Finite Group Databasebeta

← #225262

Subgroups — #225262

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22640C211yesmaximal, characteristic
2#22633C211yesmaximal, characteristic
3#20818C211yesmaximal, characteristic
4#9600C211yesmaximal, characteristic, complemented
5#9600C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#9536C211yesmaximal, characteristic, complemented
9#9533C211yesmaximal, characteristic, complemented
10#8711C211yesmaximal, characteristic
11#8711C211yesmaximal, characteristic
12#8596C211yesmaximal, characteristic
13#8289C211yesmaximal, characteristic, complemented
14#8289C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic
16C22×C4×C8C2211yescharacteristic
17#2283C2211yescharacteristic
18#2283C2211yescharacteristic
19#1939C2211yescharacteristic
20#1939C2211yescharacteristic
21#1939C2211yescharacteristic
22#1939C2211yescharacteristic
23#1924—21no—
24#1924—21no—
25#1923—21no—
26#1923—21no—
27#1846C411yescharacteristic
28#1846C411yescharacteristic
29#1840—21no—
30#1835C411yescharacteristic
31#1835C411yescharacteristic
32#1832—21no—
33#1808C411yescharacteristic
34#1808C411yescharacteristic
35#1807C411yescharacteristic
36#1807C411yescharacteristic
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×#107C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C4×#106C2211yescharacteristic
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1768—21no—
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1746—21no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#322C2211yescharacteristic
54C2×#322C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8C2×C411yescharacteristic
68C23×C8C2×C411yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107C2×C411yescharacteristic
71C2×#107C2×C411yescharacteristic
72C2×#107C2×C411yescharacteristic
73C2×#107C2×C411yescharacteristic
74C2×#106C2×C411yescharacteristic
75C2×#106C2×C411yescharacteristic
76C2×#106C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8D411yescharacteristic
90C2×C22⋊C8D411yescharacteristic
91C2×#97—21no—
92C2×#97—21no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2×C411yescharacteristic
98C2×C4×C8C2×C411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#383—21no—
106#383—21no—
107#382C2×C411yescharacteristic
108#382C2×C411yescharacteristic
109#382C2×C411yescharacteristic
110#382C2×C411yescharacteristic
111#381—21no—
112#381—21no—
113#380C2×C411yescharacteristic
114#380C2×C411yescharacteristic
115#380C2×C411yescharacteristic
116#380C2×C411yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123#376—21no—
124#376—21no—
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#335—22no—
147#335—22no—
148#322—22no—
149#322—22no—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4111yescharacteristic
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—41no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8C22×C411yescharacteristic
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C22×C411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42C2411yescharacteristic
197#107—22no—
198#107—22no—
199#107—41no—
200#107—41no—
201#106—41no—
202#106—41no—
203#106—41no—
204#106—41no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—42no—
214C4⋊C8—42no—
215C4⋊C8—81no—
216C4⋊C8—81no—
217C22⋊C8—24no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220C22⋊C8—81no—
221C22⋊C8—81no—
222#97—42no—
223#97—81no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—42no—
237C4×C8—81no—
238C4×C8#4112yes—
239C4×C8#4112yes—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256C24C4×D411yescharacteristic
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4C4×D411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C2×#4111yescharacteristic
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8#11712yes—
329C2×C8#11712yes—
330C2×C8#13112yes—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—81no—
350C22⋊C4—81no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C422−1+411yescharacteristic
368C422−1+411yescharacteristic
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23#44312yes—
376C23C2×C4×D411yescharacteristic
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4C2×#11711yescharacteristic
423C2×C4#51911yescharacteristic
424C2×C4#54711yescharacteristic
425C2×C4#54711yescharacteristic
426C2×C4#55211yescharacteristic
427C2×C4#55211yescharacteristic
428C2×C4C2×#13111yescharacteristic
429C2×C4#56811yescharacteristic
430C2×C4#56811yescharacteristic
431C2×C4#56811yescharacteristic
432C2×C4#57511yescharacteristic
433C2×C4#57511yescharacteristic
434C2×C4#57611yescharacteristic
435C2×C4#57811yescharacteristic
436C8—22no—
437C8—22no—
438C8—22no—
439C8—42no—
440C8—42no—
441C8—41no—
442C8—41no—
443C8—41no—
444C8—41no—
445C8—41no—
446C8—81no—
447C8—81no—
448C8—81no—
449C8—81no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22#234411yescharacteristic
458C22#293211yescharacteristic
459C22#293511yescharacteristic
460C22#294311yescharacteristic
461C22#295111yescharacteristic
462C22C2×#44311yescharacteristic
463C22#295711yescharacteristic
464C4—21no—
465C4—21no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4#292111yescharacteristic
478C4#292411yescharacteristic
479C4#298511yescharacteristic
480C4#298711yescharacteristic
481C4#334711yescharacteristic
482C4#334711yescharacteristic
483C4#334911yescharacteristic
484C4#334911yescharacteristic
485C2—42no—
486C2#1798211yescharacteristic
487C2#1799211yescharacteristic
488C2#1807511yescharacteristic
489C2#1808711yescharacteristic
490C2#2260211yescharacteristic
491C2#2260611yescharacteristic
492C2#2261611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9600C2semidirect
5. #9600C2semidirect
6. #9540C2semidirect
7. #9537C2semidirect
8. #9536C2semidirect
9. #9533C2semidirect
13. #8289C2semidirect
14. #8289C2semidirect

Maximal subgroups

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

subgroupindex
#226402
#226332
#208182
#96002
#95402
#95372
#95362
#95332
#87112
#85962
#82892
#81452