Finite Group Databasebeta

← #138321

Subgroups — #138321

Order 512.

660 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#1801C211yesmaximal, characteristic
2#14881C211yesmaximal, characteristic
3C2×#1354C212yesmaximal
4#8303C211yesmaximal, characteristic
5#7251C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C2×#423—21no—
8C22×#101—21no—
9C2×#395C411yescharacteristic
10C2×#395C2211yescharacteristic
11C2×#391C411yescharacteristic
12#1817—21no—
13#1801—24no—
14C2×#343C2212yes—
15C2×#324—21no—
16C2×#324C2212yes—
17#1446—21no—
18#1444—21no—
19#1441C411yescharacteristic
20#1441C2211yescharacteristic
21#1436—21no—
22#1436—21no—
23#1434C411yescharacteristic
24#1354—24no—
25C23×Q8C2×C411yescharacteristic
26C23×Q8D411yescharacteristic
27C22×M16—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×C4×Q8—42no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4C2311yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42#423—24no—
43#423—42no—
44C2×C4⋊C8—41no—
45C2×C4⋊C8—41no—
46C2×#101—24no—
47C2×#101—42no—
48C2×#101—42no—
49C2×C22⋊C8—22no—
50#395—22no—
51#395—22no—
52#395—42no—
53#391—41no—
54#391—41no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95C2×C411yescharacteristic
66C2×#95C2×C411yescharacteristic
67#343—22no—
68#343D414yes—
69#336—41no—
70#336—41no—
71#324—22no—
72#324—22no—
73#324—42no—
74C22×Q8—21no—
75C22×Q8—21no—
76C22×Q8—21no—
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×Q8—42no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8—81no—
87C22×Q8SD1612yes—
88C22×Q8Q1612yes—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—24no—
101C2×M16—42no—
102C2×M16—42no—
103C2×M16—42no—
104C2×M16—42no—
105C2×M16—42no—
106C22×C8—42no—
107C22×C8—42no—
108C4×Q8—44no—
109C4×Q8—44no—
110C4×Q8—44no—
111C4×Q8—44no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—81no—
126C2×C4⋊C4—81no—
127C2×C4⋊C4—81no—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4#4112yes—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—22no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—41no—
142C2×C42—41no—
143C4⋊C8—44no—
144C4⋊C8—44no—
145#101—48no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148#95—24no—
149#95—22no—
150#95—41no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—81no—
162#95—81no—
163C24C2×C22⋊C411yescharacteristic
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—82no—
183C2×Q8—84no—
184C2×Q8—84no—
185C2×Q8—82no—
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—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
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—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4#9911yescharacteristic
247C22×C4#9911yescharacteristic
248C22×C4#9912yes—
249C22×C4#10312yes—
250C22×C4#10312yes—
251C22×C4C4×D412yes—
252C22×C4#12011yescharacteristic
253C22×C4#12112yes—
254C22×C4#12311yescharacteristic
255C22×C4C2×SD1611yescharacteristic
256C22×C4C2×Q1611yescharacteristic
257C22×C4AGL(1,ℤ/8)11yescharacteristic
258C22×C4#13711yescharacteristic
259M16—44no—
260M16—44no—
261M16—82no—
262M16—82no—
263C2×C8—44no—
264C2×C8—82no—
265C2×C8—82no—
266C2×C8—82no—
267C2×C8—82no—
268C2×C8—82no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—84no—
274C4⋊C4—84no—
275C4⋊C4—82no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23#38611yescharacteristic
289C23C2×#9911yescharacteristic
290C23C2×#9911yescharacteristic
291C23C2×#10311yescharacteristic
292C23#41711yescharacteristic
293C23#42111yescharacteristic
294C23#42111yescharacteristic
295Q8—44no—
296Q8—44no—
297Q8—44no—
298Q8—44no—
299Q8—84no—
300Q8—84no—
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—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4#35612yes—
378C2×C4#45011yescharacteristic
379C2×C4#45111yescharacteristic
380C2×C4#48311yescharacteristic
381C2×C4#48411yescharacteristic
382C8—84no—
383C8—162no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22#151311yescharacteristic
397C22#151311yescharacteristic
398C22#152011yescharacteristic
399C22#152011yescharacteristic
400C22#187211yescharacteristic
401C22#189711yescharacteristic
402C22#189811yescharacteristic
403C22#212011yescharacteristic
404C22#212211yescharacteristic
405C22C2×#35611yescharacteristic
406C22#213311yescharacteristic
407C4—22no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—161no—
422C2—21no—
423C2—21no—
424C2—21no—
425C2—21no—
426C2#915111yescharacteristic
427C2#1183111yescharacteristic
428C2#1183311yescharacteristic
429C2#1190511yescharacteristic
430C2#1190811yescharacteristic
431C2#1308311yescharacteristic
432C2#1309411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#18012
#148812
C2×#13542
#83032
#72512