Finite Group Databasebeta

← #99671

Subgroups — #99671

Order 512.

507 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 102 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#23815C211yesmaximal, characteristic
2#23175C211yesmaximal, characteristic
3#9869C211yesmaximal, characteristic, complemented
4#9677C211yesmaximal, characteristic, complemented
5C4×#331C211yesmaximal, characteristic
6#7027C211yesmaximal, characteristic, complemented
7#7027C211yesmaximal, characteristic, complemented
8C2×#435—21no—
9C2×#435—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#2099—21no—
13#2071C411yescharacteristic
14#2071C411yescharacteristic
15#2041—21no—
16#2023C411yescharacteristic
17#2023C411yescharacteristic
18#1955—21no—
19#1955—21no—
20#1870—21no—
21#1869—21no—
22#1767C2211yescharacteristic
23#1767C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×#102—21no—
27C4×#102—21no—
28#1744C2211yescharacteristic
29C2×#331—21no—
30C2×Q8⋊C8—21no—
31C2×Q8⋊C8—21no—
32#1296—21no—
33#1279—21no—
34#1279—21no—
35C22×M16—21no—
36C23×C8—21no—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×#127D411yescharacteristic
39C2×C4×Q8—21no—
40C2×C4×D4—21no—
41C2×C4×D4—21no—
42C2×C4×D4D411yescharacteristic
43C2×C4×D4D411yescharacteristic
44#435—42no—
45#435—42no—
46#435—42no—
47#435—42no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D411yescharacteristic
51C2×C4⋊C8D411yescharacteristic
52C2×C4⋊C8D411yescharacteristic
53C2×C4⋊C8D411yescharacteristic
54C2×#102—21no—
55C2×#102—21no—
56C2×#102—21no—
57C2×#102—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×#97—21no—
65C2×#97—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69#391—21no—
70#390—21no—
71#390—21no—
72#390C2×C411yescharacteristic
73#390C2×C411yescharacteristic
74#389—21no—
75#389—21no—
76#384C2×C411yescharacteristic
77#384D411yescharacteristic
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C43C2311yescharacteristic
81#340—21no—
82#340—21no—
83#340—21no—
84#340—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336C2×C411yescharacteristic
91#336C2×C411yescharacteristic
92#331—24no—
93Q8⋊C8—42no—
94Q8⋊C8—42no—
95C22×Q8—21no—
96C22×D4—21no—
97C22×D4—21no—
98C22×D4C22⋊C411yescharacteristic
99C22×D4C22⋊C411yescharacteristic
100C23×C4—21no—
101C23×C4—21no—
102C2×M16—42no—
103C2×M16—42no—
104C2×M16—81no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—81no—
116C4⋊Q8—42no—
117#127—22no—
118#127—41no—
119C4×Q8—41no—
120C4×Q8—41no—
121C4×Q8—41no—
122C4×Q8—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C42—41no—
152C2×C42C22×C411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8D812yes—
170C4⋊C8SD1612yes—
171#102—42no—
172#102—42no—
173#102—42no—
174#102—42no—
175C22⋊C8—42no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—42no—
179C22⋊C8—81no—
180C22⋊C8—81no—
181#97—41no—
182#97—41no—
183#97—41no—
184#97—41no—
185C4×C8—41no—
186C4×C8—41no—
187C4×C8—41no—
188C4×C8—41no—
189C4×C8—41no—
190C4×C8—41no—
191#95—41no—
192#95—41no—
193C24—21no—
194C24—21no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—81no—
215C2×D4—81no—
216C2×D4#10412yes—
217C22×C4—21no—
218C22×C4—21no—
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×C4C2×C22⋊C411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12311yescharacteristic
240M16—84no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—84no—
262C2×C8—81no—
263C2×C8—81no—
264C2×C8—81no—
265C2×C8—81no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—41no—
298C42—41no—
299C42—41no—
300C42—81no—
301C42—81no—
302C42C2×D811yescharacteristic
303C42C2×SD1611yescharacteristic
304C42#13511yescharacteristic
305C42#13511yescharacteristic
306C42AGL(1,ℤ/8)11yescharacteristic
307C42AGL(1,ℤ/8)11yescharacteristic
308C42AGL(1,ℤ/8)11yescharacteristic
309C42#13711yescharacteristic
310C23—22no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#38611yescharacteristic
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329D4—42no—
330D4—42no—
331D4—81no—
332D4—82no—
333D4—82no—
334D4—82no—
335D4—82no—
336D4—81no—
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—21no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
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×C4C2×#10411yescharacteristic
371C2×C4#42111yescharacteristic
372C2×C4C4×D811yescharacteristic
373C2×C4C4×SD1611yescharacteristic
374C2×C4#44011yescharacteristic
375C2×C4#44211yescharacteristic
376C2×C4#44711yescharacteristic
377C2×C4#44911yescharacteristic
378C2×C4#45011yescharacteristic
379C2×C4#45211yescharacteristic
380C2×C4#45511yescharacteristic
381C2×C4#45511yescharacteristic
382C2×C4#45911yescharacteristic
383C2×C4#46111yescharacteristic
384C2×C4#46311yescharacteristic
385C2×C4#46311yescharacteristic
386C8—42no—
387C8—42no—
388C8—81no—
389C8—81no—
390C8—82no—
391C8—82no—
392C8—81no—
393C8—81no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#187111yescharacteristic
409C22#187711yescharacteristic
410C22#187911yescharacteristic
411C22#188111yescharacteristic
412C22#188111yescharacteristic
413C22#188511yescharacteristic
414C22#189111yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—82no—
427C4—81no—
428C4—81no—
429C4—161no—
430C4#163911yescharacteristic
431C4#164011yescharacteristic
432C4#164111yescharacteristic
433C4#164211yescharacteristic
434C2—82no—
435C2—161no—
436C2#898911yescharacteristic
437C2#899111yescharacteristic
438C2#900511yescharacteristic
439C2#902911yescharacteristic
440C2#904711yescharacteristic
441C2#904911yescharacteristic
442C2#905711yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC23#386
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9869C2semidirect
4. #9677C2semidirect
6. #7027C2semidirect
7. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#238152
#231752
#98692
#96772
C4×#3312
#70272