Finite Group Databasebeta

← #496902

Subgroups — #496902

Order 512.

461 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33475C211yesmaximal, characteristic, complemented
2#33421C211yesmaximal, characteristic
3#21627C211yesmaximal, characteristic
4#13265C211yesmaximal, characteristic, complemented
5#13163C211yesmaximal, characteristic, complemented
6#13162C211yesmaximal, characteristic, complemented
7#13161C211yesmaximal, characteristic, complemented
8#13125C211yesmaximal, characteristic
9#13124C211yesmaximal, characteristic
10#13122C211yesmaximal, characteristic
11#13122C211yesmaximal, characteristic
12#13106C211yesmaximal, characteristic, complemented
13#13106C211yesmaximal, characteristic, complemented
14#12787C211yesmaximal, characteristic
15#12663C211yesmaximal, characteristic, complemented
16#3076C411yescharacteristic
17#2960C2211yescharacteristic
18C4×#131C2211yescharacteristic
19#2183—21no—
20#2183—21no—
21#2183—21no—
22#2183—21no—
23#2164—21no—
24#2164—21no—
25#2163—21no—
26#2163—21no—
27C2×#364C2211yescharacteristic
28C2×#364C2211yescharacteristic
29#2151C2211yescharacteristic
30#2151C2211yescharacteristic
31#2151C2211yescharacteristic
32#2151C2211yescharacteristic
33#2150C2211yescharacteristic
34C2×C4⋊C16C2211yescharacteristic
35C2×#350C2211yescharacteristic
36C2×#350C2211yescharacteristic
37#2113C2211yescharacteristic
38#2113C2211yescharacteristic
39#2108C2211yescharacteristic
40#2108C2211yescharacteristic
41C4×M32C2211yescharacteristic
42C2×#346C2211yescharacteristic
43C2×#346C2211yescharacteristic
44C2×C4×C16C2211yescharacteristic
45#1986C411yescharacteristic
46#1950C411yescharacteristic
47#1922C411yescharacteristic
48#1922C411yescharacteristic
49#1866C411yescharacteristic
50#1866C411yescharacteristic
51#1806C411yescharacteristic
52#1772C2211yescharacteristic
53#1772C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1764C2211yescharacteristic
56#1764C2211yescharacteristic
57#1379C2211yescharacteristic
58#1379C2211yescharacteristic
59#1379C2211yescharacteristic
60#1379C2211yescharacteristic
61#1379C2211yescharacteristic
62#1379C2211yescharacteristic
63#1376C2211yescharacteristic
64#1376C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#131C2×C411yescharacteristic
68C2×#131C2311yescharacteristic
69#533C2×C411yescharacteristic
70C4×#41C2×C411yescharacteristic
71C2×M32C2311yescharacteristic
72C2×M32C2311yescharacteristic
73C2×M32C2311yescharacteristic
74C2×M32C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#479C811yescharacteristic
80#479C811yescharacteristic
81#478C811yescharacteristic
82#478C811yescharacteristic
83#477C811yescharacteristic
84#476C811yescharacteristic
85#475C811yescharacteristic
86#474C811yescharacteristic
87#446—21no—
88#446—21no—
89C8×Q8—21no—
90C8×Q8—21no—
91C2×#108C2×C411yescharacteristic
92C2×#108C2×C411yescharacteristic
93#427C2×C411yescharacteristic
94#427C2×C411yescharacteristic
95#424C2×C411yescharacteristic
96#424C2×C411yescharacteristic
97#423C2×C411yescharacteristic
98C2×C4⋊C8C2×C411yescharacteristic
99C2×#104C2×C411yescharacteristic
100C2×#104C2×C411yescharacteristic
101#416C2×C411yescharacteristic
102#416C2×C411yescharacteristic
103#405—21no—
104#405—21no—
105#405C2×C411yescharacteristic
106#405C2311yescharacteristic
107#405C2311yescharacteristic
108#405C2311yescharacteristic
109C4×M16—21no—
110C4×M16C2×C411yescharacteristic
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116#364—41no—
117#364—41no—
118C4⋊C16—21no—
119C4⋊C16—21no—
120C4⋊C16—21no—
121C4⋊C16—21no—
122C4⋊C16—21no—
123C4⋊C16—21no—
124C4⋊C16—21no—
125C4⋊C16—21no—
126C4⋊C16—21no—
127C4⋊C16—21no—
128C4⋊C16—21no—
129C4⋊C16—21no—
130#350—41no—
131#350—41no—
132C22⋊C16—21no—
133C22⋊C16—21no—
134C22⋊C16—21no—
135C22⋊C16—21no—
136#346—21no—
137#346—21no—
138#346—21no—
139#346—41no—
140#346D411yescharacteristic
141#346D411yescharacteristic
142#346D411yescharacteristic
143#346D411yescharacteristic
144C4×C16—21no—
145C4×C16—21no—
146C4×C16—21no—
147C4×C16—41no—
148#341C2×C411yescharacteristic
149#341C2×C411yescharacteristic
150#339C2×C411yescharacteristic
151#339C2×C411yescharacteristic
152#339C2×C411yescharacteristic
153#339C2×C411yescharacteristic
154#339C2×C411yescharacteristic
155#339C2×C411yescharacteristic
156#336C2×C411yescharacteristic
157#336C2×C411yescharacteristic
158C2×#41C22×C411yescharacteristic
159#131—21no—
160#131—21no—
161#131—21no—
162#131—21no—
163#131—41no—
164#131—41no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16C22×C411yescharacteristic
168C2×M16C22×C411yescharacteristic
169C2×M16C22×C411yescharacteristic
170C2×M16C22×C411yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C2411yescharacteristic
177C4⋊Q8C2×C811yescharacteristic
178C4⋊Q8C2×C811yescharacteristic
179#125C2×C811yescharacteristic
180#125C2×C811yescharacteristic
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8C2×C811yescharacteristic
186C4×Q8C2×C811yescharacteristic
187C4×D4—21no—
188C4×D4—21no—
189C4×D4C2×C811yescharacteristic
190C4×D4C2×C811yescharacteristic
191#117—21no—
192#117C22×C411yescharacteristic
193#117C22×C411yescharacteristic
194#117C22×C411yescharacteristic
195C2×C4⋊C4—21no—
196C2×C42—21no—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200M32—21no—
201M32—21no—
202M32—21no—
203M32—21no—
204M32—21no—
205M32—21no—
206M32—21no—
207M32—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×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16C2×D411yescharacteristic
223C2×C16C2×D411yescharacteristic
224C2×C16C2×D411yescharacteristic
225C2×C16C2×D411yescharacteristic
226#108—41no—
227#108—41no—
228#107C2×C811yescharacteristic
229#107C2×C811yescharacteristic
230#107C2×C811yescharacteristic
231#107C2×C811yescharacteristic
232#106C2×C811yescharacteristic
233#106C2×C811yescharacteristic
234#106C2×C811yescharacteristic
235#106C2×C811yescharacteristic
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8C2×C811yescharacteristic
243C4⋊C8C2×C811yescharacteristic
244C4⋊C8C2×C811yescharacteristic
245C4⋊C8C2×C811yescharacteristic
246#104—41no—
247#104—41no—
248#103C2×C811yescharacteristic
249#103C2×C811yescharacteristic
250#103C2×C811yescharacteristic
251#103C2×C811yescharacteristic
252#102C2×C811yescharacteristic
253#102C2×C811yescharacteristic
254#102C2×C811yescharacteristic
255#102C2×C811yescharacteristic
256#97—21no—
257#97—21no—
258#97—21no—
259#97—21no—
260#97—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8C2×D411yescharacteristic
271C4×C8C2×D411yescharacteristic
272C4×C8#4111yescharacteristic
273C4×C8#4111yescharacteristic
274#95—21no—
275#41—21no—
276#41—21no—
277#41—21no—
278#41—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8C22×C811yescharacteristic
282C2×D4—21no—
283C2×D4C22×C811yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4#13111yescharacteristic
290C22×C4#13111yescharacteristic
291C22×C4C23×C411yescharacteristic
292M16—21no—
293M16—21no—
294M16—21no—
295M16—21no—
296M16—21no—
297M16—21no—
298M16—21no—
299M16—21no—
300M16—42no—
301M16—41no—
302M16—41no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8C4×D411yescharacteristic
317C2×C8C4×D411yescharacteristic
318C2×C8C22×C811yescharacteristic
319C2×C8C22×C811yescharacteristic
320C2×C8C22×C811yescharacteristic
321C2×C8C22×C811yescharacteristic
322C2×C8C22×D411yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8H(5,𝔽2)11yescharacteristic
325C2×C82−1+411yescharacteristic
326C2×C82−1+411yescharacteristic
327C2×C82−1+411yescharacteristic
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—42no—
337C4⋊C4C22×C811yescharacteristic
338C4⋊C4C22×C811yescharacteristic
339C4⋊C4C22×C811yescharacteristic
340C4⋊C4C22×C811yescharacteristic
341C4⋊C4C22×C811yescharacteristic
342C4⋊C4C22×C811yescharacteristic
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42C4×D411yescharacteristic
358C42C4×D411yescharacteristic
359C42C22×C811yescharacteristic
360C42C22×C811yescharacteristic
361C16—21no—
362C16—21no—
363C16—21no—
364C16—21no—
365C16—41no—
366C16—41no—
367C16—41no—
368C16—41no—
369C16—41no—
370C16—41no—
371C23—21no—
372C23C2×#13111yescharacteristic
373Q8—21no—
374Q8—21no—
375Q8—81no—
376Q8—81no—
377D4—21no—
378D4—21no—
379D4—41no—
380D4—41no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—41no—
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—81no—
399C2×C4—81no—
400C2×C4C8×D411yescharacteristic
401C2×C4C8×D411yescharacteristic
402C2×C4C8×D411yescharacteristic
403C2×C4C8×D411yescharacteristic
404C2×C4C2×C4×D411yescharacteristic
405C2×C4#51911yescharacteristic
406C2×C4#52011yescharacteristic
407C2×C4C23×C811yescharacteristic
408C2×C4#56811yescharacteristic
409C2×C4#56811yescharacteristic
410C8—21no—
411C8—21no—
412C8—21no—
413C8—21no—
414C8—21no—
415C8—21no—
416C8—41no—
417C8—41no—
418C8—41no—
419C8—41no—
420C8#53611yescharacteristic
421C8#53711yescharacteristic
422C8#54111yescharacteristic
423C8#54211yescharacteristic
424C22—21no—
425C22—21no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22C2×C8×D411yescharacteristic
430C22#298511yescharacteristic
431C22#299311yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—81no—
445C4#232311yescharacteristic
446C4#293011yescharacteristic
447C4#298211yescharacteristic
448C4#298811yescharacteristic
449C2—21no—
450C2—21no—
451C2—81no—
452C2#2209811yescharacteristic
453C2#3344211yescharacteristic
454C2#3344311yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radicalC2×#41C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×#346 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33475C2semidirect
4. #13265C2semidirect
5. #13163C2semidirect
6. #13162C2semidirect
7. #13161C2semidirect
12. #13106C2semidirect
13. #13106C2semidirect
15. #12663C2semidirect

Maximal subgroups

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

subgroupindex
#334752
#334212
#216272
#132652
#131632
#131622
#131612
#131252
#131242
#131222
#131062
#127872
#126632