Finite Group Databasebeta

← #102308

Subgroups — #102308

Order 512.

513 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 110 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#33042C211yesmaximal, characteristic
2#10797C211yesmaximal, characteristic, complemented
3#9908C211yesmaximal, characteristic, complemented
4#9712C211yesmaximal, characteristic
5#8595C211yesmaximal, characteristic, complemented
6#7667C211yesmaximal, characteristic, complemented
7#7664C211yesmaximal, characteristic
8C2×#486C411yescharacteristic, complemented
9C22×C4×C8C2211yescharacteristic
10#2857C2211yescharacteristic
11#2856C2211yescharacteristic
12#1990C411yescharacteristic, complemented
13#1957C411yescharacteristic, complemented
14#1934—21no—
15#1934—21no—
16#1893—21no—
17#1893—21no—
18#1834C411yescharacteristic, complemented
19#1797C2211yescharacteristic
20#1797C2211yescharacteristic
21C2×#336—21no—
22C2×#336—21no—
23C2×#336C2211yescharacteristic
24C2×#334—21no—
25C2×#327—22no—
26C4×#95C2211yescharacteristic
27#1294—22no—
28#1277—21no—
29#1272—22no—
30#1270—22no—
31C23×C8—21no—
32C23×C8D411yescharacteristic
33C23×C8D411yescharacteristic
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×#127C2×C411yescharacteristic
36C2×C22⋊Q8—21no—
37C2×#121—21no—
38C2×#121—21no—
39C2×#121—21no—
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42#486—42no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×#102—21no—
46C2×#102—21no—
47C2×#102—21no—
48C2×#102—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C4×C8—42no—
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8D411yescharacteristic
56#391—21no—
57#390—21no—
58#390—21no—
59#390—21no—
60#389—21no—
61#389—21no—
62#384C2×C411yescharacteristic
63#384C2×C411yescharacteristic
64#380—21no—
65#380—21no—
66C2×#95—21no—
67C2×#95D412yes—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—42no—
76#336—42no—
77#336D412yes—
78#334—42no—
79#327—44no—
80C22×Q8—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4—41no—
85C23×C4—22no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8C22⋊C411yescharacteristic
107C22×C8C22⋊C411yescharacteristic
108C22×C8C22⋊C411yescharacteristic
109C22×C8C22⋊C411yescharacteristic
110C4⋊Q8—41no—
111C4⋊Q8—41no—
112#127—41no—
113#127—41no—
114C22⋊Q8—82no—
115#121—42no—
116#121—41no—
117#121—41no—
118#121—42no—
119#121—41no—
120#121—41no—
121#121—82no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42C22×C411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42#4111yescharacteristic
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151#102—42no—
152#102—42no—
153#102—42no—
154#102—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—41no—
160C4×C8—41no—
161#95—22no—
162#95—42no—
163#95—41no—
164#95—41no—
165#95SD1614yes—
166C24—41no—
167C24#12111yescharacteristic
168C2×Q8—42no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—81no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—82no—
186C2×D4—82no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12411yescharacteristic
211C22×C4#12711yescharacteristic
212C22×C4C2×SD1612yes—
213C22×C4#13511yescharacteristic
214C22×C4#13511yescharacteristic
215C22×C4#13512yes—
216C22×C4AGL(1,ℤ/8)11yescharacteristic
217C22×C4#13711yescharacteristic
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—81no—
237C2×C8—81no—
238C2×C8—81no—
239C2×C8—81no—
240C2×C8—81no—
241C2×C8—81no—
242C2×C8—81no—
243C2×C8—81no—
244C2×C8—81no—
245C2×C8—81no—
246C2×C8#9912yes—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C4⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—81no—
262C22⋊C4—81no—
263C22⋊C4—81no—
264C22⋊C4—81no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—41no—
270C42—41no—
271C42—81no—
272C42—81no—
273C42—81no—
274C42—81no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23#39011yescharacteristic
287C23#46311yescharacteristic
288C23#46411yescharacteristic
289C23#46512yes—
290C23#47212yes—
291Q8—162no—
292D4—81no—
293D4—81no—
294D4—81no—
295D4—81no—
296D4—162no—
297D4—162no—
298D4—162no—
299D4—162no—
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—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4C2×#9911yescharacteristic
339C2×C4#41011yescharacteristic
340C2×C4#41611yescharacteristic
341C2×C4#41711yescharacteristic
342C2×C4#42111yescharacteristic
343C2×C4#42111yescharacteristic
344C2×C4C4×SD1612yes—
345C2×C4#44412yes—
346C2×C4#46311yescharacteristic
347C2×C4#46411yescharacteristic
348C2×C4#49211yescharacteristic
349C2×C4#49511yescharacteristic
350C8—22no—
351C8—41no—
352C8—41no—
353C8—41no—
354C8—82no—
355C8—161no—
356C8—161no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—21no—
364C22—21no—
365C22—161no—
366C22—161no—
367C22—161no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22—161no—
372C22#167712yes—
373C22#167812yes—
374C22#196311yescharacteristic
375C22#196411yescharacteristic
376C22#196811yescharacteristic
377C22#197511yescharacteristic
378C22#197611yescharacteristic
379C22#197811yescharacteristic
380C22#198311yescharacteristic
381C4—21no—
382C4—21no—
383C4—22no—
384C4—22no—
385C4—21no—
386C4—21no—
387C4—41no—
388C4—84no—
389C4—161no—
390C4—161no—
391C4—161no—
392C4#151211yescharacteristic
393C4#151311yescharacteristic
394C4#151611yescharacteristic
395C4#152111yescharacteristic
396C2—22no—
397C2—22no—
398C2—321no—
399C2#971611yescharacteristic
400C2#973511yescharacteristic
401C2#975411yescharacteristic
402C2#976411yescharacteristic
403C2#978111yescharacteristic
404C2#981811yescharacteristic
405C2#984611yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10797C2semidirect
3. #9908C2semidirect
5. #8595C2semidirect
6. #7667C2semidirect
8. C2×#486C4semidirect
12. #1990C4semidirect
13. #1957C4semidirect
18. #1834C4semidirect

Maximal subgroups

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

subgroupindex
#330422
#107972
#99082
#97122
#85952
#76672
#76642