Finite Group Databasebeta

← #190253

Subgroups — #190253

Order 512.

913 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34128C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×#1461C211yesmaximal, characteristic
4#8769C212yesmaximal, complemented
5#8768C212yesmaximal, complemented
6#8767C212yesmaximal, complemented
7#8766C212yesmaximal, complemented
8#8092C212yesmaximal
9#7958C212yesmaximal
10C22×C4×Q8C2211yescharacteristic
11#2920C2212yes—
12C2×#431C2212yes—
13#2356C412yes—
14#2356C412yes—
15#2356C2211yescharacteristic
16#2356C2211yescharacteristic
17#2351C2211yescharacteristic
18#2351C2211yescharacteristic
19#2344—21no—
20#2344—22no—
21Q8×C22⋊C4—21no—
22#2295—21no—
23#2295—22no—
24#2283C412yes—
25#2283C2211yescharacteristic
26C42×Q8—21no—
27C4×#117C412yes—
28C4×#117C2211yescharacteristic
29#1821—21no—
30#1801C2212yes—
31#1801C2212yes—
32#1784—21no—
33C4×#101—21no—
34C4×#101—21no—
35#1461—22no—
36#1461C2212yes—
37#1461C2212yes—
38C2×#324—22no—
39#1303C2214yes—
40#1303C2214yes—
41#1286C2214yes—
42#1286C2214yes—
43C23×Q8D411yescharacteristic
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8C2311yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×#117—22no—
57C2×#117—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C42C2×C412yes—
61C22×C42C2311yescharacteristic
62#432—22no—
63#432—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431C2314yes—
70#423C2314yes—
71C2×C4⋊C8—22no—
72C2×#101—22no—
73C2×#101—22no—
74#407—22no—
75C2×C22⋊C8—22no—
76C4×M16—22no—
77C2×C4×C8—22no—
78#391—22no—
79#391—22no—
80#391—21no—
81#391—21no—
82#389—21no—
83#389—21no—
84#389—22no—
85#389—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#383C2×C412yes—
93#383C2×C412yes—
94#383C2×C412yes—
95#382C2×C412yes—
96#382C2×C412yes—
97#382C2311yescharacteristic
98#382C2311yescharacteristic
99#381—22no—
100#381—21no—
101#380—22no—
102#380—21no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2311yescharacteristic
114C4×C4⋊C4C2311yescharacteristic
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4C2×C411yescharacteristic
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4C2×C411yescharacteristic
120#376—22no—
121#376—21no—
122#376C2×C411yescharacteristic
123#376C2×C411yescharacteristic
124#376C2×C412yes—
125#376C2×C412yes—
126C43—22no—
127C43—21no—
128#343—22no—
129#343—22no—
130#324—24no—
131#324—24no—
132C22×Q8—24no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8C22⋊C412yes—
137C22×Q8C2×D411yescharacteristic
138C22×Q8C2×D411yescharacteristic
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C22×C8—42no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—44no—
159C4×Q8—44no—
160#117—44no—
161#117—48no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42C22⋊C412yes—
200C2×C42C22⋊C412yes—
201C2×C42C22×C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—24no—
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—42no—
215#101—44no—
216#101—44no—
217C22⋊C8—24no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221#97—42no—
222C4×C8—24no—
223C4×C8—42no—
224C4×C8—42no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95C22×C411yescharacteristic
240#95C22×C411yescharacteristic
241#95C22×C412yes—
242#95C22×C412yes—
243#95C22×C411yescharacteristic
244#95C22×C411yescharacteristic
245C24C2×C22⋊C411yescharacteristic
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—42no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8#13514yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4#11711yescharacteristic
282C22×C4#11711yescharacteristic
283C22×C4#11711yescharacteristic
284C22×C4#11711yescharacteristic
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4#12312yes—
288C22×C4#12312yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—44no—
293M16—44no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—84no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C22⋊C4—24no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—42no—
324C42—44no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—44no—
329C42—44no—
330C42—44no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42#11712yes—
335C42#11712yes—
336C42#12312yes—
337C42#12312yes—
338C42#12312yes—
339C42#12312yes—
340C42C2×#4112yes—
341C42C2×#4112yes—
342C23—22no—
343C23—21no—
344C23—21no—
345C23—41no—
346C23#38111yescharacteristic
347C23#38111yescharacteristic
348C23C22×C22⋊C411yescharacteristic
349Q8—28no—
350Q8—48no—
351Q8—44no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4#38111yescharacteristic
398C2×C4#38112yes—
399C2×C4#38112yes—
400C2×C4#38111yescharacteristic
401C2×C4#41614yes—
402C2×C4C2×#11711yescharacteristic
403C2×C4C2×#11711yescharacteristic
404C2×C4C2×#12312yes—
405C2×C4C2×#12312yes—
406C2×C4C2×#13512yes—
407C2×C4#57512yes—
408C8—84no—
409C8—84no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—42no—
418C22—42no—
419C22#182712yes—
420C22C2×#38111yescharacteristic
421C22C2×#41611yescharacteristic
422C22#289611yescharacteristic
423C4—22no—
424C4—24no—
425C4—22no—
426C4—22no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—81no—
436C4—82no—
437C4—82no—
438C4—81no—
439C4#183012yes—
440C4#309114yes—
441C2—21no—
442C2—21no—
443C2—22no—
444C2—41no—
445C2#2069911yescharacteristic
446C2C2×#182711yescharacteristic
447C2C2×#183011yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#381
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8769C2semidirect
5. #8768C2semidirect
6. #8767C2semidirect
7. #8766C2semidirect

Maximal subgroups

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

subgroupindex
#341282
#207272
C2×#14612
#87692
#87682
#87672
#87662
#80922
#79582