Finite Group Databasebeta

← #99484

Subgroups — #99484

Order 512.

463 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22178C211yesmaximal, characteristic
2#11326C211yesmaximal, characteristic, complemented
3#8944C211yesmaximal, characteristic
4#8833C211yesmaximal, characteristic, complemented
5#7565C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic, complemented
7#7026C211yesmaximal, characteristic, complemented
8C2×#436—21no—
9C22×#108—21no—
10C2×#423C2211yescharacteristic
11C4×#121C2211yescharacteristic
12#1923—21no—
13#1916—21no—
14#1870—21no—
15#1870—21no—
16#1851—21no—
17#1797C2211yescharacteristic
18C4×C22⋊C8C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#341—21no—
21#1728C2211yescharacteristic
22C2×#336C2211yescharacteristic
23C2×#334—21no—
24C2×D4⋊C8—21no—
25#1279—21no—
26#1277—21no—
27#1271C411yescharacteristic
28#1271C411yescharacteristic
29#1270C411yescharacteristic
30#1270C411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16D411yescharacteristic
34C22×M16D411yescharacteristic
35C23×C8—21no—
36C2×#121Q811yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4D411yescharacteristic
39C22×C42C2311yescharacteristic
40#436—42no—
41#436—42no—
42C2×#108—42no—
43C2×#108—42no—
44C2×#108—42no—
45#423—42no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60#390C2×C411yescharacteristic
61#389C2×C411yescharacteristic
62#388—21no—
63#386—21no—
64#380Q811yescharacteristic
65C4×C4⋊C4D411yescharacteristic
66C4×C22⋊C4—21no—
67#341—42no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336C2×C411yescharacteristic
76#336C2×C411yescharacteristic
77#336C2×C411yescharacteristic
78#336C2×C411yescharacteristic
79#334—42no—
80D4⋊C8—42no—
81C22×D4—21no—
82C22×D4C4⋊C411yescharacteristic
83C23×C4—21no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×Q811yescharacteristic
86C23×C4#4111yescharacteristic
87C2×M16—21no—
88C2×M16—22no—
89C2×M16—21no—
90C2×M16—21no—
91C2×M16—42no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—81no—
102C2×M16—81no—
103C2×M16SD1612yes—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120#121—22no—
121#121—41no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4C4⋊C411yescharacteristic
130C2×C4⋊C4C4⋊C411yescharacteristic
131C2×C4⋊C4C4⋊C411yescharacteristic
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C42—21no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42C22×C411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144#108—48no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—42no—
152C22⋊C8—22no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—81no—
159C22⋊C8—81no—
160C4×C8—41no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—41no—
164C4×C8—42no—
165#95—21no—
166#95—21no—
167#95—21no—
168C24—21no—
169C24C22⋊Q811yescharacteristic
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—81no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C2×C4⋊C411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4C4×D411yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4C22⋊Q811yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4C2×SD1611yescharacteristic
206C22×C4#13511yescharacteristic
207C22×C4AGL(1,ℤ/8)11yescharacteristic
208C22×C4AGL(1,ℤ/8)11yescharacteristic
209M16—42no—
210M16—42no—
211M16—42no—
212M16—42no—
213M16—81no—
214M16—84no—
215M16—81no—
216M16—81no—
217M16—81no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—42no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—42no—
247C2×C8—41no—
248C2×C8—81no—
249C2×C8—81no—
250C2×C8—81no—
251C2×C8—81no—
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4#10812yescomplemented
258C22⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—81no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23#38511yescharacteristic
288C23#47611yescharacteristic
289C23#47811yescharacteristic
290D4—42no—
291D4—81no—
292D4—82no—
293D4—82no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4C2×#10811yescharacteristic
332C2×C4#43011yescharacteristic
333C2×C4C4×SD1611yescharacteristic
334C2×C4#44211yescharacteristic
335C2×C4#44311yescharacteristic
336C2×C4#44411yescharacteristic
337C2×C4#44911yescharacteristic
338C2×C4#45011yescharacteristic
339C2×C4#45311yescharacteristic
340C2×C4#45611yescharacteristic
341C2×C4B(4,𝔽2)11yescharacteristic
342C2×C4#45811yescharacteristic
343C8—42no—
344C8—42no—
345C8—42no—
346C8—42no—
347C8—81no—
348C8—82no—
349C8—81no—
350C8—82no—
351C8—81no—
352C8—81no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#185511yescharacteristic
371C22#185811yescharacteristic
372C22#185911yescharacteristic
373C22#186111yescharacteristic
374C22#186411yescharacteristic
375C22#186611yescharacteristic
376C22#186711yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—42no—
386C4—81no—
387C4—81no—
388C4#160011yescharacteristic
389C4#160411yescharacteristic
390C4#160911yescharacteristic
391C4#161311yescharacteristic
392C2—41no—
393C2—41no—
394C2—161no—
395C2#886011yescharacteristic
396C2#887311yescharacteristic
397C2#887411yescharacteristic
398C2#889911yescharacteristic
399C2#890811yescharacteristic
400C2#891611yescharacteristic
401C2#893211yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC2×#121Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11326C2semidirect
4. #8833C2semidirect
6. #7029C2semidirect
7. #7026C2semidirect
257. C4⋊C4#108semidirect

Maximal subgroups

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

subgroupindex
#221782
#113262
#89442
#88332
#75652
#70292
#70262