Finite Group Databasebeta

← #224455

Subgroups — #224455

Order 512.

689 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36556C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic
3#21345C211yesmaximal, characteristic
4#9893C212yesmaximal
5#9893C212yesmaximal
6#9890C211yesmaximal, characteristic
7#9890C211yesmaximal, characteristic
8#8582C212yesmaximal
9#8580C211yesmaximal, characteristic
10#8225C212yesmaximal
11C4×#330C211yesmaximal, characteristic
12#2864C2212yes—
13#2862C2211yescharacteristic
14#2588—22no—
15#2567C411yescharacteristic
16#2546C411yescharacteristic
17#2520—22no—
18#2511—21no—
19#2509C412yes—
20#2508C412yes—
21#2446—21no—
22#2369C411yescharacteristic
23#2365—21no—
24#2365C411yescharacteristic
25#2355—21no—
26C4×#125C2211yescharacteristic
27C4×#125C2212yes—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1917—22no—
34#1917—22no—
35#1767C2211yescharacteristic
36#1767C2211yescharacteristic
37#1767C2212yes—
38#1767C2212yes—
39#1745C2212yes—
40#1745C2212yes—
41C4×#97C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×#330C2211yescharacteristic
44C2×#330C2212yes—
45C2×#330C2211yescharacteristic
46#1295C2211yescharacteristic
47#1295C2211yescharacteristic
48#1295C2212yes—
49#1295C2212yes—
50#1295C2212yes—
51#1295C2212yes—
52#1295C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2311yescharacteristic
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×#97—21no—
66C2×#97—21no—
67C2×#97C2311yescharacteristic
68C2×#97C2311yescharacteristic
69C2×#97C2312yes—
70C2×#97C2312yes—
71C2×#97C2311yescharacteristic
72C2×#97C2311yescharacteristic
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#400—22no—
76#400—22no—
77#400—22no—
78#398—22no—
79#398—22no—
80#395—22no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391—21no—
88#391—22no—
89#389—22no—
90#389—21no—
91#389C2×C412yes—
92#389C2×C411yescharacteristic
93#389C2×C411yescharacteristic
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#387—22no—
100#387—21no—
101#387—22no—
102#387—22no—
103#387—21no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#384—21no—
111#382—22no—
112#382C2×C412yes—
113#382C2312yes—
114#382C2311yescharacteristic
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D411yescharacteristic
130C43C2311yescharacteristic
131C43C2312yes—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#330—22no—
141#330—24no—
142#330Q814yescomplemented
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8C22⋊C411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—44no—
155#125—21no—
156#125—21no—
157#125C2×Q812yes—
158#125#4112yes—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—44no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2411yescharacteristic
204C4⋊C8—42no—
205C4⋊C8—42no—
206C4⋊C8—44no—
207C4⋊C8—44no—
208#97—21no—
209#97—21no—
210#97—22no—
211#97—22no—
212#97—21no—
213#97—21no—
214#97—21no—
215#97—21no—
216#97—22no—
217#97—22no—
218#97—42no—
219#97—42no—
220#97C2×Q812yes—
221#97C2×Q812yes—
222C4×C8—42no—
223C4×C8—42no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—81no—
253C2×Q8—81no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4C4×Q812yes—
316C4⋊C4C4×Q812yes—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42—41no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—41no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—44no—
330C42—42no—
331C42—42no—
332C42—82no—
333C42C22×Q811yescharacteristic
334C42C2×#4111yescharacteristic
335C422−1+411yescharacteristic
336C422−1+411yescharacteristic
337C23C22×C22⋊C411yescharacteristic
338Q8—44no—
339Q8—84no—
340Q8—162no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4#41312yes—
387C2×C4#42112yes—
388C2×C4#42112yes—
389C2×C4C2×C4×Q811yescharacteristic
390C2×C4#51911yescharacteristic
391C2×C4#54811yescharacteristic
392C2×C4#54811yescharacteristic
393C2×C4D4×Q811yescharacteristic
394C2×C4D4×Q811yescharacteristic
395C8—42no—
396C8—42no—
397C8—42no—
398C8—42no—
399C8—82no—
400C8—82no—
401C22Q8×C22⋊C411yescharacteristic
402C22C2×#41311yescharacteristic
403C22#288911yescharacteristic
404C22C2×#42111yescharacteristic
405C22C2×#42111yescharacteristic
406C22#290111yescharacteristic
407C22#290111yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—42no—
420C4—42no—
421C4—41no—
422C4—42no—
423C4—81no—
424C4—81no—
425C4—82no—
426C4—82no—
427C4#153514yes—
428C2C2×#153511yescharacteristic
429C2#1768211yescharacteristic
430C2#1768311yescharacteristic
431C2#1769111yescharacteristic
432C2#2251711yescharacteristic
433C2#2252911yescharacteristic
434C2#2252911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
142. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#365562
#213452
#98932
#98902
#85822
#85802
#82252
C4×#3302