Finite Group Databasebeta

← #103283

Subgroups — #103283

Order 512.

627 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 96 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#33147C211yesmaximal, characteristic
2#29715C211yesmaximal, characteristic
3#15193C211yesmaximal, characteristic
4#9885C211yesmaximal, characteristic
5#8707C211yesmaximal, characteristic
6#7575C211yesmaximal, characteristic
7#7039C211yesmaximal, characteristic
8C2×#451—21no—
9#2857C2211yescharacteristic
10C2×#391—21no—
11C2×#391C2211yescharacteristic
12C2×#389—21no—
13C2×C4×C4⋊C4C2211yescharacteristic
14#2072—21no—
15#2056—22no—
16#2016—21no—
17#1988—21no—
18#1948—21no—
19#1919—22no—
20#1872—21no—
21#1853—21no—
22#1824—21no—
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26#1445—22no—
27#1445—21no—
28#1444—21no—
29C4⋊#95C2211yescharacteristic
30#1294—22no—
31#1279—22no—
32#1272—22no—
33C23×Q8D411yescharacteristic
34C22×Q16—22no—
35C2×C4⋊Q8D411yescharacteristic
36C2×C22⋊Q8—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C42—21no—
43C22×C42C2311yescharacteristic
44#451—42no—
45#451—42no—
46C2×#106—22no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D412yes—
49C2×#103—22no—
50C2×#103—22no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C4×C8—21no—
55#391—21no—
56#391—21no—
57#391—21no—
58#391—21no—
59#391—21no—
60#391—44no—
61#389—21no—
62#389—21no—
63#389—44no—
64#384D411yescharacteristic
65#380—21no—
66#380—21no—
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95D412yes—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#336—21no—
84#336—21no—
85#336D412yes—
86C22×Q8—21no—
87C22×Q8—21no—
88C22×Q8—21no—
89C22×Q8—21no—
90C22×Q8—21no—
91C22×Q8—21no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×Q8SD1612yes—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C2×Q16—44no—
104C2×Q16—82no—
105C2×Q16—82no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—41no—
109C4⋊Q8—42no—
110C22⋊Q8—41no—
111C22⋊Q8—41no—
112C22⋊Q8—41no—
113C22⋊Q8—41no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—82no—
138C2×C4⋊C4Q1612yescomplemented
139C2×C22⋊C4—41no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—41no—
144C2×C42—41no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42#4111yescharacteristic
149#106—44no—
150C4⋊C8—42no—
151C4⋊C8—42no—
152#103—44no—
153#103—82no—
154C22⋊C8—22no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C4×C8—42no—
161#95—42no—
162#95—44no—
163#95—44no—
164#95—44no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172#95—41no—
173#95—41no—
174#95—41no—
175C24#12011yescharacteristic
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—44no—
179C2×Q8—44no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—81no—
191C2×Q8—81no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
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—42no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4#12011yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12711yescharacteristic
239C22×C4C2×SD1611yescharacteristic
240C22×C4C2×Q1611yescharacteristic
241C22×C4#13511yescharacteristic
242C22×C4#13511yescharacteristic
243C22×C4AGL(1,ℤ/8)11yescharacteristic
244C22×C4#13711yescharacteristic
245C22×C4#13711yescharacteristic
246C22×C4#13711yescharacteristic
247Q16—88no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—81no—
256C2×C8—81no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—88no—
276C4⋊C4—81no—
277C4⋊C4—81no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C42—22no—
281C42—22no—
282C42—41no—
283C42—41no—
284C42—82no—
285C42—82no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23#39211yescharacteristic
291C23#44811yescharacteristic
292C23#44911yescharacteristic
293C23#45111yescharacteristic
294C23#45211yescharacteristic
295C23#45511yescharacteristic
296C23#45611yescharacteristic
297Q8—44no—
298Q8—82no—
299Q8—82no—
300Q8—82no—
301Q8—82no—
302Q8—82no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4#45011yescharacteristic
363C2×C4#45111yescharacteristic
364C2×C4#45211yescharacteristic
365C2×C4#45211yescharacteristic
366C2×C4#45312yes—
367C2×C4#45812yes—
368C2×C4#46511yescharacteristic
369C2×C4#46711yescharacteristic
370C2×C4#46811yescharacteristic
371C2×C4#47011yescharacteristic
372C8—82no—
373C8—82no—
374C8—82no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22#164011yescharacteristic
388C22#164511yescharacteristic
389C22#164811yescharacteristic
390C22#165311yescharacteristic
391C22#200011yescharacteristic
392C22#200111yescharacteristic
393C22#200211yescharacteristic
394C22#200711yescharacteristic
395C22#200911yescharacteristic
396C22#201011yescharacteristic
397C22#202211yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—84no—
407C4—82no—
408C4—82no—
409C4—82no—
410C4—161no—
411C4—161no—
412C4—161no—
413C4#161312yes—
414C4#161812yes—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2#992011yescharacteristic
420C2#993711yescharacteristic
421C2#999911yescharacteristic
422C2#1000811yescharacteristic
423C2#1023411yescharacteristic
424C2#1024811yescharacteristic
425C2#1032511yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
138. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#331472
#297152
#151932
#98852
#87072
#75752
#70392