Finite Group Databasebeta

← #100598

Subgroups — #100598

Order 512.

427 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 98 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#29864C211yesmaximal, characteristic
2#22798C211yesmaximal, characteristic
3#9705C211yesmaximal, characteristic, complemented
4#9692C211yesmaximal, characteristic, complemented
5#7574C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic, complemented
7#7022C211yesmaximal, characteristic, complemented
8C2×#474—21no—
9C2×#433—21no—
10#2868C2211yescharacteristic
11#2293C2211yescharacteristic
12#2083—21no—
13#2083C412yes—
14#2071—21no—
15#2071C412yes—
16#1988—21no—
17#1955—21no—
18#1924—21no—
19#1923—21no—
20#1842C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1798—21no—
24#1771—21no—
25#1767C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C2×#332—21no—
28C2×#332—21no—
29C2×#332—21no—
30C2×#332—21no—
31#1271—21no—
32#1270—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4×D4D411yescharacteristic
37C2×#117—21no—
38#474—42no—
39#474—42no—
40#433—42no—
41#433—42no—
42C2×#107—22no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×C4⋊C8D411yescharacteristic
52C2×#102—22no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×#97—21no—
58C2×#97—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61#390C2×C411yescharacteristic
62#390D411yescharacteristic
63#389C2×C411yescharacteristic
64#389D411yescharacteristic
65#388—21no—
66#388—21no—
67#384—21no—
68#384—21no—
69#384—21no—
70#384C2×C412yes—
71C4×C4⋊C4D411yescharacteristic
72C4×C22⋊C4—21no—
73C43C2311yescharacteristic
74#340—22no—
75#340—22no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336C2×C412yes—
84#332—42no—
85#332—42no—
86#332—42no—
87#332—42no—
88C22×Q8—41no—
89C22×D4C22⋊C411yescharacteristic
90C23×C4—21no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—41no—
97C22×C8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—41no—
106C4⋊Q8—41no—
107C4⋊Q8—41no—
108C4⋊Q8—41no—
109C4⋊Q8—41no—
110C4×D4—21no—
111C4×D4—21no—
112C4×D4—21no—
113C4×D4—21no—
114#117—41no—
115#117—41no—
116#117—41no—
117#117—41no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C42—21no—
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142#107—44no—
143C4⋊C8—22no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8D812yes—
159#102—44no—
160C22⋊C8—42no—
161C22⋊C8—42no—
162C22⋊C8—81no—
163C22⋊C8—81no—
164#97—42no—
165#97—41no—
166#97—41no—
167C4×C8—42no—
168C4×C8—81no—
169#95—21no—
170#95—21no—
171#95—21no—
172C24—21no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×Q8—81no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4C4×D411yescharacteristic
196C22×C4#12011yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12311yescharacteristic
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—81no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42C2×D811yescharacteristic
260C42#13511yescharacteristic
261C42#13511yescharacteristic
262C42#13511yescharacteristic
263C42AGL(1,ℤ/8)11yescharacteristic
264C42AGL(1,ℤ/8)11yescharacteristic
265C42AGL(1,ℤ/8)11yescharacteristic
266C42AGL(1,ℤ/8)11yescharacteristic
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23#38611yescharacteristic
275Q8—161no—
276Q8—161no—
277Q8—161no—
278Q8—161no—
279D4—42no—
280D4—81no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4#41611yescharacteristic
311C2×C4#41811yescharacteristic
312C2×C4C4×D811yescharacteristic
313C2×C4#44211yescharacteristic
314C2×C4#44711yescharacteristic
315C2×C4#44911yescharacteristic
316C2×C4#44911yescharacteristic
317C2×C4#44911yescharacteristic
318C2×C4#45311yescharacteristic
319C2×C4#45411yescharacteristic
320C2×C4#45911yescharacteristic
321C2×C4#46311yescharacteristic
322C2×C4#47111yescharacteristic
323C2×C4#47311yescharacteristic
324C2×C4#48211yescharacteristic
325C2×C4#48211yescharacteristic
326C8—42no—
327C8—82no—
328C8—82no—
329C8—81no—
330C8—81no—
331C8—81no—
332C8—161no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22#187511yescharacteristic
341C22#187911yescharacteristic
342C22#188311yescharacteristic
343C22#190211yescharacteristic
344C22#190811yescharacteristic
345C22#190911yescharacteristic
346C22#191511yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—41no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—162no—
362C4#162111yescharacteristic
363C4#162411yescharacteristic
364C4#162511yescharacteristic
365C4#162811yescharacteristic
366C2—161no—
367C2#898011yescharacteristic
368C2#917211yescharacteristic
369C2#919611yescharacteristic
370C2#927111yescharacteristic
371C2#929111yescharacteristic
372C2#933911yescharacteristic
373C2#937711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9705C2semidirect
4. #9692C2semidirect
6. #7040C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#298642
#227982
#97052
#96922
#75742
#70402
#70222