Finite Group Databasebeta

← #185233

Subgroups — #185233

Order 512.

3218 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3499C211yesmaximal, characteristic
2#21851C212yesmaximal, complemented
3C2×#1558C211yesmaximal, characteristic, complemented
4C2×#1551C212yesmaximal, complemented
5C22×#333C211yesmaximal, characteristic, complemented
6#8344C214yesmaximal, complemented
7#8342C214yesmaximal, complemented
8C22×2−1+4C412yes—
9#3499—28no—
10#3499—24no—
11#3499—24no—
12C2×D4×Q8—22no—
13C2×#547—24no—
14C2×#547—22no—
15C2×#536C412yes—
16C2×#536C414yes—
17C2×#532C2212yes—
18C2×#532C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#432C2212yes—
23C22×C4⋊C8C2212yes—
24#1985—22no—
25#1955—22no—
26#1869—24no—
27#1797C2218yes—
28#1558—24no—
29#1551—28no—
30C2×#333—24no—
31C2×#333C2214yes—
32C2×#333C2214yes—
33C2×#333C2214yes—
34C2×#331C2214yes—
35C2×2−1+4—28no—
36C2×2−1+4—28no—
37C2×2−1+4—416no—
38C22×#41—22no—
39C22×#41—24no—
40C22×#41C2×C414yes—
41C22×#41C2×C412yes—
42C22×#41C2×C411yescharacteristic
43C23×Q8—22no—
44C23×Q8C2×C414yes—
45C23×Q8C2×C412yes—
46C23×C8—22no—
47D4×Q8—216no—
48D4×Q8—216no—
49#547—216no—
50#547—216no—
51#547—232no—
52#547—232no—
53#536—28no—
54#536—28no—
55#536—216no—
56#536—216no—
57#532—24no—
58#532—24no—
59#532—28no—
60#532—22no—
61#532—22no—
62#532D418yes—
63C2×C4⋊Q8—28no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—24no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—24no—
69C2×C4⋊Q8C2×C414yes—
70C2×C4⋊Q8C2×C412yes—
71C2×C4⋊Q8D414yes—
72C2×C4⋊Q8D414yes—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2312yes—
75C2×#127C2311yescharacteristic
76C2×#124C2×C414yes—
77C2×#124C2×C412yes—
78C2×#123—24no—
79C2×#123—28no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—28no—
82C2×#121—24no—
83C2×#121—22no—
84C4×#41—24no—
85C4×#41—22no—
86C4×#41—22no—
87C2×C4×Q8D412yes—
88C2×C4×D4—24no—
89C2×C4×D4—22no—
90C2×C4×D4D414yes—
91C2×C4×D4D412yes—
92C2×#117C2×C411yescharacteristic
93C2×#117C2×C412yes—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—22no—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98#432—24no—
99#432—24no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—24no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8C2314yes—
104C2×C4⋊C8C2314yes—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—28no—
107C2×#97—22no—
108#336—28no—
109#333—28no—
110#333—28no—
111#333—28no—
112#331—28no—
1132−1+4—464no—
1142−1+4—464no—
115C2×#41—216no—
116C2×#41—232no—
117C2×#41—28no—
118C2×#41—28no—
119C2×#41—28no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—24no—
123C2×#41—48no—
124C2×#41—48no—
125C2×#41—48no—
126C2×#41—416no—
127C2×#41—416no—
128C2×#41—416no—
129C2×#41C22⋊C418yes—
130C22×Q8—216no—
131C22×Q8—28no—
132C22×Q8—28no—
133C22×Q8—22no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—48no—
137C22×Q8—48no—
138C22×Q8—48no—
139C22×Q8—416no—
140C22×Q8C22⋊C418yes—
141C22×Q8C22⋊C411yescharacteristic
142C22×Q8C22×C414yes—
143C22×Q8C22×C412yes—
144C22×D4—24no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4C22⋊C411yescharacteristic
148C22×D4C22⋊C412yes—
149C22×D4C22×C412yes—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D414yes—
157C23×C4C2×D412yes—
158C22×C8—24no—
159C22×C8—24no—
160C22×C8—44no—
161C22×C8—84no—
162C4⋊Q8—216no—
163C4⋊Q8—216no—
164C4⋊Q8—28no—
165C4⋊Q8—28no—
166C4⋊Q8—24no—
167C4⋊Q8—28no—
168C4⋊Q8—28no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—24no—
172C4⋊Q8C22⋊C418yes—
173C4⋊Q8C2×D414yes—
174C4⋊Q8C2×D414yes—
175C4⋊Q8C2×D414yes—
176#127—21no—
177#127—21no—
178#127C2×D414yes—
179#124—28no—
180#124—28no—
181#124—24no—
182#124C22⋊C418yes—
183#123—232no—
184#123—264no—
185C22⋊Q8—232no—
186C22⋊Q8—264no—
187#121—216no—
188#121—216no—
189#121—28no—
190#121—28no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×D4—216no—
194C4×D4—216no—
195C4×D4—28no—
196C4×D4—28no—
197C4×D4—28no—
198C4×D4—28no—
199C4×D4—24no—
200C4×D4—24no—
201#117—22no—
202#117—22no—
203#117—24no—
204#117—24no—
205C2×C4⋊C4—216no—
206C2×C4⋊C4—216no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C414yes—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C414yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42C22×C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D414yes—
225C2×C42C2×D414yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2411yescharacteristic
230C4⋊C8—24no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—44no—
235C22⋊C8—44no—
236C22⋊C8—88no—
237#97—44no—
238C24—22no—
239C24C2×C22⋊C412yes—
240C24C2×C22⋊C411yescharacteristic
241#41—216no—
242#41—216no—
243#41—432no—
244#41—432no—
245#41—416no—
246#41—416no—
247#41—464no—
248#41—4128no—
249C2×Q8—216no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—28no—
254C2×Q8—24no—
255C2×Q8—48no—
256C2×Q8—48no—
257C2×Q8—432no—
258C2×Q8—432no—
259C2×Q8—416no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—416no—
264C2×Q8—464no—
265C2×Q8—48no—
266C2×Q8C2×C22⋊C418yes—
267C2×Q8C2×C22⋊C414yes—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—216no—
272C2×D4—28no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—44no—
278C2×D4—48no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4#10018yes—
284C2×D4C2×C22⋊C414yes—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—216no—
293C22×C4—216no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4C2×C22⋊C414yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4#12014yes—
315C22×C4#12014yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D412yes—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—48no—
322C2×C8—44no—
323C2×C8—88no—
324C4⋊C4—232no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—28no—
329C4⋊C4—216no—
330C4⋊C4—28no—
331C4⋊C4—24no—
332C22⋊C4—232no—
333C22⋊C4—24no—
334C22⋊C4—216no—
335C22⋊C4—28no—
336C42—22no—
337C42—22no—
338C42—24no—
339C42—24no—
340C42C2×C22⋊C412yes—
341C42C2×C22⋊C412yes—
342C42#12014yes—
343C42#12014yes—
344C42C22×D411yescharacteristic
345C42C22×D411yescharacteristic
346C42C22×D412yes—
347C23—24no—
348C23—22no—
349C23—28no—
350C23—22no—
351C23—24no—
352C23—44no—
353C23—48no—
354C23#37912yes—
355C23C2×#10014yes—
356C23C2×#10014yes—
357C23C22×C22⋊C411yescharacteristic
358Q8—28no—
359Q8—44no—
360Q8—48no—
361Q8—44no—
362Q8—48no—
363Q8—416no—
364Q8—432no—
365D4—28no—
366D4—216no—
367D4—44no—
368D4—44no—
369D4—432no—
370D4—416no—
371D4—44no—
372D4—44no—
373D4—48no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—24no—
388C2×C4—28no—
389C2×C4—24no—
390C2×C4—416no—
391C2×C4—432no—
392C2×C4—432no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—416no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4#37914yes—
401C2×C4#37912yes—
402C2×C4C2×#10014yes—
403C2×C4C22×C22⋊C411yescharacteristic
404C2×C4C22×C22⋊C411yescharacteristic
405C2×C4C2×#12012yes—
406C2×C4C2×#12012yes—
407C2×C4#57714yes—
408C2×C4#57814yes—
409C8—84no—
410C8—84no—
411C22—28no—
412C22—24no—
413C22—24no—
414C22—28no—
415C22—44no—
416C22—44no—
417C22—44no—
418C22—48no—
419C22—88no—
420C22#178314yes—
421C22C2×#37911yescharacteristic
422C22C22×#10012yes—
423C22#289814yes—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—24no—
428C4—24no—
429C4—44no—
430C4—48no—
431C4—44no—
432C4#301012yes—
433C4#301112yes—
434C4#301314yes—
435C2—24no—
436C2—44no—
437C2—84no—
438C2#1777914yes—
439C2C2×#178311yescharacteristic
440C2#2060312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3499C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21851C2semidirect
3. C2×#1558C2semidirect
4. C2×#1551C2semidirect
5. C22×#333C2semidirect
6. #8344C2semidirect
7. #8342C2semidirect

Maximal subgroups

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

subgroupindex
C2×#34992
#218512
C2×#15582
C2×#15512
C22×#3332
#83442
#83422