Finite Group Databasebeta

← #136142

Subgroups — #136142

Order 512.

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

normal subgroups only

This group has 511 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#22181C211yesmaximal, characteristic
2#12991C211yesmaximal, characteristic
3#12919C211yesmaximal, characteristic, complemented
4C2×#1347C211yesmaximal, characteristic
5#11328C211yesmaximal, characteristic, complemented
6#7258C211yesmaximal, characteristic, complemented
7#7183C211yesmaximal, characteristic, complemented
8C2×#466C2211yescharacteristic
9C22×#107C2211yescharacteristic
10C4×#121—21no—
11C2×#389—21no—
12C2×#366—21no—
13C2×#357—21no—
14C2×C22⋊C16C2211yescharacteristic
15C2×C22⋊C16C2211yescharacteristic
16#1925—21no—
17#1918—21no—
18#1893—21no—
19#1879—21no—
20#1862—21no—
21#1808C2211yescharacteristic
22C2×#340—21no—
23C2×#340C2211yescharacteristic
24C2×#336C2211yescharacteristic
25#1380—21no—
26#1380—21no—
27#1347C414yescomplemented
28C22×D8D411yescharacteristic
29C23×C8D411yescharacteristic
30C23×C8C2311yescharacteristic
31C2×#121—21no—
32C2×C4×D4—21no—
33C2×C4×D4—41no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—41no—
37C22×C42D411yescharacteristic
38C22×C16—21no—
39C22×C16—21no—
40#466—21no—
41#466—21no—
42#466C2×C412yes—
43#466Q812yes—
44C2×#107—21no—
45C2×#107—21no—
46C2×#107—42no—
47C2×#107D411yescharacteristic
48C2×#107D411yescharacteristic
49C2×#107D411yescharacteristic
50C2×#102—21no—
51C2×#102—21no—
52C2×#102—21no—
53#390—21no—
54#389—21no—
55#389—42no—
56#389—42no—
57#388—41no—
58#386—41no—
59#380—21no—
60C4×C4⋊C4—21no—
61C4×C22⋊C4—41no—
62C2×#95—41no—
63#366—42no—
64#357—42no—
65C22⋊C16—21no—
66C22⋊C16—21no—
67C22⋊C16—21no—
68C22⋊C16C2×C412yescomplemented
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—42no—
74#340C2×C412yescomplemented
75#336—21no—
76#336—21no—
77C22×D4—21no—
78C22×D4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4D811yescharacteristic
83C23×C4D811yescharacteristic
84C23×C4C2×D411yescharacteristic
85C2×D8—21no—
86C2×D8—41no—
87C2×D8—41no—
88C2×D8C4⋊C412yescomplemented
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8C22×C411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C22×C8C2×Q811yescharacteristic
100C22×C8#4111yescharacteristic
101C22×C8#4111yescharacteristic
102#121—22no—
103#121—22no—
104#121—41no—
105#121—41no—
106C4×D4—81no—
107C4×D4—81no—
108C4×D4—82no—
109C4×D4—82no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4—82no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C16—22no—
140C2×C16—41no—
141C2×C16—41no—
142C2×C16—41no—
143C2×C16—41no—
144C2×C16—41no—
145#107—21no—
146#107—22no—
147#107—22no—
148#107—41no—
149#107—41no—
150#107—41no—
151#107—41no—
152#107C4⋊C412yescomplemented
153#102—22no—
154#102—22no—
155#102—81no—
156#102—81no—
157#95—41no—
158#95—41no—
159#95—44no—
160#95—41no—
161C24—41no—
162C24C2×D811yescharacteristic
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4D1612yes—
209C22×C4C4×D411yescharacteristic
210C22×C4#12011yescharacteristic
211C22×C4C22⋊Q811yescharacteristic
212C22×C4#13511yescharacteristic
213C22×C4AGL(1,ℤ/8)11yescharacteristic
214C22×C4#13711yescharacteristic
215D8—42no—
216D8—81no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—81no—
230C2×C8C2×C4⋊C411yescharacteristic
231C2×C8C4×D411yescharacteristic
232C2×C8C22⋊Q811yescharacteristic
233C2×C8#12311yescharacteristic
234C4⋊C4—24no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—81no—
247C4⋊C4—82no—
248C4⋊C4—88no—
249C4⋊C4—81no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—81no—
256C22⋊C4—81no—
257C42—41no—
258C42—41no—
259C42—81no—
260C42—82no—
261C16—42no—
262C16—81no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—42no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23C4×D811yescharacteristic
275C23#44711yescharacteristic
276C23#47411yescharacteristic
277C23C2×D1611yescharacteristic
278C23#50811yescharacteristic
279C23#50911yescharacteristic
280C23#51011yescharacteristic
281D4—42no—
282D4—42no—
283D4—82no—
284D4—82no—
285D4—82no—
286D4—82no—
287D4—81no—
288D4—81no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—164no—
329C2×C4—161no—
330C2×C4—161no—
331C2×C4—161no—
332C2×C4—161no—
333C2×C4#38511yescharacteristic
334C2×C4#44011yescharacteristic
335C2×C4#45211yescharacteristic
336C2×C4#45311yescharacteristic
337C2×C4#45611yescharacteristic
338C2×C4B(4,𝔽2)11yescharacteristic
339C2×C4#45811yescharacteristic
340C2×C4#47811yescharacteristic
341C8—21no—
342C8—21no—
343C8—21no—
344C8—21no—
345C8—81no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—161no—
362C22—161no—
363C22—161no—
364C22—161no—
365C22#159511yescharacteristic
366C22#161011yescharacteristic
367C22#186211yescharacteristic
368C22C4×D1611yescharacteristic
369C22#217511yescharacteristic
370C22#218411yescharacteristic
371C22#218911yescharacteristic
372C22#219011yescharacteristic
373C22#219111yescharacteristic
374C22#222411yescharacteristic
375C22#222811yescharacteristic
376C4—21no—
377C4—21no—
378C4—42no—
379C4—84no—
380C4—81no—
381C4—82no—
382C4—161no—
383C4—162no—
384C4—161no—
385C4#160311yescharacteristic
386C4#161811yescharacteristic
387C4#185911yescharacteristic
388C4#186711yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—21no—
393C2—162no—
394C2#886611yescharacteristic
395C2#1210211yescharacteristic
396C2#1210711yescharacteristic
397C2#1228811yescharacteristic
398C2#1229311yescharacteristic
399C2#1293511yescharacteristic
400C2#1294311yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12919C2semidirect
5. #11328C2semidirect
6. #7258C2semidirect
7. #7183C2semidirect
27. #1347C4semidirect
68. C22⋊C16C2×C4semidirect
74. #340C2×C4semidirect
88. C2×D8C4⋊C4semidirect
152. #107C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#221812
#129912
#129192
C2×#13472
#113282
#72582
#71832