Finite Group Databasebeta

← #504136

Subgroups — #504136

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2225C211yesmaximal, characteristic, complemented
2C2×C4×SD32C211yesmaximal, characteristic
3#26484C211yesmaximal, characteristic
4C4⋊C4×Q16C211yesmaximal, characteristic, complemented
5#13527C211yesmaximal, characteristic
6#13506C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic
8#13452C211yesmaximal, characteristic, complemented
9#13166C211yesmaximal, characteristic, complemented
10#13104C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic
12#13013C211yesmaximal, characteristic
13#12938C211yesmaximal, characteristic, complemented
14#12938C211yesmaximal, characteristic, complemented
15#12937C211yesmaximal, characteristic
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2452—21no—
23Q8×C4⋊C4—21no—
24#2225—24no—
25C4×SD32—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#2067—21no—
40#2023—21no—
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1863—21no—
48#1863—21no—
49#1862—21no—
50#1845C2211yescharacteristic
51#1803—21no—
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C22⋊Q8—41no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×SD32—22no—
70C2×SD32—22no—
71C2×SD32Q814yescomplemented
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500—22no—
77#500—41no—
78C4×Q16—22no—
79C4×Q16—22no—
80C4×Q16—22no—
81C4×Q16—22no—
82C4×Q16—41no—
83C4×Q16—41no—
84C4×D8—22no—
85C4×D8—41no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#396—41no—
105#393—41no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—41no—
113#389—41no—
114#385—41no—
115#384—21no—
116#384—41no—
117C4×C4⋊C4—41no—
118C4×C4⋊C4D411yescharacteristic
119#367—22no—
120#367—22no—
121#367—22no—
122#367—22no—
123#367—41no—
124C4⋊C16—22no—
125#358—22no—
126#358—22no—
127#358D414yes—
128#357—22no—
129#357—22no—
130#357—41no—
131C4×C16—22no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—41no—
140C22×D4—21no—
141C23×C4—41no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16C2×D412yes—
146C2×Q16C2×Q812yes—
147C2×Q16#4112yes—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8C2×Q812yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2411yescharacteristic
158C4⋊Q8—42no—
159C4⋊Q8—81no—
160C22⋊Q8—84no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—81no—
166C4×Q8—81no—
167C4×D4—42no—
168C4×D4—81no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
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⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4D811yescharacteristic
186C2×C4⋊C4D811yescharacteristic
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198SD32—28no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16C2×D412yes—
210C2×C16C2×Q812yes—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—42no—
222#107—81no—
223#107C2×D412yes—
224#107#4112yes—
225C4⋊C8—41no—
226#103—24no—
227#103—42no—
228#103—42no—
229#102—42no—
230#102—42no—
231#102—81no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24—41no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—161no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—161no—
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C22×D411yescharacteristic
277Q16—22no—
278Q16—22no—
279Q16—22no—
280Q16C22⋊Q814yescomplemented
281D8—22no—
282D8—22no—
283D8—41no—
284D8—41no—
285C2×C8—21no—
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×C8C22×D411yescharacteristic
295C2×C8C22×Q811yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8C2×#4111yescharacteristic
298C2×C8H(5,𝔽2)11yescharacteristic
299C2×C82−1+411yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—44no—
304C4⋊C4—41no—
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—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—161no—
321C4⋊C4—161no—
322C4⋊C4—161no—
323C4⋊C4SD3214yescomplemented
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—161no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—44no—
336C42—82no—
337C42—82no—
338C16—22no—
339C16—22no—
340C16—41no—
341C16—41no—
342C23—42no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23C22×D811yescharacteristic
350Q8—24no—
351Q8—42no—
352Q8—42no—
353Q8—42no—
354Q8—164no—
355D4—42no—
356D4—42no—
357D4—81no—
358D4—81no—
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—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
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—81no—
394C2×C4—81no—
395C2×C4—164no—
396C2×C4C2×SD3212yes—
397C2×C4C2×SD3212yes—
398C2×C4C2×SD3212yes—
399C2×C4#52511yescharacteristic
400C2×C4D4211yescharacteristic
401C2×C4D4×Q811yescharacteristic
402C2×C4#57511yescharacteristic
403C2×C4#57611yescharacteristic
404C2×C4#57811yescharacteristic
405C8—41no—
406C8—41no—
407C8—41no—
408C8C2×C22⋊Q811yescharacteristic
409C8#53911yescharacteristic
410C8#54711yescharacteristic
411C8#55411yescharacteristic
412C22—82no—
413C22—82no—
414C22—82no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22#310011yescharacteristic
420C22D4×D811yescharacteristic
421C22Q8×D811yescharacteristic
422C22C22×SD3211yescharacteristic
423C22#341411yescharacteristic
424C22#341611yescharacteristic
425C22#341611yescharacteristic
426C4—22no—
427C4—22no—
428C4—22no—
429C4—44no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—82no—
434C4—82no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4#244911yescharacteristic
440C4#312411yescharacteristic
441C4#326911yescharacteristic
442C4#338911yescharacteristic
443C2—162no—
444C2#2636811yescharacteristic
445C2#3366311yescharacteristic
446C2#3368511yescharacteristic
447C2#3383711yescharacteristic
448C2D4×SD3211yescharacteristic
449C2Q8×SD3211yescharacteristic
450C2#3395511yescharacteristic
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×#2225C2semidirect
4. C4⋊C4×Q16C2semidirect
6. #13506C2semidirect
8. #13452C2semidirect
9. #13166C2semidirect
10. #13104C2semidirect
13. #12938C2semidirect
14. #12938C2semidirect
71. C2×SD32Q8semidirect
280. Q16C22⋊Q8semidirect
323. C4⋊C4SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#22252
C2×C4×SD322
#264842
C4⋊C4×Q162
#135272
#135062
#134762
#134522
#131662
#131042
#130132
#129382
#129372