Finite Group Databasebeta

← #509707

Subgroups — #509707

Order 512.

546 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2254C211yesmaximal, characteristic, complemented
2C2×#2226C211yesmaximal, characteristic
3#28813C211yesmaximal, characteristic, complemented
4#26484C211yesmaximal, characteristic
5#13527C211yesmaximal, characteristic
6#13516C211yesmaximal, characteristic, complemented
7#13506C211yesmaximal, characteristic, complemented
8#13452C211yesmaximal, characteristic, complemented
9#13452C211yesmaximal, characteristic, complemented
10#13363C211yesmaximal, characteristic
11#13147C211yesmaximal, characteristic
12#13131C211yesmaximal, characteristic
13#13110C211yesmaximal, characteristic, complemented
14#13104C211yesmaximal, characteristic, complemented
15#12937C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2551—21no—
21#2452—21no—
22#2254—24no—
23#2226—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×C4×C16C2211yescharacteristic
36#2067—21no—
37#2057—21no—
38#2057—21no—
39#2023—21no—
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1947—21no—
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1862—21no—
48#1845C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—41no—
66C2×C22⋊Q8—41no—
67C2×C4×D4—21no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#500—22no—
73#500—22no—
74#500—22no—
75#500—41no—
76#500—41no—
77#500—41no—
78C4×D8—22no—
79C4×D8—41no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#396—41no—
99#395—41no—
100#395—41no—
101#393—41no—
102#390—21no—
103#390—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—41no—
109#385—41no—
110#384—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D411yescharacteristic
113#367—22no—
114#367—22no—
115#367—22no—
116#367—22no—
117#367—41no—
118#367D414yes—
119C4⋊C16—22no—
120#357—22no—
121#357—22no—
122#357—41no—
123#357Q814yescomplemented
124C4×C16—22no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#336D411yescharacteristic
131#336D411yescharacteristic
132C22×Q8—41no—
133C22×Q8—41no—
134C22×Q8—41no—
135C22×D4—21no—
136C23×C4—41no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8C2×Q812yes—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2411yescharacteristic
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—81no—
151C4⋊Q8—81no—
152C4⋊Q8—81no—
153C4⋊Q8—84no—
154C22⋊Q8—84no—
155C4×D4—42no—
156C4×D4—81no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4D811yescharacteristic
174C2×C4⋊C4D811yescharacteristic
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C42—41no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16C2×D412yes—
195C2×C16C2×Q812yes—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—42no—
208#107—42no—
209#107—42no—
210#107—81no—
211#107C2×D412yes—
212#107C2×D412yes—
213#107C2×Q812yes—
214C4⋊C8—22no—
215#102—42no—
216#102—42no—
217#102—81no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95D811yescharacteristic
230#95D811yescharacteristic
231C24—41no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×Q8—161no—
239C2×Q8—161no—
240C2×Q8—161no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—161no—
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C22×D411yescharacteristic
264D8—22no—
265D8—22no—
266D8—41no—
267D8—41no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8#13712yes—
277C2×C8C22×D411yescharacteristic
278C2×C8C22×Q811yescharacteristic
279C2×C8H(5,𝔽2)11yescharacteristic
280C2×C82−1+411yescharacteristic
281C2×C82−1+411yescharacteristic
282C2×C82−1+411yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—161no—
307C4⋊C4SD3214yes—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—161no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—82no—
320C16—22no—
321C16—22no—
322C16—41no—
323C16—41no—
324C23—42no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23C22×D811yescharacteristic
332Q8—164no—
333Q8—164no—
334Q8—164no—
335D4—42no—
336D4—42no—
337D4—81no—
338D4—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—164no—
375C2×C4C2×SD3212yes—
376C2×C4C2×SD3212yes—
377C2×C4C2×SD3212yes—
378C2×C4#50912yes—
379C2×C4#53611yescharacteristic
380C2×C4D4211yescharacteristic
381C2×C4D4×Q811yescharacteristic
382C2×C4C2×#13711yescharacteristic
383C2×C4#57611yescharacteristic
384C2×C4#57811yescharacteristic
385C8—22no—
386C8—41no—
387C8—41no—
388C8#53611yescharacteristic
389C8#54411yescharacteristic
390C8D4×Q811yescharacteristic
391C8#56311yescharacteristic
392C22—82no—
393C22—82no—
394C22—82no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22#322611yescharacteristic
400C22D4×D811yescharacteristic
401C22Q8×D811yescharacteristic
402C22C22×SD3211yescharacteristic
403C22C2×#50911yescharacteristic
404C22#341611yescharacteristic
405C22#341611yescharacteristic
406C4—22no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—41no—
411C4—41no—
412C4—82no—
413C4—82no—
414C4—82no—
415C4—161no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4#267711yescharacteristic
420C4#324311yescharacteristic
421C4#327011yescharacteristic
422C4#338511yescharacteristic
423C2—162no—
424C2#3094611yescharacteristic
425C2#3379311yescharacteristic
426C2#3381011yescharacteristic
427C2#3383111yescharacteristic
428C2D4×SD3211yescharacteristic
429C2Q8×SD3211yescharacteristic
430C2#3395111yescharacteristic
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×#2254C2semidirect
3. #28813C2semidirect
6. #13516C2semidirect
7. #13506C2semidirect
8. #13452C2semidirect
9. #13452C2semidirect
13. #13110C2semidirect
14. #13104C2semidirect
123. #357Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22542
C2×#22262
#288132
#264842
#135272
#135162
#135062
#134522
#133632
#131472
#131312
#131102
#131042
#129372