Finite Group Databasebeta

← #498276

Subgroups — #498276

Order 512.

508 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
1#33630C211yesmaximal, characteristic
2#23164C211yesmaximal, characteristic, complemented
3#22686C211yesmaximal, characteristic
4#13289C211yesmaximal, characteristic, complemented
5#13185C211yesmaximal, characteristic, complemented
6#13174C211yesmaximal, characteristic, complemented
7#13174C211yesmaximal, characteristic, complemented
8#13119C211yesmaximal, characteristic, complemented
9#13119C211yesmaximal, characteristic, complemented
10#12955C211yesmaximal, characteristic
11#12955C211yesmaximal, characteristic
12#12666C211yesmaximal, characteristic, complemented
13#12650C211yesmaximal, characteristic
14#12650C211yesmaximal, characteristic
15C4×#360C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×#444C2211yescharacteristic
18C2×#444C2211yescharacteristic
19C4×#135C2211yescharacteristic
20#2906—21no—
21#2370—21no—
22#2229C412yescomplemented
23#2229C412yescomplemented
24#2228C412yescomplemented
25#2228C412yescomplemented
26#2159C2211yescharacteristic
27#2159C2211yescharacteristic
28C2×#365C2211yescharacteristic
29C2×#365C2211yescharacteristic
30#2150C2211yescharacteristic
31C2×#360C2211yescharacteristic
32C2×#360C2211yescharacteristic
33#2139C2211yescharacteristic
34#2139C2211yescharacteristic
35C2×#347C2211yescharacteristic
36C2×#347C2211yescharacteristic
37C4×M32C2211yescharacteristic
38#1989—21no—
39#1946C2211yescharacteristic
40#1946C2211yescharacteristic
41#1943C2211yescharacteristic
42#1943C2211yescharacteristic
43#1920—21no—
44#1919—21no—
45#1866—21no—
46#1866—21no—
47#1848C2211yescharacteristic
48#1806—21no—
49#1776C2211yescharacteristic
50#1776C2211yescharacteristic
51C4×#106C2211yescharacteristic
52#1770C2211yescharacteristic
53#1387C2211yescharacteristic
54#1387C2211yescharacteristic
55#1387C2211yescharacteristic
56#1387C2211yescharacteristic
57#1386C2211yescharacteristic
58#1386C2211yescharacteristic
59#1383C2211yescharacteristic
60#1383C2211yescharacteristic
61#1375C2211yescharacteristic
62#1375C2211yescharacteristic
63C2×#135C2311yescharacteristic
64C2×#131—21no—
65C2×#131—21no—
66C2×#125—21no—
67C4×#41—21no—
68C2×M32C2311yescharacteristic
69C2×M32C2311yescharacteristic
70C2×M32C2311yescharacteristic
71C2×M32C2311yescharacteristic
72#499—21no—
73#499—21no—
74#499C2×C412yes—
75#499C2×C412yes—
76#444—22no—
77#444—22no—
78#444—41no—
79#444—41no—
80C4×Q16C2×C411yescharacteristic
81C4×Q16C2×C411yescharacteristic
82C4×SD16—21no—
83C4×SD16—21no—
84C4×D8C2×C411yescharacteristic
85C4×D8C2×C411yescharacteristic
86C2×#108—21no—
87C2×#108C2311yescharacteristic
88C2×#108C2311yescharacteristic
89#427C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×#106C2311yescharacteristic
94C2×#106C2311yescharacteristic
95#424—21no—
96#424—21no—
97#424—41no—
98#424—41no—
99#423—21no—
100#423—41no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8D411yescharacteristic
103C2×#104—21no—
104C2×#104—21no—
105#416—21no—
106C2×#97C2311yescharacteristic
107C2×#97C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#389—21no—
110#389—21no—
111#389—41no—
112#387—41no—
113#387—41no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—41no—
118C4×C4⋊C4D411yescharacteristic
119#367C2×C412yes—
120#367C2×C412yes—
121#366C2×C412yes—
122#366C2×C412yes—
123#365—41no—
124#365—41no—
125C4⋊C16C2×C412yes—
126C4⋊C16C2×C412yes—
127#360—22no—
128#360Q814yes—
129#358C2×C412yes—
130#358C2×C412yes—
131#357C2×C412yes—
132#357C2×C412yes—
133#347—41no—
134#347—41no—
135#346—21no—
136C4×C16—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#339—21no—
143#339—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146C2×#41—21no—
147#135—21no—
148#135—21no—
149#135C2×Q812yes—
150#135#4112yes—
151C2×Q16C22×C411yescharacteristic
152C2×SD16—21no—
153C2×D8C22×C411yescharacteristic
154#131—42no—
155#131—42no—
156#131—81no—
157#131—81no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—41no—
161C2×M16—41no—
162C22×C8—41no—
163C22×C8—41no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2411yescharacteristic
168#125—22no—
169#125—22no—
170#125—41no—
171#125—41no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—41no—
177C4×D4—41no—
178#117—21no—
179#117—41no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196M32—21no—
197M32—21no—
198M32—21no—
199M32—21no—
200M32—21no—
201M32—21no—
202M32—21no—
203M32C2×Q812yes—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16C22×C412yes—
207C2×C16C22×C412yes—
208#108—21no—
209#108—21no—
210#108—21no—
211#108—21no—
212#108—81no—
213#107—21no—
214#107—21no—
215#107C22×C412yes—
216#107C22×C412yes—
217#107C22×C411yescharacteristic
218#107C22×C411yescharacteristic
219#106—22no—
220#106—22no—
221#106—21no—
222#106—21no—
223#106—21no—
224#106—21no—
225#106C2×Q812yes—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—41no—
229C4⋊C8—41no—
230C4⋊C8—41no—
231C4⋊C8—41no—
232C4⋊C8—41no—
233C4⋊C8—41no—
234#104—42no—
235#104—42no—
236#103—21no—
237#103—21no—
238#102—21no—
239#102—21no—
240#97—21no—
241#97—21no—
242#97—21no—
243#97—21no—
244C4×C8C22×C411yescharacteristic
245C4×C8C22×C411yescharacteristic
246C4×C8#4111yescharacteristic
247C4×C8#4111yescharacteristic
248#95—21no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#41—22no—
255#41—22no—
256#41—41no—
257#41—41no—
258C2×Q8—21no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4AGL(1,ℤ/8)11yescharacteristic
271C22×C4AGL(1,ℤ/8)11yescharacteristic
272C22×C4C22×D411yescharacteristic
273Q16—21no—
274Q16C4×Q812yes—
275SD16—22no—
276SD16—22no—
277D8—21no—
278D8C4×Q812yes—
279M16—41no—
280M16—41no—
281M16—41no—
282M16—41no—
283M16—82no—
284M16—82no—
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—41no—
294C2×C8—41no—
295C2×C8—82no—
296C2×C8—82no—
297C2×C8C4×D411yescharacteristic
298C2×C8C4×D411yescharacteristic
299C2×C8C23×C411yescharacteristic
300C2×C8C22×Q811yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C2×C8C2×#4111yescharacteristic
303C2×C8H(5,𝔽2)11yescharacteristic
304C2×C82−1+411yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—81no—
322C4⋊C4—82no—
323C4⋊C4—161no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—82no—
335C42—161no—
336C42—161no—
337C42C4×D411yescharacteristic
338C42C4×D411yescharacteristic
339C42AGL(1,ℤ/8)11yescharacteristic
340C42AGL(1,ℤ/8)11yescharacteristic
341C16—22no—
342C16—22no—
343C16—22no—
344C16—41no—
345C23—41no—
346C23C2×AGL(1,ℤ/8)11yescharacteristic
347Q8—22no—
348Q8—41no—
349D4—22no—
350D4—42no—
351D4—42no—
352D4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—161no—
380C2×C4#44211yescharacteristic
381C2×C4#44211yescharacteristic
382C2×C4#44211yescharacteristic
383C2×C4#44211yescharacteristic
384C2×C4C2×C4×D411yescharacteristic
385C2×C4#54611yescharacteristic
386C2×C4D4×Q811yescharacteristic
387C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
388C2×C4#57711yescharacteristic
389C2×C4#57811yescharacteristic
390C8—21no—
391C8—21no—
392C8—41no—
393C8—41no—
394C8—81no—
395C8—81no—
396C8C2×C4×Q811yescharacteristic
397C8#52011yescharacteristic
398C8#55211yescharacteristic
399C8#55411yescharacteristic
400C22—21no—
401C22—21no—
402C22—42no—
403C22—81no—
404C22—81no—
405C22C2×#44211yescharacteristic
406C22#333011yescharacteristic
407C22#338411yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—42no—
413C4—81no—
414C4—81no—
415C4—82no—
416C4—81no—
417C4—82no—
418C4—82no—
419C4—81no—
420C4—161no—
421C4#236011yescharacteristic
422C4#294711yescharacteristic
423C4#332411yescharacteristic
424C4#338611yescharacteristic
425C2—21no—
426C2—21no—
427C2—82no—
428C2#2296811yescharacteristic
429C2#3346811yescharacteristic
430C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#135C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23164C2semidirect
4. #13289C2semidirect
5. #13185C2semidirect
6. #13174C2semidirect
7. #13174C2semidirect
8. #13119C2semidirect
9. #13119C2semidirect
12. #12666C2semidirect
22. #2229C4semidirect
23. #2229C4semidirect
24. #2228C4semidirect
25. #2228C4semidirect

Maximal subgroups

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

subgroupindex
#336302
#231642
#226862
#132892
#131852
#131742
#131192
#129552
#126662
#126502
C4×#3602