#11712yes—
298C42#12312yes—
299C42#12312yes—
300C42#12312yes—
301C42#12312yes—
302C42C2×#4112yes—
303C42C2×#4112yes—
304C23—22no—
305C23—22no—
306C23—41no—
307C23#38111yescharacteristic
308C23#38111yescharacteristic
309C23C22×C22⋊C411yescharacteristic
310Q8—28no—
311Q8—48no—
312Q8—44no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#38111yescharacteristic
353C2×C4#38111yescharacteristic
354C2×C4#38112yes—
355C2×C4#38112yes—
356C2×C4#42114yes—
357C2×C4C2×#11711yescharacteristic
358C2×C4C2×#11711yescharacteristic
359C2×C4C2×#12312yes—
360C2×C4C2×#12312yes—
361C8—84no—
362C8—84no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22#183012yes—
372C22C2×#38111yescharacteristic
373C22C2×#42112yes—
374C4—22no—
375C4—22no—
376C4—24no—
377C4—22no—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4—82no—
388C4—82no—
389C4—82no—
390C2—22no—
391C2—22no—
392C2—41no—
393C2#2070511yescharacteristic
394C2#2070811yescharacteristic
395C2C2×#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
#341922
#207272
C2×#14672
#87752
#87742
#80692
#79772