Finite Group Databasebeta

← #510051

Subgroups — #510051

Order 512.

551 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2227C211yesmaximal, characteristic, complemented
2C2×#2224C211yesmaximal, characteristic
3#26485C211yesmaximal, characteristic, complemented
4#26484C211yesmaximal, characteristic
5#13477C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic
7#13475C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic
9#13451C211yesmaximal, characteristic, complemented
10#13451C211yesmaximal, characteristic, complemented
11#13238C211yesmaximal, characteristic
12#13168C211yesmaximal, characteristic, complemented
13#13160C211yesmaximal, characteristic, complemented
14#13014C211yesmaximal, characteristic
15#13013C211yesmaximal, characteristic
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2452—21no—
22#2451—21no—
23#2227—24no—
24#2224—24no—
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#346C2211yescharacteristic
40#2068—21no—
41#2067—21no—
42#2024—21no—
43#2023—21no—
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1969C2211yescharacteristic
47#1969C2211yescharacteristic
48#1942C2211yescharacteristic
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1942C2211yescharacteristic
52#1863—21no—
53#1862—21no—
54C8×C4⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C22⋊Q8—41no—
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C2×SD32—41no—
69C2×SD32—41no—
70C2×SD32Q814yescomplemented
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#500—22no—
76#500—22no—
77#500—41no—
78#500—41no—
79C4×Q16—22no—
80C4×Q16—41no—
81C4×D8—22no—
82C4×D8—41no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#398—41no—
102#396—41no—
103#395—41no—
104#393—41no—
105#391—21no—
106#391—21no—
107#391—41no—
108#390—21no—
109#390—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#387—41no—
115#385—41no—
116C4×C4⋊C4D411yescharacteristic
117#367—41no—
118#366—22no—
119#366—22no—
120#366—41no—
121#366D414yes—
122C4⋊C16—22no—
123C4⋊C16—22no—
124C4⋊C16—41no—
125#358—22no—
126#358—22no—
127#358—41no—
128#357—22no—
129#357—22no—
130#357—41no—
131#346—41no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—41no—
140C22×Q8—41no—
141C22×D4—21no—
142C23×C4—41no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16C2×Q812yes—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8C2×Q812yes—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2411yescharacteristic
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—81no—
162C4⋊Q8—81no—
163C22⋊Q8—84no—
164C4×Q8—42no—
165C4×Q8—81no—
166C4×D4—42no—
167C4×D4—81no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4D811yescharacteristic
182C2×C4⋊C4D811yescharacteristic
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192SD32—44no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16C2×D412yes—
204C2×C16C2×Q812yes—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—42no—
216#107—42no—
217#107C2×D412yes—
218#107C2×D412yes—
219C4⋊C8—41no—
220#103—42no—
221#103—42no—
222#103—81no—
223#102—42no—
224#102—42no—
225#102—81no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95D811yescharacteristic
241#95D811yescharacteristic
242C24—41no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—161no—
254C2×Q8—161no—
255C2×D4—22no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—161no—
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C22×D411yescharacteristic
278Q16—22no—
279Q16—22no—
280Q16—41no—
281Q16—41no—
282D8—22no—
283D8—22no—
284D8—41no—
285D8—41no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8C22×D411yescharacteristic
296C2×C8C22×Q811yescharacteristic
297C2×C8H(5,𝔽2)11yescharacteristic
298C2×C8H(5,𝔽2)11yescharacteristic
299C2×C82−1+411yescharacteristic
300C2×C82−1+411yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—161no—
321C4⋊C4—161no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—161no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—82no—
334C16—22no—
335C16—22no—
336C16—41no—
337C16—41no—
338C23—42no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23C22×D811yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—81no—
349Q8—81no—
350Q8—164no—
351Q8—164no—
352D4—42no—
353D4—42no—
354D4—81no—
355D4—81no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—164no—
394C2×C4#50912yes—
395C2×C4#51012yes—
396C2×C4#53711yescharacteristic
397C2×C4D4211yescharacteristic
398C2×C4D4×Q811yescharacteristic
399C2×C4#57511yescharacteristic
400C2×C4#57611yescharacteristic
401C2×C4#57811yescharacteristic
402C8—41no—
403C8—41no—
404C8—41no—
405C8#53711yescharacteristic
406C8#54311yescharacteristic
407C8D4×Q811yescharacteristic
408C8#56311yescharacteristic
409C22—82no—
410C22—82no—
411C22—82no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22#324511yescharacteristic
417C22D4×D811yescharacteristic
418C22Q8×D811yescharacteristic
419C22C2×#50911yescharacteristic
420C22C2×#51011yescharacteristic
421C22#341511yescharacteristic
422C22#341711yescharacteristic
423C4—22no—
424C4—22no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4—82no—
431C4—82no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#268311yescharacteristic
437C4#326211yescharacteristic
438C4#327711yescharacteristic
439C4#338911yescharacteristic
440C2—162no—
441C2#3123411yescharacteristic
442C2#3382011yescharacteristic
443C2#3382811yescharacteristic
444C2#3383011yescharacteristic
445C2#3383611yescharacteristic
446C2#3395211yescharacteristic
447C2#3395311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2227C2semidirect
3. #26485C2semidirect
5. #13477C2semidirect
7. #13475C2semidirect
9. #13451C2semidirect
10. #13451C2semidirect
12. #13168C2semidirect
13. #13160C2semidirect
70. C2×SD32Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22272
C2×#22242
#264852
#264842
#134772
#134762
#134752
#134742
#134512
#132382
#131682
#131602
#130142
#130132