Finite Group Databasebeta

← #190293

Subgroups — #190293

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34197C211yesmaximal, characteristic
2#20637C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8771C212yesmaximal, complemented
5#8770C212yesmaximal, complemented
6#8767C212yesmaximal, complemented
7#8766C212yesmaximal, complemented
8#8093C212yesmaximal, complemented
9#7962C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11#2920C2212yes—
12C2×#423C2212yes—
13C2×#413—21no—
14#2357C412yes—
15#2357C412yes—
16#2357C2211yescharacteristic
17#2357C2211yescharacteristic
18#2351C2211yescharacteristic
19#2351C2211yescharacteristic
20#2346—21no—
21#2346—22no—
22#2341—21no—
23#2319—22no—
24#2319—21no—
25#2283C412yes—
26#2283C412yes—
27#2283C2211yescharacteristic
28#2283C2211yescharacteristic
29#2276—21no—
30#1801—21no—
31#1801C2212yes—
32#1800—21no—
33#1800C2212yes—
34#1742—21no—
35#1465—22no—
36#1464—22no—
37#1461C2212yes—
38#1460C2212yes—
39#1303C2214yes—
40#1303C2214yes—
41#1286C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44C22×M16—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C4×#41—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8C2311yescharacteristic
53C2×C4×D4C2×C412yes—
54C2×C4×D4D412yes—
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C2×#117D411yescharacteristic
58C2×#117D412yes—
59C22×C42C2×C412yes—
60C22×C42C2311yescharacteristic
61#432—22no—
62#432—22no—
63#431—22no—
64#431C2314yes—
65#423—22no—
66#423—22no—
67#423—22no—
68#423—22no—
69#423C2314yes—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72#413—44no—
73C2×#101—22no—
74C2×#100—22no—
75#407—22no—
76C4×M16—22no—
77#390—22no—
78#390—22no—
79#390—21no—
80#390—21no—
81#389—21no—
82#389—21no—
83#389—22no—
84#389—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#383C2×C411yescharacteristic
92#383C2×C412yes—
93#383C2×C411yescharacteristic
94#382—22no—
95#382—21no—
96#382C2×C412yes—
97#382C2×C412yes—
98#382C2311yescharacteristic
99#382C2311yescharacteristic
100#381—22no—
101#381—21no—
102#381C2×C411yescharacteristic
103#381C2×C411yescharacteristic
104#381C2×C412yes—
105#381C2×C412yes—
106#380—22no—
107#380—22no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—22no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4C2×C412yes—
124C4×C22⋊C4C2×C412yes—
125#376—22no—
126#376—22no—
127#376—21no—
128#376—21no—
129#343—22no—
130#343—22no—
131#324—24no—
132#323—24no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—42no—
138C22×Q8C2×D411yescharacteristic
139C22×D4C22⋊C412yes—
140C22×D4C2×D411yescharacteristic
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C2×M16—44no—
151C2×M16—44no—
152C22×C8—42no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—44no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168#117—22no—
169#117—22no—
170#117—24no—
171#117—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C2×D412yes—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42C22⋊C412yes—
206C2×C42C22⋊C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221C4⋊C8—42no—
222#101—44no—
223#100—44no—
224C22⋊C8—24no—
225C22⋊C8—42no—
226#97—42no—
227C4×C8—24no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95C22×C412yes—
243#95C22×C412yes—
244#95C22×C411yescharacteristic
245#95C22×C411yescharacteristic
246#95C22×C411yescharacteristic
247#95C22×C411yescharacteristic
248C24C2×C22⋊C412yes—
249C24C2×C22⋊C411yescharacteristic
250#41—44no—
251#41—44no—
252#41—44no—
253#41—44no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8AGL(1,ℤ/8)12yes—
257C2×Q8#13712yes—
258C2×D4—24no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—42no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4#11712yes—
293C22×C4#11712yes—
294C22×C4#12312yes—
295C22×C4#12312yes—
296C22×C4#12312yes—
297C22×C4#12312yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301M16—44no—
302M16—44no—
303M16—84no—
304M16—84no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—84no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—81no—
318C4⋊C4—81no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—44no—
338C42—41no—
339C42—41no—
340C42—42no—
341C42—44no—
342C42—82no—
343C42—82no—
344C42—82no—
345C42#11711yescharacteristic
346C42#11711yescharacteristic
347C42#11711yescharacteristic
348C42#11711yescharacteristic
349C42#12312yes—
350C42#12312yes—
351C42#12312yes—
352C42#12312yes—
353C42C2×#4112yes—
354C42C2×#4112yes—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23#38112yes—
365C23C22×C22⋊C411yescharacteristic
366Q8—24no—
367Q8—24no—
368D4—48no—
369D4—42no—
370D4—42no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#38111yescharacteristic
416C2×C4#38111yescharacteristic
417C2×C4#38111yescharacteristic
418C2×C4#38111yescharacteristic
419C2×C4#38112yes—
420C2×C4#41714yes—
421C2×C4C2×#11711yescharacteristic
422C2×C4C2×#11711yescharacteristic
423C2×C4C2×#12312yes—
424C2×C4C2×#12312yes—
425C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
426C2×C4C2×#13711yescharacteristic
427C2×C4#57512yes—
428C8—84no—
429C8—84no—
430C22—21no—
431C22—21no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—82no—
441C22—82no—
442C22C2×#38111yescharacteristic
443C22C2×#41711yescharacteristic
444C22#289611yescharacteristic
445C4—21no—
446C4—21no—
447C4—24no—
448C4—22no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—22no—
454C4—42no—
455C4—42no—
456C4—81no—
457C4—82no—
458C4—81no—
459C4—82no—
460C4—81no—
461C4—81no—
462C4#309412yes—
463C4#309512yes—
464C2—21no—
465C2—21no—
466C2—42no—
467C2—82no—
468C2#2070211yescharacteristic
469C2#2070811yescharacteristic
470C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8771C2semidirect
5. #8770C2semidirect
6. #8767C2semidirect
7. #8766C2semidirect
8. #8093C2semidirect
9. #7962C2semidirect

Maximal subgroups

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

subgroupindex
#341972
#206372
#173662
#87712
#87702
#87672
#87662
#80932
#79622