Finite Group Databasebeta

← #101679

Subgroups — #101679

Order 512.

515 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29157C211yesmaximal, characteristic
2#23116C211yesmaximal, characteristic
3#9679C211yesmaximal, characteristic, complemented
4#8577C211yesmaximal, characteristic
5#7668C211yesmaximal, characteristic, complemented
6#7029C211yesmaximal, characteristic, complemented
7#7025C211yesmaximal, characteristic, complemented
8C2×C8×D4—21no—
9#2857C2211yescharacteristic
10C4×#121C2211yescharacteristic
11#2099C411yescharacteristic, complemented
12#2093C411yescharacteristic, complemented
13#2057C411yescharacteristic, complemented
14#2023C411yescharacteristic, complemented
15#1923—21no—
16#1918—21no—
17#1916—21no—
18#1893—21no—
19#1869—21no—
20#1869—21no—
21#1862—21no—
22#1808—21no—
23#1797C2211yescharacteristic
24C8×C22⋊C4—21no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#328—21no—
29C2×#328—21no—
30C2×#327—21no—
31C4⋊#95C2211yescharacteristic
32#1294—21no—
33#1271—21no—
34#1271—21no—
35C23×C8—21no—
36C23×C8—21no—
37C23×C8—21no—
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×C22⋊Q8—21no—
40C2×#121D411yescharacteristic
41C2×C4×D4—21no—
42C2×C4×D4D411yescharacteristic
43C22×C4⋊C4—21no—
44C22×C42C2311yescharacteristic
45C8×D4—42no—
46C8×D4—42no—
47C2×#107—21no—
48C2×#107—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×#102—21no—
52C2×#102—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8D411yescharacteristic
58C2×C22⋊C8D411yescharacteristic
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61#391—21no—
62#390C2×C411yescharacteristic
63#389—21no—
64#389—21no—
65#389C2×C411yescharacteristic
66#388—21no—
67#386—21no—
68#384C2×C411yescharacteristic
69#380—21no—
70#380—21no—
71#380D411yescharacteristic
72C4×C4⋊C4D411yescharacteristic
73C4×C22⋊C4—21no—
74C2×#95—21no—
75C2×#95—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79#340—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336C2×C411yescharacteristic
87#336C2×C411yescharacteristic
88#336D411yescharacteristic
89#336D411yescharacteristic
90#328—42no—
91#328—42no—
92#327—42no—
93C22×Q8—21no—
94C22×D4—21no—
95C22×D4C22⋊C411yescharacteristic
96C23×C4—21no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8—42no—
110C22×C8—42no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—81no—
117C22×C8—81no—
118C22×C8—81no—
119C4⋊Q8—81no—
120C22⋊Q8—41no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124#121—21no—
125#121—21no—
126#121SD1612yes—
127#121Q1612yes—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—81no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—81no—
143C2×C4⋊C4—81no—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—41no—
153C2×C42—21no—
154C2×C42—81no—
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42#4111yescharacteristic
159#107—42no—
160#107—81no—
161C4⋊C8—42no—
162C4⋊C8—81no—
163#102—42no—
164#102—81no—
165C22⋊C8—21no—
166C22⋊C8—21no—
167C22⋊C8—21no—
168C22⋊C8—42no—
169C22⋊C8—42no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—81no—
173C22⋊C8D812yes—
174C4×C8—42no—
175C4×C8—81no—
176#95—21no—
177#95—21no—
178#95—21no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185C24—21no—
186C24#12111yescharacteristic
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—81no—
190C2×Q8—81no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4#10312yes—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4C2×D811yescharacteristic
235C22×C4C2×SD1611yescharacteristic
236C22×C4C2×Q1611yescharacteristic
237C22×C4#13511yescharacteristic
238C22×C4#13511yescharacteristic
239C22×C4#13511yescharacteristic
240C22×C4AGL(1,ℤ/8)11yescharacteristic
241C22×C4AGL(1,ℤ/8)11yescharacteristic
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—84no—
261C2×C8—82no—
262C2×C8—82no—
263C2×C8—82no—
264C2×C8—82no—
265C2×C8—81no—
266C2×C8—81no—
267C2×C8—81no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—81no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4#10312yes—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—41no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—22no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23#38611yescharacteristic
316C23#45911yescharacteristic
317C23#46311yescharacteristic
318C23#46511yescharacteristic
319C23#46711yescharacteristic
320C23#47111yescharacteristic
321C23#47311yescharacteristic
322Q8—161no—
323Q8—161no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—161no—
378C2×C4—161no—
379C2×C4—161no—
380C2×C4—161no—
381C2×C4C2×#10311yescharacteristic
382C2×C4#41611yescharacteristic
383C2×C4C4×D811yescharacteristic
384C2×C4#44211yescharacteristic
385C2×C4#44311yescharacteristic
386C2×C4#44411yescharacteristic
387C2×C4#44711yescharacteristic
388C2×C4#44911yescharacteristic
389C2×C4#46511yescharacteristic
390C2×C4#46711yescharacteristic
391C8—42no—
392C8—42no—
393C8—42no—
394C8—81no—
395C8—81no—
396C8—81no—
397C8—161no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#166511yescharacteristic
418C22#166911yescharacteristic
419C22#167911yescharacteristic
420C22#168311yescharacteristic
421C22#187611yescharacteristic
422C22#187911yescharacteristic
423C22#188411yescharacteristic
424C22#189211yescharacteristic
425C22#190611yescharacteristic
426C22#190811yescharacteristic
427C22#191311yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—42no—
439C4—41no—
440C4—81no—
441C4—81no—
442C4—162no—
443C4—161no—
444C4—161no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—82no—
450C2#905311yescharacteristic
451C2#920711yescharacteristic
452C2#921911yescharacteristic
453C2#926311yescharacteristic
454C2#927711yescharacteristic
455C2#938311yescharacteristic
456C2#943111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9679C2semidirect
5. #7668C2semidirect
6. #7029C2semidirect
7. #7025C2semidirect
11. #2099C4semidirect
12. #2093C4semidirect
13. #2057C4semidirect
14. #2023C4semidirect

Maximal subgroups

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

subgroupindex
#291572
#231162
#96792
#85772
#76682
#70292
#70252