Finite Group Databasebeta

← #140926

Subgroups — #140926

Order 512.

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

normal subgroups only

This group has 674 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#15163C211yesmaximal, characteristic
2C2×#1346C212yesmaximal, complemented
3C2×#1345C212yesmaximal
4#10847C211yesmaximal, characteristic, complemented
5C4×#324C211yesmaximal, characteristic, complemented
6C2×#423—21no—
7C2×#398C2211yescharacteristic
8C2×#396—21no—
9C2×#394C2211yescharacteristic
10C2×#390C411yescharacteristic
11C2×#388—21no—
12C2×C4×C4⋊C4C2211yescharacteristic
13C4×C22⋊C8—21no—
14C2×#343—21no—
15C2×#343C2212yes—
16C2×#324C2212yes—
17#1445C411yescharacteristic
18#1439—21no—
19#1438C411yescharacteristic
20#1437—21no—
21C4⋊#95C411yescharacteristic
22#1346—24no—
23#1345—24no—
24#1280—21no—
25C23×D4C2×C411yescharacteristic
26C22×M16—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C4⋊C4C2311yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C42C2×C411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41#423—24no—
42#423—42no—
43C2×C4⋊C8—41no—
44C2×C4⋊C8—41no—
45C2×C22⋊C8—22no—
46C2×C4×C8—41no—
47#398—22no—
48#398—22no—
49#396—42no—
50#396—42no—
51#394—22no—
52#394—22no—
53#390—41no—
54#390—41no—
55#388—42no—
56#388—42no—
57C4×C4⋊C4—41no—
58C4×C4⋊C4—41no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95C2×C411yescharacteristic
64C2×#95C2×C411yescharacteristic
65C2×#95C2×C411yescharacteristic
66C2×#95D411yescharacteristic
67#343—22no—
68#343—22no—
69#343—42no—
70#336—41no—
71#324—22no—
72#324D414yes—
73C25—21no—
74C22×D4—22no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4—42no—
78C22×D4—81no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C2×M16—24no—
90C2×M16—42no—
91C2×M16—42no—
92C2×M16—42no—
93C2×M16—42no—
94C2×M16—42no—
95C22×C8—42no—
96C22×C8—42no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—81no—
113C2×C4⋊C4D812yes—
114C2×C4⋊C4SD1612yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4#4112yes—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—81no—
138C2×C22⋊C4—81no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—22no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—81no—
147C2×C42—81no—
148C2×C42D812yes—
149C2×C42SD1612yes—
150C4⋊C8—44no—
151C4⋊C8—44no—
152C22⋊C8—24no—
153C22⋊C8—42no—
154C4×C8—82no—
155#95—22no—
156#95—22no—
157#95—41no—
158#95—41no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95—42no—
163C24—22no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24—41no—
174C24—41no—
175C24—41no—
176C24—41no—
177C24—41no—
178C24C2×C22⋊C411yescharacteristic
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—82no—
182C2×D4—84no—
183C2×D4—82no—
184C2×D4—84no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4#9911yescharacteristic
233C22×C4#9911yescharacteristic
234C22×C4#10212yes—
235C22×C4#10212yes—
236C22×C4#10212yes—
237C22×C4#10212yes—
238C22×C4#10412yes—
239C22×C4C4×D412yes—
240C22×C4#12112yes—
241C22×C4#12411yescharacteristic
242C22×C4#12711yescharacteristic
243C22×C4C2×D811yescharacteristic
244C22×C4C2×D811yescharacteristic
245C22×C4C2×SD1611yescharacteristic
246C22×C4C2×SD1611yescharacteristic
247M16—44no—
248M16—44no—
249M16—82no—
250M16—82no—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—82no—
254C2×C8—82no—
255C2×C8—82no—
256C2×C8—82no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—82no—
262C4⋊C4—84no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C22⋊C4—48no—
268C22⋊C4—84no—
269C22⋊C4—84no—
270C22⋊C4—84no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—42no—
282C42—81no—
283C42—81no—
284C42—81no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23#39011yescharacteristic
325C23C2×#9911yescharacteristic
326C23#41011yescharacteristic
327C23C2×#10211yescharacteristic
328C23C2×#10211yescharacteristic
329C23C2×#10411yescharacteristic
330C23#42111yescharacteristic
331D4—84no—
332D4—84no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4#32712yes—
398C2×C4#32712yes—
399C2×C4#48611yescharacteristic
400C2×C4#48611yescharacteristic
401C2×C4#49211yescharacteristic
402C2×C4#49311yescharacteristic
403C8—84no—
404C8—84no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22C2×#32711yescharacteristic
433C22#150211yescharacteristic
434C22#151711yescharacteristic
435C22#151911yescharacteristic
436C22#195711yescharacteristic
437C22#196411yescharacteristic
438C22#198011yescharacteristic
439C22#212011yescharacteristic
440C22#212111yescharacteristic
441C22#213211yescharacteristic
442C22#213311yescharacteristic
443C4—22no—
444C4—22no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—161no—
456C4—161no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—162no—
462C2#968011yescharacteristic
463C2#1169711yescharacteristic
464C2#1169811yescharacteristic
465C2#1187911yescharacteristic
466C2#1188411yescharacteristic
467C2#1322211yescharacteristic
468C2#1322811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1346C2semidirect
4. #10847C2semidirect
5. C4×#324C2semidirect

Maximal subgroups

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

subgroupindex
#151632
C2×#13462
C2×#13452
#108472
C4×#3242