Finite Group Databasebeta

← #102301

Subgroups — #102301

Order 512.

585 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 114 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
2C8×#95C211yesmaximal, characteristic, complemented
3#9712C211yesmaximal, characteristic
4#9711C211yesmaximal, characteristic, complemented
5#8596C211yesmaximal, characteristic, complemented
6#7665C211yesmaximal, characteristic, complemented
7#7664C211yesmaximal, characteristic
8C2×#486C411yescharacteristic, complemented
9C22×C4×C8—21no—
10C22×C4×C8C2211yescharacteristic
11#2857C2211yescharacteristic
12#2856C2211yescharacteristic
13#1990C411yescharacteristic, complemented
14#1957C411yescharacteristic, complemented
15#1934—21no—
16#1934—21no—
17#1893—21no—
18#1893—21no—
19#1834C411yescharacteristic, complemented
20#1797C2211yescharacteristic
21#1797C2211yescharacteristic
22C2×#336—21no—
23C2×#336C2211yescharacteristic
24C2×#335—21no—
25C2×#328—22no—
26C2×#327—22no—
27C4×#95C2211yescharacteristic
28#1277—21no—
29#1271—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—24no—
52C2×C4×C8—42no—
53C2×C4×C8—42no—
54C2×C4×C8—42no—
55C2×C4×C8C2×C411yescharacteristic
56C2×C4×C8C2×C411yescharacteristic
57C2×C4×C8D411yescharacteristic
58C2×C4×C8D411yescharacteristic
59#391—21no—
60#390—21no—
61#390—21no—
62#390—21no—
63#389—21no—
64#389—21no—
65#384C2×C411yescharacteristic
66#384C2×C411yescharacteristic
67#380—21no—
68#380—21no—
69C2×#95—21no—
70C2×#95D412yes—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—24no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336D412yes—
80#335—42no—
81#328—44no—
82#327—44no—
83C22×Q8—21no—
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—41no—
88C23×C4—22no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4#4111yescharacteristic
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—22no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8C22⋊C411yescharacteristic
110C22×C8C22⋊C411yescharacteristic
111C22×C8C22⋊C411yescharacteristic
112C22×C8C22⋊C411yescharacteristic
113C4⋊Q8—41no—
114C4⋊Q8—41no—
115#127—41no—
116#127—41no—
117C22⋊Q8—42no—
118C22⋊Q8—42no—
119#121—42no—
120#121—41no—
121#121—41no—
122#121—42no—
123#121—41no—
124#121—41no—
125#121—82no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42—41no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42#4111yescharacteristic
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155#102—42no—
156#102—42no—
157#102—42no—
158#102—42no—
159C22⋊C8—42no—
160C22⋊C8—42no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—41no—
164C4×C8—41no—
165C4×C8—48no—
166#95—24no—
167#95—22no—
168#95—41no—
169#95—41no—
170#95D814yes—
171C24—41no—
172C24#12111yescharacteristic
173C2×Q8—42no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—81no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4—81no—
190C2×D4—82no—
191C2×D4—82no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12411yescharacteristic
216C22×C4#12711yescharacteristic
217C22×C4C2×D812yes—
218C22×C4#13511yescharacteristic
219C22×C4#13511yescharacteristic
220C22×C4#13512yes—
221C22×C4AGL(1,ℤ/8)11yescharacteristic
222C22×C4#13711yescharacteristic
223C2×C8—24no—
224C2×C8—24no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—24no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—81no—
244C2×C8—81no—
245C2×C8—81no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8—81no—
249C2×C8#9912yes—
250C2×C8#10412yes—
251C2×C8#10412yes—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C22⋊C4—82no—
266C22⋊C4—82no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—42no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23#39011yescharacteristic
293C23#46311yescharacteristic
294C23#46411yescharacteristic
295C23#46612yes—
296C23#47112yes—
297Q8—162no—
298D4—81no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—162no—
303D4—162no—
304D4—162no—
305D4—162no—
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—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—28no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4C2×#9911yescharacteristic
345C2×C4#41011yescharacteristic
346C2×C4#41611yescharacteristic
347C2×C4#41711yescharacteristic
348C2×C4C2×#10411yescharacteristic
349C2×C4C2×#10411yescharacteristic
350C2×C4C4×D812yes—
351C2×C4#44312yes—
352C2×C4#46311yescharacteristic
353C2×C4#46411yescharacteristic
354C2×C4#49311yescharacteristic
355C2×C4#49511yescharacteristic
356C8—22no—
357C8—22no—
358C8—22no—
359C8—44no—
360C8—41no—
361C8—161no—
362C8—161no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—21no—
370C22—21no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22—161no—
378C22#168112yes—
379C22#168212yes—
380C22#196211yescharacteristic
381C22#196411yescharacteristic
382C22#197011yescharacteristic
383C22#197411yescharacteristic
384C22#197611yescharacteristic
385C22#197711yescharacteristic
386C22#198311yescharacteristic
387C4—21no—
388C4—22no—
389C4—21no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—48no—
394C4—41no—
395C4—161no—
396C4—161no—
397C4—161no—
398C4#150011yescharacteristic
399C4#150111yescharacteristic
400C4#150411yescharacteristic
401C4#150911yescharacteristic
402C2—22no—
403C2—22no—
404C2—321no—
405C2#971311yescharacteristic
406C2#973811yescharacteristic
407C2#975111yescharacteristic
408C2#976111yescharacteristic
409C2#978311yescharacteristic
410C2#981311yescharacteristic
411C2#984411yescharacteristic
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. C8×#95C2semidirect
4. #9711C2semidirect
5. #8596C2semidirect
6. #7665C2semidirect
8. C2×#486C4semidirect
13. #1990C4semidirect
14. #1957C4semidirect
19. #1834C4semidirect

Maximal subgroups

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

subgroupindex
#330422
C8×#952
#97122
#97112
#85962
#76652
#76642