Finite Group Databasebeta

← #224298

Subgroups — #224298

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2#22629C212yesmaximal
3#9882C214yesmaximal
4#9879C212yesmaximal
5#8558C212yesmaximal
6#8558C211yesmaximal, characteristic
7#8153C212yesmaximal
8C4×#324C211yesmaximal, characteristic
9C22×C4⋊Q8C411yescharacteristic
10#2369—24no—
11#2369—22no—
12#2369—22no—
13C2×#391C412yes—
14C2×#391C2212yes—
15C2×#389C412yes—
16C2×#389C411yescharacteristic
17C2×#389C2211yescharacteristic
18C2×C4×C4⋊C4C411yescharacteristic
19C2×C4×C4⋊C4C411yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#1939—22no—
24#1917—24no—
25#1797C2212yes—
26#1797C2214yes—
27#1770—22no—
28#1749C2214yes—
29C4×C22⋊C8C2212yes—
30C2×#324C2211yescharacteristic
31C2×#324C2212yes—
32C2×#324C2211yescharacteristic
33#1280C2214yes—
34#1280C2212yes—
35#1280C2214yes—
36#1280C2212yes—
37C23×Q8C2×C412yes—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—24no—
41C2×C4⋊Q8—24no—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C42C2×C412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C412yes—
55C22×C42C2311yescharacteristic
56C22×C42C2312yes—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—24no—
59C2×C22⋊C8C2312yes—
60C2×C22⋊C8C2314yes—
61C2×C22⋊C8C2312yes—
62C2×#97—24no—
63C2×C4×C8—22no—
64#391—24no—
65#391—24no—
66#391—24no—
67#391—24no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—24no—
73#389—24no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D414yes—
85C4×C4⋊C4D412yes—
86C4×C4⋊C4D412yes—
87C2×#95C2×C412yes—
88C2×#95C2×C412yes—
89C2×#95C2×C412yes—
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C412yes—
92C2×#95C2312yes—
93C2×#95C2311yescharacteristic
94#336—22no—
95#336—24no—
96#336—22no—
97#336—24no—
98#324—22no—
99#324—24no—
100#324Q814yescomplemented
101C22×Q8—24no—
102C22×Q8—24no—
103C22×Q8—24no—
104C22×Q8—82no—
105C22×Q8—82no—
106C23×C4C22×C412yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C22×C411yescharacteristic
109C23×C4C22×C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C2411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—24no—
120C4⋊Q8—216no—
121C4⋊Q8—216no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4M1614yes—
149C2×C4⋊C4C2×Q812yes—
150C2×C4⋊C4#4112yes—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—24no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42—82no—
165C2×C42—82no—
166C2×C42M1614yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D414yes—
170C2×C42C2×D414yes—
171C4⋊C8—44no—
172C4⋊C8—84no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—24no—
176C22⋊C8—22no—
177C22⋊C8—24no—
178C22⋊C8C2×Q814yes—
179#97—84no—
180C4×C8—44no—
181#95—24no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—24no—
188#95—22no—
189#95C22⋊C414yes—
190#95C22⋊C414yes—
191C24C23×C411yescharacteristic
192C2×Q8—216no—
193C2×Q8—84no—
194C2×Q8—84no—
195C2×Q8—88no—
196C2×Q8—88no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C414yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C4×Q812yes—
246C22×C4C4×Q812yes—
247C22×C4C2×M1612yes—
248C22×C4C2×M1612yes—
249C22×C4C2×M1612yes—
250C22×C4C2×M1612yes—
251C22×C4C2×M1614yes—
252C22×C4#13112yes—
253C22×C4C22×D412yes—
254C22×C4C22×Q811yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C42−1+412yes—
257C2×C8—24no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—44no—
266C4⋊C4—216no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4#10114yes—
281C42—24no—
282C42—24no—
283C42—24no—
284C42—44no—
285C42—44no—
286C42—42no—
287C42—44no—
288C42—44no—
289C42—42no—
290C42—42no—
291C42—82no—
292C42—82no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23C22×C22⋊C411yescharacteristic
298C23C2×C4×Q811yescharacteristic
299C23#51911yescharacteristic
300C23C22×M1611yescharacteristic
301C23C22×M1611yescharacteristic
302C23C2×#13111yescharacteristic
303C23#56811yescharacteristic
304Q8—816no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4#40712yes—
363C2×C4#40712yes—
364C2×C4#41012yes—
365C2×C4C2×#10114yes—
366C2×C4C2×#10112yes—
367C2×C4#44614yes—
368C2×C4#54812yes—
369C2×C4D4×Q812yes—
370C8—44no—
371C8—82no—
372C8—84no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22Q8×C22⋊C411yescharacteristic
386C22C2×#40711yescharacteristic
387C22#288011yescharacteristic
388C22C2×#41011yescharacteristic
389C22#288311yescharacteristic
390C22C22×#10111yescharacteristic
391C22#288811yescharacteristic
392C22C2×#44611yescharacteristic
393C22Q8×M1611yescharacteristic
394C22#298311yescharacteristic
395C22#298411yescharacteristic
396C4—24no—
397C4—22no—
398C4—24no—
399C4—22no—
400C4—22no—
401C4—42no—
402C4—42no—
403C4—44no—
404C4—44no—
405C4—42no—
406C4—42no—
407C4—82no—
408C4—81no—
409C4#146714yes—
410C2—21no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2C2×#146711yescharacteristic
415C2#1736911yescharacteristic
416C2#1741811yescharacteristic
417C2#1742611yescharacteristic
418C2#2251011yescharacteristic
419C2#2251311yescharacteristic
420C2Q8×#10111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
100. #324Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
#226292
#98822
#98792
#85582
#81532
C4×#3242