Finite Group Databasebeta

← #499498

Subgroups — #499498

Order 512.

621 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2175C211yesmaximal, characteristic, complemented
2C2×#2164C211yesmaximal, characteristic, complemented
3#21525C211yesmaximal, characteristic, complemented
4#21524C211yesmaximal, characteristic
5#13515C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13449C211yesmaximal, characteristic, complemented
10#13383C211yesmaximal, characteristic, complemented
11#13381C211yesmaximal, characteristic
12#13305C211yesmaximal, characteristic
13#12674C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2324—21no—
23#2323—21no—
24#2175—24no—
25#2164—24no—
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#1970C2211yescharacteristic
42#1969C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1936—21no—
46#1935—21no—
47#1895—21no—
48#1894—21no—
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C4×#107C2211yescharacteristic
53#1763—21no—
54#1762—21no—
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C22×M16D411yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×#117D411yescharacteristic
70C2×SD32—22no—
71C2×SD32—22no—
72C2×SD32—22no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#467—22no—
78#467—22no—
79#466—22no—
80#466—22no—
81C4×Q16—22no—
82C4×Q16—22no—
83C4×D8—22no—
84C4×D8—22no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97#405—22no—
98#405—22no—
99C2×#97D411yescharacteristic
100C2×#97D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#391—21no—
104#390—21no—
105#390—41no—
106#389—21no—
107#389—21no—
108#389—41no—
109#388—41no—
110#388—41no—
111#388—41no—
112#387—41no—
113#386—41no—
114#385—41no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119#367—22no—
120#366—22no—
121#366—22no—
122C4⋊C16—22no—
123C4⋊C16—22no—
124#358—22no—
125#358—22no—
126#358—22no—
127#357—22no—
128#357—22no—
129#357—22no—
130C22⋊C16—22no—
131C22⋊C16—22no—
132#346D414yescomplemented
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4—41no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16#4112yes—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8#4112yes—
157C2×M16—22no—
158C2×M16—41no—
159C2×M16—41no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170C22⋊Q8—24no—
171C22⋊Q8—42no—
172#121—24no—
173#121—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×D4—42no—
177C4×D4—42no—
178#117—22no—
179#117—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4D811yescharacteristic
191C2×C4⋊C4D811yescharacteristic
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4D811yescharacteristic
200C2×C22⋊C4D811yescharacteristic
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206SD32—28no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16C2×D412yes—
218C2×C16C2×D412yes—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107#4112yes—
230#107#4112yes—
231#103—24no—
232#103—42no—
233#103—42no—
234#102—24no—
235#102—42no—
236#102—42no—
237#97—22no—
238#97—22no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8C2×D412yes—
242C4×C8#4112yes—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—41no—
252C24C2×D811yescharacteristic
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—22no—
260C2×D4—44no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—161no—
267C2×D4—161no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—82no—
282C22×C4—161no—
283C22×C4—161no—
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C22×D411yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—41no—
293Q16—41no—
294D8—22no—
295D8—22no—
296D8—41no—
297D8—41no—
298M16—44no—
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—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8#12112yes—
312C2×C8C22×D411yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—44no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—82no—
334C4⋊C4—161no—
335C4⋊C4—161no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C22⋊C4—161no—
342C22⋊C4—161no—
343C22⋊C4—161no—
344C22⋊C4—161no—
345C42—21no—
346C42—21no—
347C42—82no—
348C42—161no—
349C42—161no—
350C42#12112yes—
351C42#13512yes—
352C16—22no—
353C16—22no—
354C16—41no—
355C16—41no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—42no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23C22×D811yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—81no—
374Q8—81no—
375D4—42no—
376D4—42no—
377D4—81no—
378D4—81no—
379D4—164no—
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—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—164no—
419C2×C4#46612yes—
420C2×C4#46612yes—
421C2×C4C2×#12111yescharacteristic
422C2×C4#52511yescharacteristic
423C2×C4#52511yescharacteristic
424C2×C4C2×#13511yescharacteristic
425C2×C4#57511yescharacteristic
426C2×C4#57511yescharacteristic
427C8—22no—
428C8—41no—
429C8—41no—
430C8#52511yescharacteristic
431C8#52911yescharacteristic
432C8#52911yescharacteristic
433C8#53211yescharacteristic
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22C2×#46611yescharacteristic
449C22#310011yescharacteristic
450C22#310011yescharacteristic
451C22#341511yescharacteristic
452C22#341611yescharacteristic
453C22#341611yescharacteristic
454C22#341711yescharacteristic
455C4—22no—
456C4—22no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—82no—
461C4—82no—
462C4—82no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4#239211yescharacteristic
468C4#305411yescharacteristic
469C4#310211yescharacteristic
470C4#310211yescharacteristic
471C2—81no—
472C2—162no—
473C2#2369611yescharacteristic
474C2#3361211yescharacteristic
475C2#3361311yescharacteristic
476C2#3367911yescharacteristic
477C2#3368111yescharacteristic
478C2#3368311yescharacteristic
479C2#3368411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2175C2semidirect
2. C2×#2164C2semidirect
3. #21525C2semidirect
5. #13515C2semidirect
6. #13492C2semidirect
7. #13477C2semidirect
8. #13474C2semidirect
9. #13449C2semidirect
10. #13383C2semidirect
13. #12674C2semidirect
14. C4×#358C2semidirect
15. C4×#357C2semidirect
132. #346D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21752
C2×#21642
#215252
#215242
#135152
#134922
#134772
#134742
#134492
#133832
#133812
#133052
#126742
C4×#3582
C4×#3572